optimization of north sea transnational grid solutionsnorthsea transnational grid work package5...

North Sea Transnational GridWork Package 5

Optimization of North Sea TransnationalGrid solutions

AuthorsTUDelft: Silvio RodriguesTUDelft: Pavol BauerECN: Edwin WiggelinkhuizenECN: Jan Pierik

October, 2013

This document was prepared using LATEX.

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 1

1.1 Work Package 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main conclusions of the Work Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Reports and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Transmission Systems Design 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Component Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Offshore Wind Farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.2 AC Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 DC Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.4 Voltage Source Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.5 Other Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 MTdc Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 MTdc network Power Flow Optimization 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Optimal power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 MTdc Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 MTdc network single objective optimization: load flow control to minimize losses . . . . . 18

3.4 MTdc network multi-objective optimization: load flow control for minimal losses and max-

imum social welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

i

4 Conclusions 33

4.0.1 MTdc Control: Single-Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.0.2 MTdc Control: Multi-Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.0.3 Transmission Systems Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.0.4 Multi-Objective Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 39

Appendix A MTdc network technical and economic aspects 40

A.1 MTdc networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.1.1 Applications of a MTdc grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.1.2 Challenges towards a VSC-based MTdc network . . . . . . . . . . . . . . . . . . . 42

A.2 Market Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.2.1 Pool Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.2.2 Transmission Loss Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.3 MTdc Network Control Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A.3.1 Information flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.3.2 Social Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Appendix B Multi-Objective Optimization Algorithms 50

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

B.2 Proposed Multi-Objective Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . 52

B.2.1 Population Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B.2.2 Fitness evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B.2.3 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B.2.4 Population Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.2.5 Population Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.2.6 Archive Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.2.7 Ending criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B.2.8 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.2.9 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.2.10 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.2.11 Regeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.2.12 Chromosome composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

ii

B.2.13 Population composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Appendix C Benchmark Results 64

iii

List of Tables

3.1 System Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 MTdc Network Cables length and Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Description of the analyzed case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 MTdc Network Rated Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


3.6 Power flow results and loss allocation coefficients for the analyzed case studies. . . . . . . 25

3.7 OWFs market bids, capacity factors and power productions. . . . . . . . . . . . . . . . . . 28

3.8 Onshore ac networks market bids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


3.10 Direct voltages, Power flow and loss allocation coefficients for the analyzed case studies. . 30

B.1 Possible rank values and their description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.2 Description of the population sorting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.3 Description of the field Origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C.1 Contribution of each genetic operator for the non dominated solutions in the final Archive. 68

iv

List of Figures

1.1 Multi-criteria decision making process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Optimization platform for the design of transmission systems. . . . . . . . . . . . . . . . . 5

2.2 Weibull distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Offshore wind farm normalized power curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Losses variation of three different AC Cables for a 100 km transmission system. . . . . . . 7

2.5 Losses variation of three different DC Cables for a 100 km transmission system. . . . . . . 8

2.6 Losses variation of the voltage-source converter. . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Configuration of the HVAC and HVDC transmission systems. . . . . . . . . . . . . . . . . 9

2.8 Variation of the transmission system losses percentage for both technologies with the trans-

mission distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.9 Transmission system losses percentage breakdown through the components of the DC

system when a distance of 100 km and minimization of losses are considered. . . . . . . . 11

2.10 Transmission system losses percentage breakdown through the components of the AC sys-

tem when a distance of 100 km and minimization of losses are considered. . . . . . . . . . 11

2.11 Variation of the total investment cost for both technologies with the transmission distance. 12

2.12 Transmission system losses percentage breakdown through the components of the DC

system when a distance of 100 km and minimization of total investment costs are considered. 12

2.13 Optimal Pareto Fronts for both transmission technologies when a distance of 50 km is

considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Fitness evaluation flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Constraints incorporated in the genetic algorithm. . . . . . . . . . . . . . . . . . . . . . . 17

3.3 The 19 node-meshed connected offshore MTdc network used in the simulations. . . . . . . 18

3.4 Wind farms power production during the simulation. . . . . . . . . . . . . . . . . . . . . . 19

v

3.5 Transmission power losses during the simulations for the three tested cases. . . . . . . . . 21

3.6 Direct voltage at the OWFs and onshore nodes during the simulations for the three tested

cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.7 Active power at the onshore nodes during the simulations for the three tested cases. . . . 22

3.8 Meshed MTdc network used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . 24

3.9 Network power losses and social welfare. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.10 Pareto Front and Market Clearing Price (disregarding losses) for the analyzed case studies. 26

3.11 Pareto Fronts between Social Welfare and Power Losses. . . . . . . . . . . . . . . . . . . . 29

3.12 Available Pareto front between social welfare and power losses and respective buyers and

sellers welfare for the considered case studies. . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.1 Diagram of a pool market applied to a future offshore MTdc network. . . . . . . . . . . . 44

A.2 Flow chart of the distributed voltage control method. . . . . . . . . . . . . . . . . . . . . . 47

A.3 Information flow between the OWFs, the GA and the onshore VSC stations. . . . . . . . 47

A.4 Social welfare of two areas, 1 and 2, for three distinctive cases: (a) no trade; (b) limited

trade; (c) unlimited trade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

B.1 Pareto front of a Multi-Objective Minimization Problem. . . . . . . . . . . . . . . . . . . 51

B.2 Flow chart of MOOA implemented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.3 Population initialization process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B.4 Pareto front of a Multi-Objective Minimization Problem. . . . . . . . . . . . . . . . . . . 55

B.5 Archiving process of the non dominated solutions. . . . . . . . . . . . . . . . . . . . . . . 57

B.6 Crowding competition among the non dominated solutions present in the archive. . . . . . 58

B.7 Selection of solutions from the current population and archive to population the mating

pool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.8 Possible effects of the crossover operator on the genes of the new solution. . . . . . . . . . 60

B.9 Desired effect of the mutation operator on the genes of the new solution. . . . . . . . . . . 61

B.10 Possible effects of the renegeration operator on the genes of the new solution. . . . . . . . 62

B.11 Composition of the chromosomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.12 Composition of the population after the genetic operators. . . . . . . . . . . . . . . . . . . 63

C.1 Contribution of each genetic operator to the population with new solutions. . . . . . . . . 64

C.2 Non dominated solutions of the final Archive obtained by the developed MOOA. . . . . . 68

vi

Chapter 1

Introduction

1.1 Work Package 5

The design of complex engineering systems, such as the North Sea Transnational Grid, requires application

of knowledge from several disciplines (multidisciplinary) of engineering and economy (electrical aspects

of multi-terminal converter, control aspects, grid aspects, costs). The interdisciplinary nature of complex

systems presents challenges associated with modeling, simulation, computation time and integration of

models from different disciplines. The increased complexity of systems as well as the increased number of

design parameters in the optimization results in the necessity to use mathematical system models and the

application of optimization techniques. An adequate system model is built and optimization algorithms

are employed to determine the optimal parameters for the system based on model results.

A well-known optimization method used gradient search algorithms. A major barrier to the use of gradient

based search methods is that complex multidisciplinary design spaces tend to have many apparent local

optima. Therefore a new method, based on multi-objective optimization and multi-criteria decision

making, called Progressive Design Methodology (PDM), will be used.

The Progressive Design Methodology is a three-step design process. In the first step a simple model of the

components of a system is developed and the problem is reformulated as a Multi-Objective Optimization

Problem (MOOP). In the second step the results obtained in the MOOP process are analyzed and a small

set of feasible solutions is selected. In the final step, a detailed model of the variants of the system, as

selected from the previous set, is developed and the optimization variables of the system are fine-tuned.

The objective of optimization will be to determine a configuration of the North Sea Transnational Grid

with low levelized production cost, high power quality, high reliability and high low voltage ride through

capability. The set of optimization objectives can be extended and the optimization variables will be the

result of inventory. Fig. 1.1 shows a diagram of the three-step Progressive Design Methodology.

1

System requirementsanalysis

Definition of boundaries

Development of a system’s model

Multi-objective optimization

Set of Pareto Fronts

New constrains added

Multi-criteria decision making

Reduced set of solutions

Detailed model

Reduced set of solutions

Multi-objective optimization

Final decision

Synthesis Intermediate analysis Final design

Figure 1.1: Multi-criteria decision making process.

1.2 Main conclusions of the Work Package

• A multi-objective optimization algorithm was developed and applied in different case studies;

• Single- and multi-objective optimal power-flows inside a large MTdc network were achieved using

the proposed control strategy;

• The performed dynamic numerical simulations show that the power flow control is accurate and

fast, i.e. a new operating point can be achieved within 200 ms;

• Optimal pareto fronts were achieved for the design of a wind farm transmission system for both

AC and DC options.

2

1.3 Reports and Publications

1. S. Rodrigues, “Multi-Objective Optimization Problems and Algorithms - A Survey,” Report, EPP

- TU Delft, December 2011.

2. S. Rodrigues, “Market in Electric Power Systems and Transmission Losses Allocation,” Report,

EPP - TU Delft, February 2012.

3. S. Rodrigues, R. Teixeira Pinto, P. Bauer, and J. Pierik, “Optimization of social welfare and

transmission losses in offshore mtdc networks through multi-objective genetic algorithm,” in Power

Electronics and Motion Control Conference (IPEMC), 2012 7th International, 2012, pp. 1287-1294.

4. S. Rodrigues, P. Bauer, and J. Pierik, “Comparison of offshore power transmission technologies:

A multi-objective optimization approach,” in 15th International Power Electronics and Motion

Control Conference (EPE/PEMC), 2012.

5. S. Rodrigues, R. Pinto, P. Bauer, E. Wiggelinkhuizen, and J. Pierik, “Optimal power flow of vsc-

based multi-terminal dc networks using genetic algorithm optimization,” in Energy Conversion

Congress and Exposition (ECCE), 2012.

6. R. Teixeira Pinto, P. Bauer, S. Rodrigues, E. Wiggelinkhuizen, J. Pierik, and B. Ferreira, “A

novel distributed direct-voltage control strategy for grid integration of offshore wind energy systems

through mtdc network,” IEEE Transactions on Industrial Electronics, vol. 60, no. 6, pp. 2429-2441,

2013.

7. R. Teixeira Pinto, S. Rodrigues, E. J. Wiggelinkhuizen, R. Scherrer, P. Bauer and J. Pierik: Opera-

tion and Power Flow Control of Multi-Terminal DC Networks for Grid Integration of OffshoreWind

Farms using Genetic Algorithms; ”Energies Journal,” submitted.

8. R. Teixeira Pinto, S. Rodrigues and P. Bauer: Optimal Control Tuning of Grid-Connected Voltage-

Source Converters using a Multi-Objective Genetic Algorithm; abstract submitted to PCIM 2013

Conference.

9. S. Rodrigues, R. T. Pinto , P. Bauer, and J. Pierik, “Optimal power flow control of vsc-based

multi-terminal dc network for offshore wind integration in the north sea,” IEEE Transactions on

Emerging and Selected Topics in Power Electronics, 2013.

3

Chapter 2

Transmission Systems Design

2.1 Introduction

Most of the existing offshore wind farms make use of AC transmission technology [1]. The Horns Rev

wind farm, built in 2002, in Denmark, has an installed capacity of 160 MW and was the first wind farm

to make use of an offshore transformer substation. The connection to shore is made via a three-core AC

cable with a rated voltage of 150 kV. On the other hand, the first Dutch offshore wind farm, Egmond aan

Zee (108 MW), built in 2006; and the first UK offshore wind farm, North Hoyle (60 MW), are examples

of systems which did not require an offshore transformer substation. The latter is connected to the UK

onshore grid through 33-kV AC cables [2].

Offshore wind farms are increasingly being built further from the coast and in deeper waters. In 2011 the

average water depth of offshore wind farms was circa 23 m and the average distance to shore was around

24 km [1]. For projects under construction, the average depth is 25 m, whereas the distance to shore is

33 km. According to the study in [1] this trend will continue in the short future.

In this way, it is expected that investment costs of future large offshore wind farms will increase with

the longer distances to shore. Therefore, it is important to study and compare the possible transmission

technologies in order to reduce investment costs and energy losses as much as possible.

Several studies which compared the costs of AC and DC transmissions systems for different wind farm

installed capacities and distances to shore are possible to be found in the related literature [3, 4, 5, 6].

However, they only optimize one criterion at a time, e.g. either energy losses or investment costs or LPC.

Examples of MOOAs application to the optimization design of electrical power transmission systems and

design of offshore structures may be found in [7] and [8], respectively.

The main objective of the this chapter is to implement a multi-objective approach to compare HVAC

and HVDC transmission alternatives. The components of the transmission system will be selected from

a database which contains real component data and costs.

4

2.2 Optimization Method

In order to apply the MOOA to the optimization of the design of offshore transmission systems it is

necessary to identify the variables of the problem. In this way, each solution X of the population is a

vector composed by several variables as shown as follows:

X =[α c1 · · · cn

](2.1)

The first field of each solution, α, determines whether x represents a HVAC or HVDC transmission

system. If α = 1, an AC transmission system is considered. On the other hand, if α = 2, the solution X

corresponds to a DC system. Every chromosome X holds one variable per component of the respective

transmission system. Such variables indicate which components of the database are used. Finally, a

variable representing the amount of elements for each component is present in the chromosomes. In Fig.

2.1 it is show the flow chart of the optimization method and its integration in the MOOA.

6. Terminate?

1. Population Initialisation 2. Finess Evaluation

3. RankPopulation

4. SortPopulation

5. ArchiveUpdate

7. Selection 8. Crossover 9. Mutation

11. Output Archive

Yes

No

10. Regeneration

Database with component information

Component models

Constraints

Total Investment Costs and Energy Losses

Figure 2.1: Optimization platform for the design of transmission systems.

2.2.1 Constraints

In order to obtain feasible solutions, constraints were implemented in the MOOA. For the AC configura-

tion the following constraints were used:

⎧⎪⎨⎪⎩

0.7V LT1 ≤ V cable

AC ≤ 1.3V LT1

0.7V HT1 ≤ V cable

HVAC ≤ 1.3V HT1

0.7V HT2 ≤ V cable

HVAC ≤ 1.3V HT2

(2.2)

For the DC technology the constraints used were as follows:

5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0.7V LT1 ≤ V cable

AC ≤ 1.3V LT1

0.7V HT1 ≤ V REC

AC ≤ 1.3V HT1

0.7V HT2 ≤ V INV

AC ≤ 1.3V HT2

mconvertersi ≤ 1.1

0.7V RECDC ≤ V cable

HVDC ≤ 1.3V RECDC

0.7V INVDC ≤ V cable

HVDC ≤ 1.3V INVDC

(2.3)

Margins are given to the transformers’ voltages. In this way, it is considered that the transformers have

a tap changer which allows a variation of 30% in the voltage. Moreover, it is made sure that the power

ratings of all the components are respected.

2.3 Component Models

In order to optimize the design of offshore transmission systems it is necessary to model every component.

2.3.1 Offshore Wind Farm

In order to obtain realistic results for the energy produced at the offshore wind farm, a Weibull distribution

for the offshore wind and a normalized wind farm power curve are used (see Fig. 2.2).

The Weibull distribution used was based on measurement data collected during 4 years in a platform,

Fino 1, located in the North Sea [9].

The wind farm power curve used was obtained by smoothing the normalized turbine power curve and

it is represented in Fig. 2.3. The power curve includes factors such as the turbine design and future

developments in this field up to the year 2030 [10].

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Wind Speed [m/s]

Rela

tive

Freq

uenc

y

Offshore Wind Weibull Distribution

Figure 2.2: Weibull distribution.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Wind Speed [m/s]

Win

d Fa

rm O

utpu

t Pow

er [p

u]

Wind Farm Normalized Power Curve

Figure 2.3: Offshore wind farm normalized powercurve.

When the multi-turbine power curve and the wind distribution are known, the transmission system loss

percentage can be calculated as in (2.4). For the sake of simplification the wind farm was considered to

be always available.

6

Elosses[%] =

∑υf(υ)× Plosses(υ)∑

υf(υ)× Pprod(υ)

× 100 (2.4)

Using (2.4), it is possible to fairly compare both HVAC and HVDC technologies in terms of the energy

lost in the transmission system itself.

2.3.2 AC Cables

The losses variation with the transmitted power for three AC cables with different rated voltages is shown

in Figure 2.4. The losses are present in pu, having the rated power of each cable as the power base.

The cable with lowest rated voltage (30-kV) is the one that presents higher power losses. The 150-kV

cable shows higher losses when compared to the cable of 138-kV one. This is due to the fact that the

150-kV cable present in the database has a higher resistance per unit of length when compared with the

138-kV cable. However, the 150-kV cable has a higher rated power and, in this way, it can transport

more power per cable ashore; thus, less cables are needed.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

Transmitted Power [pu]

Pow

er L

osse

s [%

]

AC Cables Losses (100 km)

ACcable - 30 kV (40 MVA)



Figure 2.4: Losses variation of three different AC Cables for a 100 km transmission system.

Due to the high transmission distances, in order to model the HVAC cables, shorter sections of the cable

were considered to be connected to each other. Thereafter, each section was modeled through a π-model.

Submarine and underground cables present larger distributed capacitances when compared to overhead

lines. The HVAC cable is modeled as several sections, each constituted of a lumped-π-model, in order to

minimize the error due to simplifications.

2.3.3 DC Cables

The DC cables were also modeled through a π-model. The power losses were scaled by a factor of two

since all the DC cables present in the database are bipolar. In Figure 2.5, the variation of the power

losses with the transmitted power is shown for three DC cables with different power ratings. Once again

the losses are present in pu, having the rated power of each cable as the power base. All the cables have

a rated voltage of 320 kV.

7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Pow

er L

osse

s [%

]

DC Cables Losses (100 km)

DCcable - 320 kV (500 MW)



Figure 2.5: Losses variation of three different DC Cables for a 100 km transmission system.

2.3.4 Voltage Source Converters

The power losses of the voltage-source converter are due to the switching components; the conduction

losses, which vary in a quadratic proportion with the current; and the “no-load” losses. In Figure 2.6,

the contribution of each one of these factors and the total power losses of the converter are shown. The

power losses are shown in percentage of the converter rated power (570 MW).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Pow

er L

osse

s [%

]

Converter Losses

Plosses

Psw itchingPconduction

Pno load

Figure 2.6: Losses variation of the voltage-source converter.

2.3.5 Other Components

Offshore wind farms need a collection network. The medium voltage AC cables that constitute the

offshore wind farm collection network are modeled through a π-model. Since the distance considered

for the cables is much shorter than the ones for the HV cables, one π-section is applied to model the

collection system.

HVAC cables produce reactive power due to their dominant capacitance. The reactive charging current

redistributes over the cable length depending on the voltage amplitudes and angles at both ends of

the cable. In this way, inductors were installed at both ends of the cables to provide reactive power

compensation.

8

2.4 MTdc Design

In this case study, the objective functions used to compare the offshore transmission systems are the total

investment cost and the percentage of energy lost during transmission.

F

⎧⎪⎪⎨⎪⎪⎩

Investment cost =n∑

i=1

Costcompi

Energy losses [%] =n∑

i=1

Elosses[%]

where n is the number of components of each transmission system.

Case Study

The objective is to compare the AC and DC transmission systems in a multi-objective basis. The layout

and the components, of both HVAC and HVDC transmission systems, are depicted in Figure 2.7. The

AC cables that constitute the collection network are 5 km long. The rated power of the wind farm is 500

MW.

OffshoreWind farm AC cables Trafo 1 HVAC Cable

AC cables Trafo 1 HVDC Cable Inverter

==

RectifierOffshoreWind farm

...

...

...

...

...

...

Trafo 2

Trafo 2

Inductor 1 Inductor 2

Figure 2.7: Configuration of the HVAC and HVDC transmission systems.

Results

Initially a comparison between the two transmission systems was made to take into consideration one

criterion at the time. In the last part, a comparison based on multi-objective optimization was performed.

Single-Objective Optimization Comparison

Firstly, the transmission system losses percentage was optimized. Thereafter, the investment costs was

minimized once again for both AC and DC systems.

9

Energy Losses

In Figure 2.8, it is shown the transmission system losses percentages according to the distance between

the offshore wind farm to the shore. The break-even point, i.e. the distance for which the DC technology

presents lower energy losses when compared to the AC option, is circa 100 km. This result is in tune with

the findings of several other studies on the subject, which serves as a confirmation of the model accuracy

[2, 3].

The losses for the DC technology increase slowly with the distance. This is due to the fact that the

genetic algorithm chose the same configuration for all the considered distances. In this way, the losses

only depend on the increase of the DC cable’s length. The DC technology presents an energy loss of circa

5.44% at 100 km.

The AC alternative presents lower percentage losses for shorter distances. However, the increase rate is

much higher when compared to the DC one. This has to due with the fact that the reactive power in the

HVAC cables are highly dependent on cable rated voltage. When a 100 km distance is considered, the

AC transmission system presents an energy loss of 5.48%.

Differently from the DC solutions, the genetic algorithm for distances shorter than 120 km chose the AC

cable with a rated voltage of 138 kV. However, for longer distances the 150-kV AC cable was chosen

instead. Although as previously shown in Figure 2.4, the 138-kV cable presents lower losses, for distances

higher than 120 km, the cable does not respect the power rate constraint and therefore it becomes

unfeasible.

AC voltages higher than 150 kV have not been used so far in practical submarine installations [11].

Moreover, such voltage level is a standard choice for submarine cables in Europe [12] and it was used, for

instance, to connect the wind farm on the Thornton Bank and the Danish Horns Rev wind farm [1].

50 60 70 80 90 100 110 120 130 140 1503.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Length [km]

Ener

gy L

osse

s [%

]

HVAC and HVDC Transmission System Losses (500 MVA)

HVACHVDC

Figure 2.8: Variation of the transmission system losses percentage for both technologies with the trans-mission distance.

An interesting study that can be done is to observe how the energy losses are distributed over the

components of each transmission system for the break-even distance (100 km). In Figure 2.9, the energy

losses breakdown for the components of the DC option is shown. The HVDC cable is the component with

the lowest share of losses. This has to do with the fact that the genetic algorithm chose the DC cable

with 1500 MW rated power. Even though the wind farm has only 500 MW of installed capacity, since the

investment costs are not being minimized in this first approach, the HVDC cable with the lowest losses

was chosen. On the other hand, the converters present the higher percentages. As previously shown in

Figure 2.6, the converter losses increase with the power transferred but they always present switching

10

and “no-load” losses. Furthermore, most of the time the offshore wind farm is not producing at its full

capacity and, thus, the converters will not be operating at their maximum efficiency.

AC Cables Trafo 1 Rectifier HVDC Cable Inverter Trafo 20

5

10

15

20

25

30

DC Transmission System Components

Ener

gy L

osse

s [%

]

Energy Losses Breakdown: Losses Minimization (100km)

Figure 2.9: Transmission system losses percentage breakdown through the components of the DC systemwhen a distance of 100 km and minimization of losses are considered.

In Figure 2.10 the energy losses breakdown for the components of the AC system is presented. The

138-kV cable was the one chosen when a distance of 100 km was considered and it is the component with

the highest share of the energy losses.

AC Cables Trafo 1 Inductor 1 HVAC Cable Inductor 2 Trafo 20

10

20

30

40

50

60Energy Losses Breakdown: AC Transmission (100km)

AC Transmission System Components

Ener

gy L

osse

s [%

]

Figure 2.10: Transmission system losses percentage breakdown through the components of the AC systemwhen a distance of 100 km and minimization of losses are considered.

Total investment costs

When the total investment costs are considered as the optimization function, the results are the ones

depicted in Figure 2.11. Once again the DC configuration chosen by the genetic algorithm was kept

constant for all the considered distances. However, it is a different configuration when compared with the

one obtained previously. In order to demonstrate it, the percentage of energy lost per component when a

distance of 100 km is considered is shown in Figure 2.12. It can be observed that the HVDC cable chosen

has higher losses percentage than the one chosen when only the losses were optimized. The HVDC cable

selected for this case has a rated power of 500 MVA, which is the power installed at the offshore wind

farm. Although the cost of the HVDC cable is lower, it presents higher losses.

For the AC configuration, the total investment cost for distances below 120 km raises with the increase of

the HVAC cables length since the configuration was kept constant. However, similarly for the case when

the losses were minimized, the HVAC cable chosen for distances longer than 110 km has a higher rated

voltage (150 kV). Such cable is much more expensive due to insulation requirements and; thus, the cost

of the transmission system is also higher.

11

50 60 70 80 90 100 110 120 130 140 15040

60

80

100

120

140

160

180

200

220

240

Length [km]

Tran

smis

sion

Sys

tem

Cos

t [M

€]

HVAC and HVDC Transmission System Costs (500 MVA)

HVACHVDC

Figure 2.11: Variation of the total investment cost for both technologies with the transmission distance.

AC Cables Trafo 1 Rectifier HVDC Cable Inverter Trafo 20

5

10

15

20

25

DC Transmission System Components

Ener

gy L

osse

s [%

]

Energy Losses Breakdown: Cost Minimization (100km)

Figure 2.12: Transmission system losses percentage breakdown through the components of the DC systemwhen a distance of 100 km and minimization of total investment costs are considered.

12

Multi-Objective Optimization Comparison

With the previous single-criterion optimization it is difficult to have a clear picture of all the possible

solutions. When the energy losses are being optimized, a set of transmission systems are obtained. On the

other hand, when the investments costs are minimized, a different set of solutions is reached. Therefore,

it is not possible, in a simple way, to have a clear picture of all the possible solutions for this particular

problem.

In Figures 2.13 to 2.15, the solutions obtained through the MOOA implemented are shown. All the non-

dominated solutions, for both AC and DC systems, are plotted as filled squares and dots, respectively.

When a distance of 50 km is used, as shown in Figure 2.13, the non-dominated AC system is the one

that presents the lowest losses and investment costs. Therefore, it is the best option for the transmission

system.

3.5 4 4.5 5 5.5 6 6.5 740

60

80

100

120

140

160

180

200

220

Energy Losses [%]

Tran

smis

sion

Sys

tem

Cos

t [M

€]

HVAC and HVDC Transmission Systems (50 km)

HVACHVDC

Figure 2.13: Optimal Pareto Fronts for both transmission technologies when a distance of 50 km isconsidered.

In Figure 2.14, the solutions for a 100 km distance are shown. One of the non-dominated DC solutions

is the one with the lowest energy losses. However, the non-dominated AC solution presents very similar

losses. Moreover, such an AC system is circa three times cheaper compared to the DC system with similar

losses. In this way, the AC system is still the best option to connect the offshore wind farm to shore.

5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 760

80

100

120

140

160

180

200

220

240

260

Energy Losses [%]

Tran

smis

sion

Sys

tem

Cos

t [M

€]


HVACHVDC


13

Finally, when a distance of 150 km is considered, the DC configurations which belong to the OPF present

lower energy losses and total investment costs when compared to the AC solutions (see Figure 2.15). In

this way, for this distance the DC technology is the most advantageous.

5.5 6 6.5 7 7.5 8 8.5 9140

160

180

200

220

240

260

280

300

Energy Losses [%]

Tran

smis

sion

Sys

tem

Cos

t [M

€]


HVACHVDC


14

Chapter 3

MTdc network Power Flow

Optimization

3.1 Introduction

There are several reasons contributing to promote the development of an European offshore grid in the

North Sea. The main drives are, amongst others, the need to integrate large amounts of renewable

energy, the will to boost transnational electricity trade, the European Union’s targets on renewable

energy generation and desire for security of supply [13][14].

Several European projects have performed studies on market, regulatory and policy challenges related

with the development of a North Sea transnational grid [15][16]. Nonetheless, hitherto, very few have

concentrated on its technical aspects, especially the challenges involved with controlling and operating

such networks [17][18].

The main goal of the present chapter is to show how to control and operate large multi-terminal dc

networks with the aid of a MOOA in order to obtain optimal and controllable power flows according to

different optimization targets.

3.2 Optimal power flow

The idea of obtaining optimal power flows with the aid of genetic algorithms is not new. Several studies

may be found in the literature [19, 20]. In [21] the OPF applied to IEEE test systems is achieved via

a genetic algorithm that makes use of both continuous and discrete control variables. The optimization

of social welfare was considered in [22] and [23]. Multi-Objective optimization to account the system

security in electricity markets may be found in [24]. The application of OPF for smart grid is more

recent, having as examples the works performed in [25] and [26].

The Independent System Operator (ISO) makes use of an OPF in order to dispatch the entities involved

in the pool market. During the optimization process, all the network constraints are taken in to account.

These constraints are usually described as a set of equality and inequality functions [27]. Using the

following security-constrained optimization problem, an OPF-based market model can be represented as

[28]:

OPF equation: f(PS , PD) (3.1)

15

Constraints: ⎧⎪⎪⎪⎨⎪⎪⎪⎩

V minDCi

≤ VDCi≤ V max

DCi

PS =∑

PWFs

0 ≤ PDi≤ Pmax

V SCi

−ImaxDCi

≤ IDCi ≤ ImaxDCi

(3.2)

where VDC represents the nodal DC voltages inside the MTdc network; PS and PD stand for supply and

demand power bids; and IDC is the current flowing in the DC cables.

Additionally to respecting the network’s voltage and current constraints, the OPF algorithm here imple-

mented will make sure that the ISO is presented with solutions for the MTdc network power flow which

are N-1 secure. In this way, the reliability and security of the grid are enhanced. As a matter of fact,

in case of an outage in any of the DC network terminals, if the OPF solution is guaranteed to be N-1

secure, the power balance inside the MTdc grid will be maintained [29].

3.2.1 MTdc Control Scheme

The variables encoded in the chromosomes, besides the other fields presented in Fig. B.11, are the direct

voltage references of all the onshore nodes without restrictions on their active power. All the encoded

variables are real valued and the composition of each individual chromosome is given in (3.3).

X =[V ∗DC1 · · · V ∗

DCn

](3.3)

where V ∗DCn is the dc system voltage reference of the n-th onshore VSC.

It is important to refer that in the distributed voltage control method, even when receiving a preestab-

lished amount of power, an onshore VSC station will always be operating as a direct voltage regulating

node. However, its voltage reference will be determined by the MOOA after solving the optimal MTdc

network load flow. The fitness evaluation (step 2 of Fig. B.2) flowchart of the MOOA applied to the

control of MTdc networks is depicted in Fig. 3.1.

The step 2.3 from Fig. 3.1 is where the population of solutions are tested to verify their performance

with respect to the optimization targets set by the system operator.

Constraints

In order to guarantee that the solutions provided are feasible, several constraints were added to the

MOOA. These constraints guarantee that there are no dc cables overloaded, that the dc voltages of all

the nodes inside the MTdc grid respect the predefined boundaries and that the load flow solutions are N-1

secure, i.e. the power control will still be achieved even if an outage occurs in any of the VSC terminals

[30]. In Fig. 3.2 the implemented constraints are depicted.

16

6. Terminate?

1. Population Initialisation 2. Finess Evaluation

3. RankPopulation

4. SortPopulation

5. ArchiveUpdate

7. Selection 8. Crossover 9. Mutation

11. Output Archive

Yes

No

10. Regeneration

Population checked?2.1 Individual of the Population

YesNo

2.2 Determination of the Power Flow

2.3 Fitness evaluation 2.4 Verification of constraints

2.5 Augmented fitness function

2. Fitness evaluation

Step 3

Figure 3.1: Fitness evaluation flowchart.

Genetic Algorithm

VSC stations powerratings

DC voltageboundaries

N-1redundancy

Load flowOWFs powerproduction

DC cablescapability

Figure 3.2: Constraints incorporated in the genetic algorithm.

17

UK1

UK2

BE1

NL1

NL2

DE1DE2

DK1

DK2

HUB1

HUB2

HUB3NL

UK

BE

DK

DE

HUB4

HUB5

Figure 3.3: The 19 node-meshed connected offshore MTdc network used in the simulations.

3.3 MTdc network single objective optimization: load flow con-

trol to minimize losses

The performance of the algorithm and the DVC strategy in operating a MTdc network is tested via

numerical simulations. The MTdc network used is depicted in Fig. 3.3. It is composed of 19 nodes, inter-

connecting 9 OWFs to 5 different onshore ac networks representing each a northern European country;

namely, the UK, Belgium, the Netherlands, Germany and Denmark. Out of the total number of nodes, 5

are hub nodes, which means there are no converters installed in these locations and their main function

is to allow for cable joining.

As previously mentioned, there are 9 OWFs installed in the MTdc network and their aggregated installed

capacity is of 3 GW, or 15 pu (1 pu corresponds to 200 MVA) in the system base (see Table 3.4). The dc

system voltage is of ±250 kV and all the onshore VSCs are assumed to be connected at ac substations

with short-circuit ratio three times higher than the converter installed capacity.

In Table 3.2 the characteristics of the transmission lines of the MTdc network are given. It shows the

cable length in kilometers and also the installed capacity of the departing VSC node. For instance, Table

3.2 shows that the UK1 installed capacity is 3 pu, whereas 2 pu for UK2 and 5 pu for the UK.

The dc cables rated power was chosen to match the rated power of the nodes that they are connected

to. Hence, the dc cables have either the installed capacity of their respective OWF or onshore node. An

exception is made for the dc cables connecting the hubs, which have an installed capacity of 1 pu.

Table 3.1: System Parameters.

Parameter Unit Value

System base Rated Power Sb MVA 200DC System Voltage base Rated Voltage Vb kV ±250Short-circuit Ratio SCR - - 3

MTdc NetworkRated DC System Voltage Vdc kV ±250

Cable impedance ZdcΩ/km 0.02+j0.06nF 220

In performing the case studies real wind data was used to represent the power production of the different

18

Table 3.2: MTdc Network Cables length and Capacity.

Lines Length [km] Size [pu] Lines Length [km] Size [pu]

UK1 - HUB1 100 3 HUB3 - HUB4 250 1UK2 - HUB1 40 2 DE1 - HUB4 40 2UK - HUB1 120 5 DE2 - HUB4 70 2HUB1 - HUB2 300 1 DE - HUB4 150 4BE1 - HUB2 50 1 HUB4 - HUB5 120 1BE - HUB2 100 1 DK1 - HUB5 40 1HUB2 - HUB3 120 1 DK2 - HUB5 50 1NL1 - HUB3 100 2 DK - HUB5 150 2NL2 - HUB3 40 1 HUB1 - HUB5 380 1NL - HUB3 70 3

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

time [s]

Act

ive

Pow

er [p

u]

Wind Farms Power ProductionPUK1PUK2PBE1PNL1PNL2PDE1PDE2PDK1PDK2

Figure 3.4: Wind farms power production during the simulation.

19

OWFs. Wind data measured from a wind turbine was smoothed in order to simulate the power production

of the complete wind farm [31]. In Fig. 3.4 the OWFs power production employed during the simulation

is given. The wind power production curves are plotted in the system power base as given in Table 3.4.

The three different case studies analyzed are described in details in Table 3.3.

Table 3.3: Description of the analyzed case studies

Case Description

Case i The MTdc network will be controlled accordingto a flat voltage profile, i.e. all the onshore VSCvoltage references are set to 1.0 pu.

Case ii The MTdc network voltage references are set inorder to minimize the power losses during the en-tire simulation.

Case iii In the first part (5 ≤ t ≤ 10 s), priority in receiv-ing power is given to the UK and Germany. TheUK is willing to receive 1.5 pu and Germany 2pu. In the second part (10 ≤ t ≤ 20 s), the VSCvoltage references are set such that Belgium willreceive 1 pu; while the Netherlands and Germanywill receive 2 pu each. In the last part (t ≥ 20 s),the Netherlands and Denmark will receive 1 pu ofpower each.

In all case studies a three-phase fault was applied in the ac grid of the British converter station. The

fault occurs at t = 25 s and it is cleared one second later. The fault lasts for a complete second to show

that, when a converter station is definitely lost, the system is able to reach a new feasible operating point

after the transients decay. After 1 second, the fault is cleared to demonstrate that the system is again

able return to its previous operating point.

Results

The case studies described in Table 3.3 were analyzed in a simulation model. For the dynamic simulations

the ac grids are modeled as ideal voltage sources behind short circuit impedances whereas the MTdc

network is modeled as lumped impedances in a state-space matrix representation [32, 33]. The VSC

model employed is an averaged model which includes the dynamics of the HVdc terminals controls but

neglects the switching phenomena. The OWFs are modeled as current sources with the output power

given by Fig. 3.4 [34].

The transmission losses comparison between the three analyzed cases is displayed in Fig. 3.5. The MTdc

network voltages are shown in Fig. 3.6, while the actual active power at the onshore nodes is given in

Fig. 3.7.

Case i - No optimization

As described in Table 3.3, during Case i, the MTdc network is controlled with a flat voltage profile. In

that case, the MTdc network losses stay at around 2.75% of the OWFs produced power (see Fig. 3.5).

The first column in Fig. 3.6 displays the MTdc network onshore node voltages and the upper plot in

Fig. 3.7 shows the onshore nodes dc power for Case i. As seen from the results, all onshore VSCs are

able to keep the direct voltage constant at 1 pu by slowly varying the input dc power according to the

variations at the OWFs side. However, in this scenario the load flow distribution is not controlled and it

is not assured whether the MTdc network is N-1 secure.

20

0 5 10 15 20 25 302

2.53

3.54

4.55

5.56

6.57

7.58

time [s]

P loss

es [%

]

Transmission Power losses

Case iCase iiCase iii

Figure 3.5: Transmission power losses during the simulations for the three tested cases.

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case i - United Kingdom

VUK1

VUK2

VUK

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - United Kingdom

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case iii - United Kingdom

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case i - Belgium

VBE1

VBE

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - Belgium

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case iii - Belgium

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case i - Netherlands

VNL1

VNL2

VNL

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - Netherlands

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - Netherlands

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case i - Germany

VDE1

VDE2

VDE

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - Germany

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case iii - Germany

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case i - Denmark

VDK1

VDK2

VDK

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case ii - Denmark

0 5 10 15 20 25 300.8

0.840.880.920.96

11.041.081.121.16

time [s]

V DC [p

u]

Case iii - Denmark

Figure 3.6: Direct voltage at the OWFs and onshore nodes during the simulations for the three testedcases.

21

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5

−4.5−4

−3.5−3

−2.5−2

−1.5−1

−0.50

0.51

1.5

time [s]

Act

ive

Pow

er [p

u]

Case iPUK PBE PNL PDE PDK

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5

−4.5−4

−3.5−3

−2.5−2

−1.5−1

−0.50

0.51

1.5

time [s]

Act

ive

Pow

er [p

u]

Case ii

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5

−4.5−4

−3.5−3

−2.5−2

−1.5−1

−0.50

0.51

1.5

time [s]

Act

ive

Pow

er [p

u]

Case iii

Figure 3.7: Active power at the onshore nodes during the simulations for the three tested cases.

Case ii - Losses optimization

From the transmission losses optimization results, as displayed in Fig. 3.5, it is straightforward to realize

that the MTdc network losses are only marginally lower than the case with no optimization. The dc

system voltages and active power for the flat profile are very close to the optimal load flow found in Case

ii (see Fig. 3.7). However, to minimize the transmission losses, the genetic algorithm reaches a load-flow

scenario with higher dc system voltage reference values (circa 1.02 pu). In this way, the transmission

power losses are slightly decreased. Additionally, with higher operating direct voltages it is expected

that the converter losses would decrease. Finally, the system security is also enhanced, since the genetic

algorithm assures that the network flow is N-1 secure, which is not certain with a flat profile.

Case iii - Controlled power flow

In this last case study, the power flow inside the MTdc network is controlled according to the specifications

given in last row of Table 3.3. It is possible to observe, from the lower plot in Fig. 3.7, that the DVC

strategy implemented via the genetic algorithm has correctly set the onshore node voltages to establish

the desired power flow. The load-flow results obtained are as expected, even though the optimal dc

power flow implemented by the genetic algorithm works with the wind power production averaged over

5 seconds. As previously mentioned, the active power variability is highly reduced since several OWFs

are interconnected over a wide area.

In order to control the power flow inside the MTdc network, the dc system voltage at the onshore nodes

has to be varied. The last column of Fig. 3.6 displays the variations in the direct voltages; which, for the

analyzed case, always lie inside the operational margin of ± 10% the rated value. Additionally, as shown

by the simulation results, if so desired, changes from one load-flow operating point to another can be

22

accomplished in less than 200 ms. This is due to the fast direct voltage control capabilities of the onshore

VSC terminals. Finally, in the last case study, the optimization goal is to minimize the losses while still

respecting the load-flow constraints set by a possible TSO. As a result of these extra constraints, the

MTdc network transmission losses in this case will always be higher than optimal, as shown in Fig. 3.5.

The results shown in the right column of Fig. 3.6 demonstrate that the onshore VSC stations are able

to withstand the onshore ac three-phase fault at the British station while keeping the power balance

inside the MTdc grid. The steady-state direct voltages inside the MTdc network are kept under the

pre-established 1.10 pu limit. The British onshore and offshore converter stations are, as expected, the

ones that suffer the highest variations in the direct voltage, while the remaining onshore nodes present

lower variations due to their higher distance to the fault. Between 25 s ≤ t ≤26 s, the power flow inside

the network is not optimized to minimize the power losses. However, it is guaranteed that the converter

stations rated power is respected.

During the fault the British converter station is not receiving any power, whereas the Belgian onshore

node does not alter its power transmission during the fault since it was previously at its maximum

transfer capacity of 1 pu. The Dutch onshore VSC station was the node that increased the most its

power reception, from circa 1.5 pu to 2.5 pu because the Netherlands is the closest onshore node with

available margin to increase its power reception. The remaining two countries, Denmark and Germany,

also increased their power consumption to maintain the direct voltage stable inside the MTdc network.

After fault clearance, at t = 26 s, the onshore stations are able to once again operate under the TSO’s

desired power flow.

3.4 MTdc network multi-objective optimization: load flow con-

trol for minimal losses and maximum social welfare

The main objective of this case study is to obtain a Pareto front showing a trade-off between the mini-

mization of the transmission losses and the maximization of the social welfare. The latter is explained in

A.3.2 and be calculated as in (A.6), while the transmission losses can be calculated by (A.3).

MTdc network 1

The MTdc network displayed in Fig. 3.8 is the one used during the simulations. It is a parallel-connected

meshed network, composed by 11 nodes and 11 DC cables. There are a total of 5 offshore wind farms

connected to 3 different onshore networks, which represent countries. In the case study these countries

are considered to be the United Kingdom, the Netherlands and Norway, respectively.

All the offshore wind farms are connected to hubs instead of being directly connected to its rightful owner

country. Furthermore, each hub is connected to its respective country, but also to adjacent hubs. This

type of DC network connection enhances the electricity trade amongst the interconnected onshore grids

[35].

The MTdc network parameters are given in Table 3.4. The system power base is 500 MVA. During

steady-state operation all DC voltages must lie within ±5% of the nominal value. The DC line distances

are given in Fig. 3.8. The wind farms cables (lines 2,3,5,6 and 9) are rated at 500 MW, whereas all

the other DC cables have two times the capacity of the wind farms, i.e. 1000 MW. The extra capacity

in the offshore hub cables is designed in order to increase the network flexibility and to enhance power

trade between the countries. The power being produced at the offshore wind farms during the simulation

scenarios is also given in Fig. 3.8.

23

ACnetwork 1

ACnetwork 3

ACnetwork 2

N2

WF 3

WF 4

WF 5WF 1

WF 2

N1

N6 N9

N10

N3

N7

N4

N11

N5

N8line 1 line 8 line 10

line 2

line 3line 4 line 11

line 5line 6

line 7

line 9

125 km

20 km

20 km

400 km

200 km200 km

15 km

100 km

15 km

125 km

15 km

VSC 1 VSC 3

VSC 2

1 0.5WFP pu

2 0.4WFP pu

3 0.5WFP pu

4 0.7WFP pu

5 0.9WFP pu

Figure 3.8: Meshed MTdc network used in the simulations.

In this study, the market clearing prices for the 3 interconnected countries are considered to be 60 e/MWh

in Norway, 72.8 e/MWh in the UK and 77.5 e/MWh in the Netherlands [35]. The offshore DC network

is operated in such a way that there are no congestions due to electricity trade between the countries

and, therefore, all the power being produced at the wind farms can be injected in the MTdc network.

Table 3.4: MTdc Network Rated Parameters.

Parameter Unit Value

Offshore VSC Rated Power Sowf MVA 500

Onshore VSC Rated Power Svsc MVA 1000

MTdc NetworkRated DC Voltage Vdc kV ± 200Cable impedance Zdc Ω/km 0.02

DC lines 2,3,5,6 & 9 Pdc MW 500DC lines 1,4,7,8,10 & 11 Pdc MW 1000


Case Description

Case i No restrictions applied.

Case ii At least some of the offshore power being pro-duced at the Norway wind parks has to beconsumed locally.

Case iii Idem as Case ii. In addition the Netherlandsis willing to buy power for an extra 0.5 pu ontop of its offshore wind farm production of 1.2pu.

Results

A trade-off – or Pareto Front – for the operation of the MTdc network under study is obtained by solving

the OPF problem with the MOOA previously presented. The resulting network power losses and social

welfare are shown in Fig. 3.11 (a).

24

35 40 45 50 55 60 65 70 75 801

2

3

4

5

6

7

Social Welfare [€/h]

Pow

er L

osse

s [%

]

Power losses and Social Welfare

All solutions found

35 40 45 50 55 60 65 70 75 801

2

3

4

5

6

7


Pow

er L

osse

s [%

]

Power losses and Social Welfare: N-1 secure and Pareto Front

N-1 Secure SolutionsPareto-Front Solutions

Figure 3.9: Network power losses and social welfare.

However, not all the obtained solutions are guaranteed to be N-1 secure. Therefore, all solutions are tested

against a DC load-flow algorithm. The algorithm systematically applies a fault to one network node at

a time. Afterwards, it excludes solutions that do not respect the DC voltage and current constraints for

all the possible N-1 scenarios. The new set of solutions is displayed in Fig. 3.11 (b), where the optimal

Pareto front is also highlighted.

Depending on the restrictions selected by the ISO, some MTdc operating points on the Pareto front

may become unavailable. Fig. 3.10 shows, for the selected case studies, the available operating points

on the Pareto front and the market clearing price. The hollow points on the Pareto fronts (Fig. 3.10),

represent unavailable operating points due to the ISO and Pool Market restrictions. In addition, Fig.

3.10 also displays, for each analyzed case study, the MCP obtained for the two most extreme points on

the available Pareto front. These points are equivalent to the lower losses (point A) and the higher social

welfare (point B) scenarios.

Table 3.6 contains the power flow results for the lower losses (point A) and higher social welfare (point

B) scenarios for all the examined case studies. It displays the nodal DC voltages (in per unit), the loss

allocation coefficients (in percentage of the losses), the transmission losses (in percentage of the offshore

generation) and the resultant social welfare.

Table 3.6: Power flow results and loss allocation coefficients for the analyzed case studies.

Node

Case i Case ii Case iii

A B A B A B

VDC Lk VDC Lk VDC Lk VDC Lk VDC Lk VDC Lk

WF1 1.0272 5.05 0.9983 -2.29 1.0272 5.05 1.0141 1.71 1.0021 2.46 1.0141 1.71WF2 1.0269 3.83 0.9980 -1.89 1.0269 3.83 1.0138 1.25 1.0018 1.81 1.0138 1.25WF3 1.0268 4.62 1.0064 -0.18 1.0268 4.62 1.0095 -1.17 0.9992 0.16 1.0095 -1.17WF4 1.0271 7.02 1.0068 -0.12 1.0271 7.02 1.0098 -1.32 0.9996 0.63 1.0098 -1.32WF5 1.0271 9.03 1.0352 12.70 1.0271 9.03 1.0256 15.86 1.0105 16.491 1.0256 15.86VSC1 1.0128 19.39 0.9666 42.04 1.0128 19.39 0.9986 15.22 0.9863 19.61 0.9986 15.22VSC2 1.0118 28.78 0.9811 26.97 1.0118 28.78 0.9842 68.72 0.9771 61.39 0.9842 68.72VSC3 1.0118 22.27 1.0494 22.76 1.0118 22.27 1.0237 -0.26 1.0045 -2.55 1.0237 -0.26

PL [%] 1.46 6.42 1.46 2.63 2.08 2.63SW [e/h] 59.07 77.52 59.07 71.69 68.14 71.69

25

58 60 62 64 66 68 70 72 74 76 781

2

3

4

5

6

7


Pow

er L

osse

s [%

]

Case i - No restrictions

58 60 62 64 66 68 70 72 74 76 781

2

3

4

5

6

7


Pow

er L

osse

s [%

]

Case ii - One restriction

58 60 62 64 66 68 70 72 74 76 781

2

3

4

5

6

7


Pow

er L

osse

s [%

]

Case iii - Two restrictions

Unavailable operating pointsAvailable operating points



Figure 3.10: Pareto Front and Market Clearing Price (disregarding losses) for the analyzed case studies.

Case Study i

In the first case scenario, as stated in Table 3.5, there are no extra restrictions applied by the ISO to the

load flow. In this way, the entire Pareto front (Fig. 3.10 (b)) is reachable.

For the point A, where the lowest power losses are found, the resultant load flow shows that the WFs

are sending almost all of their generation to the respective countries. In this way, the power flow inside

the grid is minimized and, thus, the power losses are optimal. Since all countries are receiving power

produced offshore, the MCP, as shown in Fig. 3.10 (d), is equal to 60 e/MWh, which corresponds to the

bid lowest bid, made by Norway.

The loss components, Lk, are all positive for point A, meaning that all the market participants are

responsible to pay part of the power lost in the transmission system. The Netherlands has the highest

loss component (28.78%) since it is the node which receives the highest amount of power from the MTdc

grid. This is because the Dutch farms have the highest combined production. Among the offshore nodes,

the highest loss allocation is given to the Norwegian WF (9.03%) since it has the highest single production

and, therefore, it is the one that injects the most power in the network.

In point B, the social welfare is being maximized. In order to do so, the countries with the highest

electricity bids have to receive the most power from the grid. As shown in Fig. 3.10 (d), both the

Netherlands and the UK are receiving power at their maximum capacity, i.e. 2 pu. Since the total power

production at all the WFs is 3 pu, Norway is transmitting the extra power for allowing the maximization

of the social welfare. The MCP for this operating point is the British bid, i.e. 72.8 e/MWh (Fig. 3.10

(d)).

In this scenario, the loss components of the Dutch and British WFs are negative. Negative loss compo-

nents represent monetary incentives to market participants that are positioned in strategic places in the

network [36], helping, in this way, to achieve the desired load flow. On the other hand, the loss component

of the Norwegian WF is positive. This has to do with the fact that this WF is transmitting power to the

UK and the Netherlands. Consequently, since the WF is located far from both countries, such injection

of power into the MTdc grid has a significant impact in the total transmission losses. Nevertheless, all the

26

onshore nodes are still allocated the highest loss components. The UK is the node receiving the highest

amount of power that is not being produced at its WFs (1.1 pu), consequently it is the node with the

highest share of power losses (42.04%).

Case Study ii

In Case ii, an additional constraint is added; at least some of the offshore power being produced at

the Norwegian wind park has to be consumed locally. Due to such constraint, approximately half of

the previous Pareto front becomes unavailable operating points. The load flows with the highest social

welfare values are not tangible in this case since they require Norway to transmit power to the MTdc

network.

The operating point A for Case ii is the same one as found in Case i. This is due to the fact that the

unavailable part of the Pareto front is located on the region where the social welfare is maximized.

On the other hand, point B is now shifted along the Pareto front. In this operating point, Norway is

neither receiving or injecting power in the MTdc network (see Fig. 3.10 (e)). Since the Dutch bid is the

highest one, the maximization of social welfare is now made through the transmission of 2 pu of power

to the Netherlands (maximum capacity of the node) and 1 pu to the UK. The MCP is once again defined

by the bid made by the UK.

Regarding the loss components for point B, it is interesting to notice that both the Dutch WFs have

negative values. The Netherlands is the node that needs to receive the most power in order to maximize

the social welfare. Consequently, since both WFs are the closest generating nodes from the Dutch grid,

they are the ones that contribute most efficiently the load flow. On the other hand, the Dutch onshore

node is allocated the highest share of transmission losses.

Once again the Norwegian WF is allocated a positive Lk. Since barely no power is being transmitted

to Norway, its offhshore production is being delivered to the Netherlands. In this way, the power has to

be transported from the WF to the Dutch onshore node. Due to this additional flow, the power losses

increase and, therefore, the loss component of the Norwegian WF have to account for the additional

power losses.

Case Study iii

In the last case study, on top of the restriction imposed in Case ii, the Netherlands requests, besides the

power being produced at its wind farms, an extra 0.5 pu of power extra from the MTdc network.

The operation point A is now shifted to the right along the subset of the Pareto front of the Case ii. In

this new point A, the Netherlands is now receiving 1.7 pu of power from the MTdc network (see Fig.

3.10 (f)). Since point A from Case i is no longer feasible due to the constraints imposed by the ISO, the

closest operation point is achieved when 1.7 pu of power is being delivered at the Dutch onshore node.

For such power flow scenario, the highest loss component is once again attributed to the Netherlands.

The Dutch WFs are producing a total 1.2 pu of power, thus, the additional 0.5 pu has to be produced

and transported from somewhere else in the MTdc network. Through the power flow results it is possible

to observe that this additional power is coming from the Norwegian WF. Hence, the latter is assigned

a positive and higher Lk, when compared to the other WFs, which are sending their generation to their

respective countries.

The extra constraint added in Case iii rules out operation points from the Pareto front were the lowest

power losses can be found. Therefore, the point that maximizes the social welfare (point B) is kept

constant and equal to the one from Case ii.

27

MTdc network 2

The analyzed MTdc network is shown in Fig. 3.3. Table 3.7 shows the OWFs bids, power production

and capacity factors [37]. In the time frame considered, the OWFs power production is assumed to be

its installed capacity times its capacity factor. The OWFs bid values were chosen in order to be within

the range given in [38]. The bids of the interconnected countries are given in Table 3.8 [39].

Table 3.7: OWFs market bids, capacity factors and power productions.

OWF UK1 UK2 BE1 NL1 NL2 DE1 DE2 DK1 DK2

Bid [/eMWh] 80 80 75 70 70 85 85 90 90Capacity factors [%] 0.35 0.4 0.5 0.4 0.45 0.35 0.4 0.5 0.4Power production [pu] 1.05 0.8 0.5 0.8 0.45 0.7 0.8 0.5 0.4

Table 3.8: Onshore ac networks market bids.

AC Network UK BL NL DE DK

Bid [/eMWh] 98.95 98.99 91.38 114.17 97.74

Three different case studies were established to analyze the influence of extra market constraints on the

optimization results. Table 3.9 contains a detailed description of each case study.


Case Description

Case i No restrictions applied.

Case iiThe dc current in the cable betweenthe Netherlands and HUB3 is atmost 1.2 pu

Case iiiIdem as Case ii. In addition, the dccurrent in the German cable (fromHUB4 to shore) has 1.5 pu as maxi-mum

Results

The Pareto fronts for the MTdc network operation with the trade-off between transmission losses and

social welfare are shown in Fig. 3.11. The fronts composed by hollow circles represent the intermediate

Pareto fronts found during the optimization process. The filled circles represent the best found Pareto

fronts.

When the N-1 network security constraint is active, the best found Pareto front is shifted towards lower

social welfare and higher transmission losses values. To make the dc network N-1 secure, as the maximum

dc voltage value is set to 1.15 pu, the MOGA has to lower the onshore dc voltage references – this explains

the higher losses – because when there is a fault in an onshore node, the dc voltages inside the grid will

rise. On the other hand, the social welfare values are lowered since, to maintain network security while

optimizing social welfare, less power goes to onshore nodes with higher prices. For instance, when

maximum social welfare is considered, the power going to the highest bid, Germany, is reduced from 2.19

pu to 1.55 pu if the MTdc network is N-1 secure. The power balancing among onshore nodes, when N-1

security is considered, happens so that in case of a fault, the offshore power can be shifted to converter

28

stations neighboring the defective one. This step in the optimization process guarantees that, in case of

a fault, the converter stations will not be overloaded.

Differently from the maximum direct voltage limitation, the 1-pu current limit in the hubs dc cables

demonstrated to be an inactive constraint. For the non-secure MTdc scenario, the maximum hub current

was 0.50 pu from HUB3 to HUB4, at the maximum social welfare operating point. For the N-1 secure

case, the same situation was encountered but now in the dc cable connecting HUB2 to HUB3. This

feature from the MOOA can help the ISO to optimize the hubs installed capacity, saving costs.

1050 1100 1150 1200 1250 1300 13506

7

8

9

10

11

12

Social Welfare [M€/year]

Pow

er L

osse

s [%

]

Network not secureNetwork N-1 secure

Figure 3.11: Pareto Fronts between Social Welfare and Power Losses.

1050 1075 1100 1125 1150 1175 1200 1225 12508

8.5

9

9.5

10

10.5

11

11.5


Pow

er L

osse

s [%

]

Case iii

Available operating pointsUnavailable operating points

1050 1075 1100 1125 1150 1175 1200 1225 12508

8.5

9

9.5

10

10.5

11

11.5


Pow

er L

osse

s [%

]

Case ii

Available operating pointsUnavailable operating points

1050 1075 1100 1125 1150 1175 1200 1225 12508

8.5

9

9.5

10

10.5

11

11.5


Pow

er L

osse

s [%

]

Case i

Available operating points

UK1+UK2

BE1

NL1+NL2

DE1+DE2

=DK1+DK2

BE

UK

DK

DE

1

1

NL

UK1+UK2

BE1

NL1+NL2

DE1+DE2

BE

UK

DK

DE

NL

UK1+UK2

BE1

NL1+NL2

DE1+DE2

BE

UK

DK

DE

NL

2A 3A3B

2B

1A

1B

=DK1+DK2 =DK1+DK2

2

2

3

3

1Aq

1Bq

2Aq

2Bq

3 3A Bq q

Figure 3.12: Available Pareto front between social welfare and power losses and respective buyers andsellers welfare for the considered case studies.

The graphic on Fig. 3.12 (a) shows the original optimization N-1 secure Pareto front without any market

constrains. The extra market constrains result in some MTdc operating points becoming unavailable

(represented by a thin line). Additionally, the second row of Fig. 3.12 displays, for each case study,

the sellers and buyers welfares for the two extremities on the available Pareto fronts. These two points

correspond to the lowest losses (point A) and the highest social welfare (point B) scenarios. The total

power received by the onshore nodes for the three different case studies is also shown.

29

Table 3.10: Direct voltages, Power flow and loss allocation coefficients for the analyzed case studies.

Node

Case i Case ii Case iii

A1 B1 A2 B2 A3 B3

Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk

UK1 1.05 1.05 14.2 1.03 1.05 9.1 1.05 1.05 14.0 1.03 1.05 9.1 1.05 1.05 14.0 1.05 1.05 13.8UK2 1.02 0.80 6.5 1.01 0.80 3.7 1.02 0.80 6.4 1.01 0.80 3.7 1.02 0.80 6.4 1.02 0.80 6.3BE1 0.98 0.50 -0.8 1.00 0.50 2.1 0.98 0.50 -0.1 1.00 0.50 2.1 0.98 0.50 -0.1 0.98 0.50 0.1NL1 1.00 0.80 3.4 1.05 0.80 8.5 1.01 0.80 4.7 1.05 0.80 8.5 1.01 0.80 4.7 1.01 0.80 5.2NL2 0.98 0.45 -0.5 1.02 0.45 3.3 0.99 0.45 0.4 1.02 0.45 3.3 0.99 0.45 0.4 0.99 0.45 0.7DE1 1.01 0.70 4.2 0.98 0.70 0.2 1.00 0.70 2.9 0.98 0.70 0.2 1.00 0.70 2.9 1.00 0.70 2.4DE2 1.02 0.80 6.6 0.99 0.80 1.7 1.01 0.80 5.1 0.99 0.80 1.7 1.01 0.80 5.1 1.01 0.80 4.5DK1 1.01 0.50 2.2 0.96 0.50 -0.9 1.00 0.50 1.8 0.96 0.50 -0.9 1.00 0.50 1.8 1.00 0.50 1.6DK2 1.01 0.40 1.8 0.96 0.40 -0.7 1.00 0.40 1.5 0.96 0.40 -0.7 1.00 0.40 1.5 1.00 0.40 1.3UK 0.94 -1.25 10.7 0.91 -1.44 14.6 0.94 -1.27 11.0 0.91 -1.44 14.6 0.94 -1.27 11.0 0.94 -1.28 10.8BE 0.92 -0.92 12.4 0.95 -0.95 3.7 0.93 -0.93 10.8 0.95 -0.95 3.7 0.93 -0.93 10.8 0.93 -0.93 10.1NL 0.93 -1.32 15.9 1.01 -0.20 -1.1 0.94 -1.13 9.3 1.01 -0.20 -1.1 0.94 -1.13 9.3 0.95 -1.05 7.0DE 0.93 -1.01 11.2 0.85 -1.55 34.2 0.90 -1.24 21.4 0.85 -1.55 34.2 0.90 -1.24 21.4 0.89 -1.33 26.0DK 0.93 -1.00 12.1 0.87 -1.19 21.7 0.93 -0.93 10.8 0.87 -1.19 21.7 0.93 -0.93 10.8 0.93 -0.90 10.1

PL [%] 8.10 11.12 8.29 11.12 8.29 8.47SW [M/eyear] 1080 1223 1128 1223 1128 1146

Table 3.10 displays the nodal direct voltages and power (in per unit), the loss allocation coefficients (in

percentage of the total losses), the transmission losses (in percentage of the offshore generation), and the

resultant social welfare (in M/eyear) for points A and B of all the examined case studies.

Case Study i

In the first case scenario there are no extra market restrictions on the load flow, thus, the entire Pareto

front on Fig. 3.12 (a) is reachable. The network losses difference for the two Pareto front extremities

(points A1 and B1) is approximately 3% of the power being generated. Considering the offshore bid as

90 e/MWh (the Danish OWF bid), and the same OWFs power production, the yearly cost difference in

the transmission losses between the two points is circa 157 Me. On the other hand, the yearly difference

in social welfare is 143 M.e In this case, if the MTdc grid is operated in point A1 that would lead to a

saving of circa 14 Me, which comes from the fact that, although, the social welfare is higher in point B1,

it does not compensate the extra power losses.

Point A1

The onshore direct voltages are approximately the same (see Table 3.10), leading to a reduction of the

power trade between the countries. Consequently, this is the point where the power losses are minimum.

Since all OWFs have entered the market, the MCP in this case, as shown in Fig. 3.12 (d), is equal to

90 e/MWh, i.e. the Danish OWFs bid.

The onshore nodes losses coefficients for point A1, are evenly distributed. The Netherlands has the

highest loss coefficient (15.9%), as it receives the highest amount of power – 1.32 pu – from the MTdc

grid. The second higher loss coefficient is given to Belgium, as the Belgian OWF (BE1) power production

is 0.5 pu and, hence, an extra 0.42 pu needs to be transmitted there. Amongst the offshore nodes, the

highest loss allocation is given to UK1 (14.2%), the highest power injecting node. Finally, two offshore

nodes, BE1 and NL2, have a negative loss coefficient which, according to the Z-bus method, means their

power production and location in the network is helping to achieve the desired load flow.

Point B1

The countries with the highest bids receive most of the power coming from the MTdc network, thus,

the social welfare is at is maximum. Table 3.10 shows that Germany receives 1.55 pu of power at B1

in comparison with only 1 pu at A1. This extra power for Germany is, for its most part, being shifted

30

from the Netherlands, whose power reception dropped from 1.32 pu to 0.20 pu, since it is the lowest

bidder. Accordingly, the German onshore node is attributed the highest loss coefficient (34.2%); because,

to maximize the social welfare, it receives more power than other countries. Amongst the offshore nodes,

again the highest loss coefficient is given to UK1 (9.1%).

Case Study ii

In Case ii, due to the additional market constraint (see Table 3.9), part of the original Pareto front – the

operating points with low losses – becomes unavailable. The Pareto extremities points, A2 and B2, have

a difference of 2.83% in their power losses, which capitalized over a year represents circa 147 Me. On the

other hand, the difference in social welfare is 95 Me. Hence, the Pareto extremity point A2 represents,

from the market participants point of view, a more economical operating point. Point A2 in comparison

with A1, represents a gain in social welfare of 48 M/eyear, while the power losses represent a loss of

circa 10 M.e Thus, point A2 is more beneficial for the market participants than its counterpart A1.

Point A2

The transmission losses in this case are 8.29%, which is 0.19% higher than for point A1. The new

active constraint reduces the Dutch power to 1.13 pu, compared to previous 1.32 pu (see Table 3.9); and

the difference is once more transferred to Germany. The power at the other onshore nodes was kept

approximately constant.

Regarding the loss components, Germany is the node with the highest losses share with 21.4%. The

remaining countries are being apportioned circa the same losses coefficients. For the OWFs, once more,

UK1 has the highest share (14%).

Point B2

Since the Pareto front region, where the social welfare is being maximized, has not changed due to the

extra market constraint in Case ii, the point B remained constant, i.e. B1 = B2.

Case Study iii

The second additional market constraint (see Table 3.9) impairs the highest social welfare value in the

Pareto front by 77 M/eyear. The transmission losses are reduced by 2.65%, when comparing points B2

and B3, leading to savings of approximately 138 M/eyear. Therefore, for the market participants point

B3 is more economical than B2.

Point A3

The extra constraint only excludes Pareto front points where the highest social welfare values are found,

thus, point A2 is kept constant, i.e. A3 = A2.

Point B3

Germany now receives 1.33 pu of power, which is 0.22 pu lower than in point B2. The decrease is a result

of the extra active constraint limiting the maximum dc current to Germany at 1.5 pu. The dc power

which could not be transferred to Germany is injected in the Netherlands. Hence, as the Netherlands

31

is helping to achieved the optimal load flow, it is rewarded with the lowest losses share amongst the

onshore nodes. On the other hand, Germany and OWF UK1 are assigned, once more, with the highest

losses coefficients amongst the onshore and offshore nodes, for being the highest buyer and producer,

respectively.

32

Chapter 4

Conclusions

4.0.1 MTdc Control: Single-Objective

Multi-terminal dc networks can help pave the way for further penetration of remotely located renewable

energy sources, such as offshore wind and wave energy. Nevertheless, before MTdc networks can be

developed, various problems, such as the control of the power flow, need to be addressed and solved.

The power-flow control inside a large MTdc network can be achieved using the distributed voltage control

(DVC) strategy. In the present work, a genetic algorithm has successfully been applied to solve an

optimal dc load-flow problem and give the DVC strategy the necessary voltage references to share the

responsibility of controlling the MTdc system voltage between all the onshore nodes. Dynamic numerical

simulations performed show that the power flow control is accurate and fast, i.e. a new operating point

can be achieved within 200 ms. Moreover, the genetic algorithm makes sure that the MTdc network is

N-1 secure and, in the event of an ac fault in any network node, the network operation is not disrupted

as shown in the simulation results.

4.0.2 MTdc Control: Multi-Objective

Several studies have suggested the construction of a transnational multi-terminal dc network for the

interconnection of offshore wind farms, specially in the North Sea. However, optimal power flow control

in MTdc networks is still a challenge. In this work, an independent system operator, which controls the

MTdc network via a market pool structure, has been proposed. The introduced ISO is responsible for

receiving the pool market participants bids and, after solving an optimal dc power flow, sending the dc

voltage references for the onshore converters so that the optimal flow can be realized. The optimization of

the MTdc network power flow was performed by a multi-objective genetic algorithm set to optimize the

network transmission losses and social welfare. Additionally, a loss allocation procedure was implemented

to distribute the transmission power losses costs in a fair way amongst the generator and consumer entities.

Through this procedure, market participants with higher power supply and demand were allocated with

higher power losses coefficients.

The results from the obtained Pareto fronts indicate the maximum social welfare is achieved when more

power is transmitted to countries with higher bids, however, this leads to an increase in transmission

losses. The addition of extra constraints to the power flow prohibits the operation in some of the Pareto

front regions. Consequently, it may not be possible for the ISO to operate the MTdc network where the

best possible cases are found. In this way, the ISO has to decide which is the best operating point for

the offshore grid.

33

The proposed approach to control MTdc network offers a high flexibility to the ISO in obtaining optimal

trade-offs. Although, in this work only the transmission losses and social welfare were analyzed, more

optimization goals may be considered simultaneously.

4.0.3 Transmission Systems Design

The MOOA implemented was able to find feasible solutions for the proposed problem. Optimal Pareto

Fronts were achieved for both AC and DC transmission options when different transmission distances

were considered. The decision maker can then be presented with these OPFs for the transmission system;

and, based on the trade-off between the two goals, it will be possible for him to decide which configuration

better suits the purposes of the project.

It is important to refer that data from real components was used. Nevertheless, as with any database,

the one used is limited in information; and, therefore, the results obtained with the algorithm have

the database as a major drawback. Moreover, the MOOA itself does not guarantee the optimal set of

solutions [40].

The next application of the multi-objective optimization method would be to include component reliabil-

ity, power quality, control and grid code aspects to transmission systems for offshore grids, which cannot

be easily taken into account with other optimization methods.

4.0.4 Multi-Objective Optimization Algorithms

MOOAs are an important instrument since, among other characteristics, they do not require any infor-

mation about the targets to be optimized besides their own evaluation. Therefore, the goal functions

are not restrict with respect to differentiation, continuity or any other particular behavior in the search

space of the problem. MOOAs are able to work simultaneously with real and integer representation of

variables [41]. A very important feature for the success of such algorithms is the fact that they treat the

multi-objective problem with non-commensurable objectives, i.e. no a priori information is needed [42].

34

Bibliography

[1] EWEA, “The European offshore wind industry key 2011 trends and statistics,” Tech. Rep. January, 2012.

[2] P. Bresesti, W. L. Kling, R. L. Hendriks, and R. Vailati, “HVDC Connection of Offshore Wind Farms to the Transmission

System,” IEEE Transactions on Energy Conversion, vol. 22, pp. 37–43, Mar. 2007.

[3] N. B. Negra, J. Todorovic, and T. Ackermann, “Loss evaluation of HVAC and HVDC transmission solutions for large offshore

wind farms,” Renewable and Sustainable Energy Reviews, vol. 76, pp. 916–927, July 2006.

[4] B. V. Eeckhout, The Economic Value of VSC HVDC compared to HVAC for offshore wind farms. PhD thesis, K.U.Leuven,

2008.

[5] S. Meier and P. C. K. R, “Benchmark of Annual Energy Production for Different Wind Farm Topologies,” in Power Electronics

Specialists Conference, pp. 2073–2080, 2005.

[6] R. Sharma, T. Rasmussen, K. H. j. Jensen, and V. Akamatov, “Modular VSC converter based HVDC power transmission

from offshore wind power plant: Compared to the conventional HVAC system,” in Electric Power and Energy Conference,

(Halifax, NS, Canada), 2010.

[7] F. Cadini, E. Zio, and C. A. Petrescu, “Multi-objective genetic algorithm optimization of electrical power transmission

networks,” tech. rep., 2010.

[8] L. Birk, G. F. Clauss, and J. Y. Lee, “Pratical Application of Global Optimization to the Design of Offshore Structures,” in

23rd Internation Conference on Offshore Mechanics and Arctic Engineering, (Vanciuver (Canada)), 2004.

[9] E. Berge, Byrkjedal, Y. Ydersbond, and D. Kindler, “Modelling of offshore wind resources. Comparison of a meso-scale model

and measurements from FINO 1 and North Sea oil rigs.,” EWEC 2009, pp. 1–8, 2009.

[10] J. R. Mclean and G. Hassan, “TradeWind - Equivalent Wind Power Curves,” Tech. Rep. October, 2008.

[11] C. A. Morgan, H. M. Snodin, and N. C. Scott, “Offshore Wind - Economies of scale, engineering resource and load factors,”

tech. rep., Garrad Hassan and Partners, Department of Trade and Industry/Carbon Trust, 2003.

[12] I. Erlich, M. Wilch, and C. Feltes, “Reactive Power Generation by DFIG Based Wind Farms with AC Grid Connection

Keywords,” tech. rep., Institute of Electrical Power Systems (EAN), Universitat Duisburg-Essen, Duisburg, Germany, 2007.

[13] N. Ahmed, S. Norrga, H.-P. Nee, A. Haider, D. Van Hertem, L. Zhang, and L. Harnefors, “HVDC SuperGrids with modular

multilevel converters - The power transmission backbone of the future,” in International Multi-Conference on Systems,

Signals and Devices, pp. 1–7, Ieee, Mar. 2012.

[14] S. Gang, C. Zhe, and C. Xu, “Overview of multi-terminal VSC HVDC transmission for large offshore wind farms,” in The

International Conference on Advanced Power System Automation and Protection, pp. 1324–1329, October 2011.

[15] OffshoreGrid, “Offshore Electricity Infrastructure in Europe,” tech. rep., OffshoreGrid, October 2011.

[16] D. Van Hertem and M. Ghandhari, “Multi-terminal VSC HVDC for the European supergrid: Obstacles,” Renewable and

Sustainable Energy Reviews, vol. 14, pp. 3156–3163, Dec. 2010.

[17] A. A. van der Meer, R. Teixeira Pinto, M. Gibescu, P. Bauer, J. Pierik, F. Nieuwenhout, R. Hendriks, W. Kling, and G. van

Kuik, “Offshore transnational grids in Europe: the North Sea Transnational grid research project in relation to other research

initiatives,” in 9th International Workshop on Large-Scale Integration of Wind Power Into Power Systems As Well As

On Transmission Networks For Offshore Wind Power Plants, (Quebec, Canada), October 2010.

[18] M. Glinkowski, J. Hou, and G. Rackliffe, “Advances in Wind Energy Technologies in the Context of Smart Grid,” Proceedings

of the IEEE, vol. 99, pp. 1083–1097, June 2011.

[19] J. Yuryevich, “Evolutionary programming based optimal power flow algorithm,” IEEE Transactions on Power Systems,

vol. 14, no. 4, pp. 1245–1250, 1999.

[20] M. Osman, M. Abo-Sinna, and A. Mousa, “A solution to the optimal power flow using genetic algorithm,” Applied Mathe-

matics and Computation (Elsevier), vol. 155, no. 2004, pp. 391–405, 2003.

[21] A. G. Bakirtzis, S. Member, P. N. Biskas, and S. Member, “Optimal Power Flow by Enhanced Genetic Algorithm,” IEEE

Transactions on Power Systems, vol. 17, no. 2, pp. 229–236, 2002.

35

[22] T. Numnonda and U. Annakkage, “Optimal power dispatch in multinode electricity market using genetic algorithm,” Electric

Power Systems Research, vol. 49, pp. 211–220, Apr. 1999.

[23] P. Roy, S. Sen, S. Sengupta, and A. Chakrabarti, “Genetic Algorithm Based Spot Pricing of Electricity in Deregulated

Environment for Consumer Welfare,” in Electrical and Computer Engineering ICECE, no. December, (Dhaka, Bangladesh),

2010.

[24] F. Milano, C. Canizares, and M. Invernizzi, “Multiobjective optimization for pricing system security in electricity markets,”

IEEE Transactions on Power Systems, vol. 18, no. 2, pp. 596–604, 2003.

[25] R. T. Pinto, P. Bauer, S. F. Rodrigues, E. J. Wiggelinkhuizen, J. Pierik, and B. Ferreira, “A Novel Distributed Direct Voltage

Control Strategy for Grid Integration of Offshore Wind Energy Systems through MTDC Network,” IEEE Transactions on

Industrial Electronics, vol. PP, no. 99, pp. 1–13, 2011.

[26] M. Kolenc, I. Papic, and B. Blazic, “Minimization of Losses in Smart Grids Using Coordinated Voltage Control,” Energies,

vol. 5, pp. 3768–3787, Sept. 2012.

[27] J. Zhu, Optimization of Power System Operation. New Jersey, NJ: Willey-IEEE Press, 2009.

[28] K. Xie, Y.-H. Song, and J. Stonham, “Decomposition model and interior point methods for optimal spot pricing of electricity

in deregulation environments,” IEEE Transactions on Power Systems, vol. 15, no. 1, pp. 39–50, 2000.

[29] A. Gomez-Exposito, A. J. Conejo, and C. Canizares, ”Electric Energy Systems”, vol. 17 of Electric Power Engineering

Series. CRC Press, 2008.

[30] A. Gomez-Exposito, A. Conejo, and C. Canizares, Electric Energy Systems - Analysys and Operation. CRC Press, 2008.

[31] P. Norgaard and H. Holttinen, “A Multi-Turbine Power Curve Approach,” in Nordic Wind Power Conference, no. March,

2004.

[32] E. Prieto-Araujo, F. Bianchi, A. Junyent-Ferre, and O. Gomis-Bellmut, “Methodology for Droop Control Dynamic Analysis

of Multiterminal VSC-HVDC Grids for Offshore Wind Farms,” IEEE TRANSACTIONS ON POWER DELIVERY, vol. 26,

no. 4, pp. 2476–2485, 2011.

[33] S. Rodrigues, Dynamic Modeling and Control of VSC-based Multi-terminal DC Networks. PhD thesis, Instituto Superior

Tecnico, 2011.

[34] V. Jalili-Marandi and V. Dinavahi, “Real-Time Simulation of Grid-Connected Wind Farms Using Physical Aggregation,”

IEEE Transactions on Industrial Electronics, vol. 57, pp. 3010–3021, Sept. 2010.

[35] OffshoreGrid, “Offshore Electricity Infrastructure in Europe,” October, pp. 1–151, 2011.

[36] A. J. Conejo, F. D. Galiana, and I. Kockar, “Z-Bus Loss Allocation,” IEEE Transactions on Power Systems, vol. 16, no. 1,

pp. 105–110, 2001.

[37] J. Kaldellis and M. Kapsali, “Shifting towards offshore wind energy – Recent activity and future development,” Energy

Policy, vol. 53, no. 0, pp. 136–148, 2013.

[38] M. I. Blanco, “The economics of wind energy,” Renewable and Sustainable Energy Reviews, vol. 13, no. 6-7, pp. 1372–1382,

2009.

[39] Europe’s Energy Portal. Last accessed in 5 February 2013.

[40] K. Deb, ”Multi-Objective Optimization using Evolutionary Algorithms”. Wiltshire, England: John Wiley & Sons, Ltd,

1st ed., 2001.

[41] Y.-x. Li and M. Gen, “Nonlinear Mixed Integer Programming Problems Using Genetic Algorithm and Penalty Function,”

IEEE International Conference on Systems, Man, and Cybernetics, vol. 4, pp. 2677–2682, 1996.

[42] C. M. Fonseca and P. J. Fleming, “Genetic Algorithms for Multiobjective Optimization : Formulation , Discussion and

Generalization,” in Proceedings of the Fifth International Conference, no. July, (San Mateo, CA), 1993.

[43] K. Meah and S. Ula, “Comparative Evaluation of HVDC and HVAC Transmission Systems,” in Power Engineering Society

General Meeting, pp. 1–5, 2007.

[44] I. Alegrıa, J. Martın, I. Kortabarria, J. Andreu, and P. Ereno, “Transmission alternatives for offshore electrical power,”

Renewable and Sustainable Energy Reviews, vol. 13, pp. 1027–1038, June 2009.

[45] N. Flourentzou, S. Member, V. G. Agelidis, S. Member, and G. D. Demetriades, “VSC-Based HVDC Power Transmission

Systems : An Overview,” IEEE TRANSACTIONS ON POWER ELECTRONICS, vol. 24, no. 3, pp. 592–602, 2009.

[46] L. Livermore, D. Liang, and D. Ekanayake, “MTDC VSC Technology and its applications for Wind Power,” in Universities

Power Engineering Conference (UPEC), 2010.

[47] L. Zhang, Modeling and Control of VSC-HVDC Links Connected to Weak AC Systems. PhD thesis, ROYAL INSTITUTE

OF TECHNOLOGY, 2010.

[48] R. Teixeira Pinto and P. Bauer, “The Role of Modularity Inside the North Sea Transnational Grid Project : Modular

Concepts for the Construction and Operation of Large Offshore,” in Renewable Energy World Europe Conference, (Milan,

Italy), 2011.

36

[49] J. F. Chozas, H. C. Soerensen, and M. Korpas, “Integration of Wave and Offshore Wind Energy in a European Offshore

Grid,” in Proceedings of the Twentieth (2010) International Offshore and Polar Engineering Conference, vol. 7, (Beijing,

China), 2010.

[50] M. Hyttinen, J. O. Lamell, and T. F. Nestli, “New Application of Voltage Source Converter (VSC) HVDC to be Installed on

the Gas Platform Troll A,” in CIGRE, (Paris), 2004.

[51] R. L. Hendriks, G. S. Member, M. Gibescu, M. M. Roggenkamp, and W. L. Kling, “Developing a Transnational Electricity

Infrastructure Offshore : Interdependence between Technical and Regulatory Solutions,” in Power and Energy Society

General Meeting, 2010.

[52] M. Tsili and S. Papathanassiou, “A review of grid code technical requirements for wind farms,” IET Renewable Power

Generation, vol. 3, no. 3, pp. 308–332, 2009.

[53] I. Erlich and U. Bachmann, “Grid code requirements concerning connection and operation of wind turbines in Germany,” in

Power Engineering Society General Meeting, pp. 2230–2234, 2005.

[54] LORC. [Last accessed 9th July 2013], 2013.

[55] D. V. Hertem and M. Ghandhari, “Multi-terminal vsc hvdc for the european supergrid: Obstacles,” Renewable and Sustain-

able Energy Reviews, vol. 14, no. 9, pp. 3156–3163, 2010.

[56] V. Lescale, A. Kumar, L.-E. Juhlin, H. Bjorklund, and K. Nyberg, “Challenges with multi-terminal uhvdc transmissions,” in

Power System Technology and IEEE Power India Conference, 2008. POWERCON 2008. Joint International Conference

on, pp. 1–7, oct. 2008.

[57] E. Koldby and M. Hyttinen, “Challenges on the Road to an Offshore HVDC Grid,” in Nordic Wind Power Conference,

2009.

[58] M. Callavik, M. Bahrman, and P. Sandeberg, “Technology developments and plans to solve operational challenges facilitating

the HVDC offshore grid,” in Power and Energy Society General Meeting, pp. 1–6, 2012.

[59] North Seas Countries’ Offshore Grid Initiative, “Final Report - Grid Configuration,” technical report, NSCOGI, November

2012.

[60] A. Yazdani and R. Iravani, “A unified dynamic model and control for the voltage-sourced converter under unbalanced grid

conditions,” IEEE Transactions on Power Delivery, vol. 21, no. 3, pp. 1620–1629, 2006.

[61] S. Cole, J. Beerten, and R. Belmans, “Generalized Dynamic VSC MTDC Model for Power System Stability Studies,” IEEE

Transactions on Power Systems, vol. 25, no. 3, pp. 1655–1662, 2010.

[62] L. Harnefors, A. Antonopoulos, S. Norrga, L. Angquist, and H.-P. Nee, “Dynamic Analysis of Modular Multilevel Converters,”

Industrial Electronics, IEEE Transactions on, vol. 60, no. 7, pp. 2526–2537, 2013.

[63] R. Adapa, “High-Wire Act,” IEEE Power and Energy Magazine, pp. 18–29, Nov./Dec. 2012.

[64] J. Hafner and B. Jacobson, “Proactive hybrid HVDC breakers – A key innovation for reliable HVDC grids,” in International

Symposium on Integrating supergrids and microgrids, (Bologna), pp. 1–8, CIGRE, September 2011.

[65] C. Franck, “HVDC Circuit Breakers: A Review Identifying Future Research Needs,” IEEE Trans. on Power Delivery,

vol. 26, no. 2, pp. 998–1007, 2011.

[66] N. Chaudhuri, R. Majumder, B. Chaudhuri, and J. Pan, “Stability Analysis of VSC MTDC Grids Connected to Multimachine

AC Systems,” vol. 26, pp. 2774–2784, Oct. 2011.

[67] L. Zhang, Modeling and Control of VSC-HVDC Links Connected to Weak AC Systems. Phd. thesis, Royal Institute of

Technology, Stockholm, April 2010. ISBN: 978-91-7415-640-9.

[68] R. Teixeira Pinto, P. Rodrigues, Sılvio Bauer, and J. Pierik, “Description and comparison of dc voltage control strategies for

offshore mtdc networks: Steady-state and fault analysis,” European Power Electronics Journal, vol. 22, no. 4, pp. 13 – 21,

2013.

[69] B. Johnson, R. Lasseter, F. Alvarado, and R. Adapa, “Expandable multiterminal dc systems based on voltage droop,” vol. 8,

pp. 1926–1932, Oct. 1993.

[70] L. Xu, L. Yao, and M. Bazargan, “Dc grid management of a multi-terminal hvdc transmission system for large offshore wind

farms,” in Sustainable Power Generation and Supply, 2009. SUPERGEN ’09. International Conference on, pp. 1–7, april

2009.

[71] L. Xu, B. Williams, and L. Yao, “Multi-terminal dc transmission systems for connecting large offshore wind farms,” in Power

and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE,

pp. 1–7, july 2008.

[72] Y. Tokiwa, F. Ichikawa, K. Suzuki, H. Inokuchi, S. Hirose, and K. Kimura, “Novel control strategies for hvdc system with

self-contained converter,” Electrical Engineering in Japan, vol. 113, no. 5, pp. 1–13, 1993.

[73] R. Teixeira Pinto, P. Bauer, S. Rodrigues, E. Wiggelinkhuizen, J. Pierik, and B. Ferreira, “A Novel Distributed Direct-Voltage

Control Strategy for Grid Integration of Offshore Wind Energy Systems Through MTDC Network,” IEEE Transactions on

Industrial Electronics, vol. 60, pp. 2429 –2441, june 2013.

37

[74] H. Rudnick, “Chile: Pioneer in deregulation of the electric power sector,” IEEE Power Engineering Review, vol. 14(6):28-3,

no. June, 1994.

[75] M. M. Martinez, Optimization models and techniques for implementation and pricing of electricity markets. PhD thesis,

University of Waterloo, Ontario, Canada, 2000.

[76] J. Momoh and L. Mili, Economic market design and planning for electric power systems, vol. 52. Wiley-IEEE Press, 2009.

[77] “European Commission, Synthesis report of the workshop - Contribution of EU technology projects (EEPR/ FP7) to the

development of the offshore grid.”

[78] A. Conejo, J. Arroyo, N. Alguacil, and A. Guijarro, “Transmission Loss Allocation: A Comparison of Different Practical

Algorithms,” IEEE Transactions on Power Systems, vol. 17, no. 3, pp. 571–576, 2002.

[79] J. Bialek, “Tracing the flow of electricity,” IEE Proceedings, vol. 143, no. 4, pp. 313–320, 1996.

[80] D. Kirschen, R. Allan, and G. Strbac, “Contributions of individual generators to loads and flows,” IEEE Transactions on

Power Systems, vol. 12, no. 1, pp. 52–60, 1997.

[81] A. van der Meer, R. Teixeira Pinto, M. Gibescu, P. Bauer, J. T. G. Pierik, F. D. J. Nieuwenhout, R. L. Hendriks, W. L.

Kling, and G. A. M. van Kuik, “Offshore Transnational Grids in Europe: The North Sea Transnational Grid Research

Project in Relation to Other Research Initiatives,” in 9th International Workshop on Large-Scale Integration of Wind

Power, (Montreal, Canada), 2010.

[82] M. J. Kurzidem, Analysis of Flow-based Market Coupling in Oligopolistic Power Markets. PhD thesis, ETH Zurich, 2010.

[83] U. Maulik, Sanghamitra Bandyopadhyay, and A. Mukhopadhyay, ”Multiobjective Genetic Algorithms for Clustering”. New

York: Springer, 1st ed., 2011.

[84] F. Y. Cheng and D. Li, “Multiobjective optimization design with pareto genetic algorithm,” Journal of Structural Engi-

neering, vol. 123, no. 9, pp. 1252–1261, 1997.

[85] C. Darwin, “On the Origin of Species,” 1859.

[86] F. Kursawe, “A Variant of Evolution Strategies for Vector Optimization,” in Proceedings of the 1st Workshop on Parallel

Problem Solving from Nature, PPSN I, pp. 193–197, 1990.

[87] C. M. Fonseca and P. J. Fleming, “Multiobjective Optimization and Multiple Constraint Handling with Evolutionary Algo-

rithms - Part I A Unified Formulation,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 28, no. 1, pp. 26–37,

1998.

[88] K. Deb, A. Member, A. Pratap, S. Agarwal, and T. Meyarivan, “A Fast and Elitist Multiobjective Genetic Algorithm :,”

IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182–197, 2002.

[89] J. Knowles and D. Corne, “The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective

optimisation,” Proceedings of the 1999 Congress on Evolutionary Computation, pp. 98–105, 1999.

[90] E. Zitzler, Evolutionary Algorithms for Multiobjective Optimization - Methods and Applications. PhD thesis, Swiss Federal

Institute of Technology (ETH), 1999.

[91] A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial,” Reliability

Engineering & System Safety, vol. 91, pp. 992–1007, Sept. 2006.

[92] J. Blondin, “Particle Swarm Optimization : A Tutorial,” tech. rep., 2009.

[93] C. A. C. Coello, G. T. Pulido, and M. S. Lechuga, “Handling Multi Objectives With Particle Swarm Optimization,” IEEE

Transactions on Evolutionary Com- putation, vol. 8, no. 3, pp. 256–279, 2004.

[94] C. Fonseca and P. J. Fleming, “An overview of evolutionary algorithms in multiobjective optimization,” Evolutionary Com-

putation, vol. 3, no. 1, pp. 1–16, 1995.

[95] C. A. C. Coello, “A Survey of Constraint Handling Techniques used with Evolutionary Algorithms,” tech. rep., Laboratorio

Nacional de Informatica Avanzada, 1999.

[96] M. Gen and R. Cheng, “A Survey of Penalty Techniques in Genetic Algorithms,” in IEEE International Conference on

Evolutionary Computation, pp. 804–809, 1996.

[97] Y. Woldesenbet, G. Yen, and B. Tessema, “Constraint Handling in Multiobjective Evolutionary Optimization,” IEEE Trans-

actions on Evolutionary Computation, vol. 13, pp. 514–525, June 2009.

[98] J. A. Joines and C. R. Houck, “On the Use of Non-Stationary Penalty Functions to Solve Nonlinear Constrained Optimization

Problems with GA’ s,” in Proceedings of the First IEEE Conference on Evolutionary Compuatation, vol. 2, (Raleigh,NC),

pp. 579–584, 1994.

[99] H. J. Barbosa and A. C. Lemonge, “An adaptive penalty scheme in genetic algorithms for constrained optimization problems,”

tech. rep.

[100] H. N. Luong and P. A. N. Bosman, “Elitist Archiving for Multi-Objective Evolutionary Algorithms : To Adapt or Not to

Adapt,” Tech. Rep. x, 2012.

38

[101] J. E. Fieldsend, R. M. Everson, and S. Singh, “Using Unconstrained Elite Archives for Multiobjective Optimization,” Com-

puter, vol. 7, no. 3, pp. 305–323, 2003.

[102] P. Bosman and D. Thierens, “The balance between proximity and diversity in multiobjective evolutionary algorithms,” IEEE

Transactions on Evolutionary Computation, vol. 7, pp. 174–188, Apr. 2003.

[103] R. L. Haupt and D. H. Werner, Genetic algorithms in electromagnetics. New Jersey: John Wiley & Sons, Inc., 2007.

[104] C. A. C. Coello, G. B. Lamont, and D. A. V. Veldhuizen, Evolutionary Algorithms for Solving Multi-Objective Problems.

New York: Springer, 2nd edt ed., 2007.

[105] A. A. Adewuya, New Methods in Genetic Search with Real-Valued Chromosomes. PhD thesis, Massachusetts Institute of

Technology, 1996.

[106] D. S. Linden, “Using a Real Chromosome in a Genetic Algorithm for Wire Antenna Optimization,” pp. 1704–1707.

[107] R. L. Haupt and S. E. Haupt, Pratical Genetic Algorithms (Second Edition). New Jersey: John Wiley & Sons, Inc., 2004.

[108] Y. Arslanoglu, GENETIC ALGORITHM FOR PERSONNEL ASSIGNMENT PROBLEM WITH MULTIPLE OBJEC-

TIVES. PhD thesis, Middle East Technical University, 2006.

[109] N. Mori, Y. Yabumoto, H. Kita, and Y. Nishikawa, “Multi-Objective Optimization by Means of the Thermodynamical Genetic

Algorithm,” Transactions of the Institute of Systems, Control and Information Engineers, vol. 11, no. 3, pp. 103–111, 1998.

[110] K. Deb, “Multi-objective genetic algorithms: problem difficulties and construction of test problems,” Jan. 1999.

[111] K. Deb, A. Pratap, and T. Meyarivan, “Constrained Test Problems for Multi-objective Evolutionary Optimization,” pp. 284–

298, 2001.

[112] M. Tanaka, H. Watanabe, Y. Furukawa, and T. Tanino, “GA-based decision support system for multicriteria optimization,”

1995 IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century, vol. 2,

no. 2, pp. 1556–1561, 1995.

[113] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable Test Problems for Evolutionary Multi-Objective Optimization,”

Tech. Rep. 1990, TIK-Technical Report 112, 2001.

39

Appendix A

MTdc network technical and

economic aspects

A.1 MTdc networks

For bulk transmission purposes (e.g. ≥ 500 MW), as well as for long submarine transmission distances

(e.g. ≥ 60 - 80 km), using HVdc systems for the transmission of the generated electricity is both

economically and technically more attractive than using HVac systems [43][44]. When it comes to HVdc

transmission two options are available: line-commutated current-source HVdc (CSC-HVdc), also referred

to as classic HVdc, or forced-commutated voltage-source HVdc transmission (VSC-HVdc). The latter

is a newer technology and, despite the fact it has higher converter losses, it has significant advantages

when it comes to bringing the generated offshore wind energy ashore [45][46]. Due to its smaller footprint

and improved controllability, the use of VSC-HVdc systems is being considered as suitable for a broad

range of multi-terminal dc applications, such as supergrids, electronic power distribution systems and

transnational networks for integration of offshore wind farms [32].

The advent of VSC-HVdc for transmission purposes has brought back the interest in building MTdc

networks; which has been proved difficult solely using CSC-HVdc technology [2]. The limitations to

2-quadrant operation, the need for reactive power supply at each terminal, harmonic content filtering

equipment size and its limited control capabilities are examples of drawbacks when it comes to the

construction of a MTdc network using classic HVdc technology [46]. Hence, most likely, high-voltage

MTdc networks will be developed using voltage-source converters, given their control flexibility and

hardware modularity [47, 48].

A.1.1 Applications of a MTdc grid

Multi-terminal HVdc transmission systems are characterized by having more than two converter stations

interconnected on the network dc side. There are several possibilities for the grid topology: shore-to-

shore, radial, meshed, cluster with multi-way interconnector, a combination of interconnectors, amongst

others [1]. The configuration choice may differ depending on the application of the MTdc network. Some

or the most prominent applications are briefly discussed next.

40

Offshore Wind & Ocean Energy

In Europe, the offshore potential consists both of wind and ocean energy. Ocean energy includes wave,

tidal current, tidal range, osmotic and ocean thermal energies. West European coasts have high wave

energy potential [49]. Since wave energy has greater predictability and less variability when compared to

wind energy; and because wave energy generally peaks 6 to 8 hours later than wind energy, a combination

of both resources would result in smoother variations of the generated power and better predictability

[49].

In the near future, offshore wind farms (OWFs) are expected to have higher power ratings and to be

situated further offshore [1]. This implies that the costs of the electrical infrastructure will constitute a

higher share in the total investment costs. New offshore projects may be interconnected with the already

existing MTdc network, which increases the existing connection utilization. Moreover, since it is not

necessary to build a new transmission systems to connect to shore, the initial capital expenditure of new

projects can be reduced [49].

Oil & Gas Platforms

In 2005, the Norwegian energy company Statoil became the first to have an offshore platform (Troll A)

powered via a VSC-HVdc transmission system [50]. In most offshore installations, generators and large

compressors are driven by onboard gas turbines or diesel engines, whose efficiencies are low, usually well

below 50% [50]. This results in large amounts of CO2 emissions and a high fuel consumption.

A MTdc network interconnecting OWFs, oil and gas platforms and onshore grids would result in reduced

operational costs, increased reliability, reduced CO2 emissions, lower maintenance requirements, longer

lifetime and higher availability when compared to gas turbines and diesel engines. Moreover, if the

transmission equipment can be located on decommissioned offshore platforms, their postponed removal

cost can be an important factor as well [50].

Energy Market

Due to the deregulation of the European energy sectors, there exists an increasing demand for cross-

border exchange capacity. The need for the higher interconnection capacity is led by market coupling

and for balancing fluctuations in the output from renewable energy. A MTdc network connecting offshore

wind power plants with inter-connectors between countries could increase the utilization of the electrical

infrastructure and, hence, reduce costs. Additionally, it could help to decrease the variability in wind

power production (which decays when aggregated over large areas), as well as enable the trade of power

imbalances, such as unscheduled surpluses or deficits of wind energy [51].

Asynchronous Systems & AC Grid Reinforcement

A dc network allows for the interconnection of asynchronous systems, differently from ac networks; where

a surplus of generating capacity in one grid it cannot be transmitted to others if they have different

frequencies (e.g. Brazil and Paraguay) or are not synchronous (e.g. the UK and mainland Europe). Every

network must have its own capacity of peak power generation, usually in the form of older, inefficient

power plants. Moreover, asynchronous HVdc links can act as a firewall, effectively protecting against

propagation of cascading outages in one network to another. Furthermore, with a dc transmission system

it is also possible to damp local area power oscillations, manage congestion and reinforce existing ac grids

[33].

41

Reliability and Redundancy

Most offshore wind power plants built so far use a single connection to shore [15]. The reliability of the

connection critically relies on some of its main components, most importantly the cable circuitry. When

malfunctioning, these components might take the complete installation out of service.

In the future, TSOs may increase the requirements on the connection availability. By having several con-

nection points to the onshore grids, properly designed MTdc networks are expected to improve reliability.

Even in the event of a MTdc network section fault, there can be an alternative path along which the

power, or part of it, can still be transmitted.

Flexibility and Controllability

Nowadays, large wind farms are required to actively participate in the power system control – both

frequency and voltage – just as any conventional power plant [52][53]. When an OWF is connected via

a dc transmission system, the responsibility of complying to the grid code requirements is passed onto

the onshore HVdc station. A VSC can generate or consume reactive power from the connected network

helping with voltage regulation. Additionally, a predefined power flow along a certain transmission cable

can be quickly achieved.

Therefore, when connecting offshore wind farms, a MTdc network may guarantee the necessary flexibility

and controllability to meet the main grid code requirements such as frequency operating range, active

and reactive power control, support during voltage dips and fault-ride through capabilities.

A.1.2 Challenges towards a VSC-based MTdc network

By 2030, circa 150 GW of wind power is predicted to be installed in the North Sea and offshore wind

farm constructions have already started. Out of a total of 76 offshore wind farms installations around

the world, 31 (circa 41%) are in the North Sea. From the latter, six OWFs – all planned to be delivered

in 2013/2014 – will use an HVdc transmission system [54].

The North Sea MTdc network can serve as a platform for offshore wind and electricity market integration

in Northern Europe. Hence, power flow control inside MTdc networks will be of vital importance [55].

However, before multi-terminal dc networks can be developed, four key aspects need to be better studied

and developed: system integration, dynamic and contingency behavior, stability and power flow control.

System Integration

System integration aspects of multi-terminal dc networks have been tackled by several studies [56, 57, 58].

The main conclusion is that MTdc networks will have to organically grow with time; from an inherently

simple initial phase to a desired later form, a hopefully much more complex meshed topology [59].

Dynamic & Contingency Behaviour

The dynamic behavior of VSCs — including multi-level topologies — and multi-terminal dc networks has

been extensively analyzed in the literature [60, 61, 62]. The behavior of MTdc networks during contin-

gencies, specially on the dc-side of the transmission system, will be vital for its successful development.

After a dc contingency the system will most likely have only a few milliseconds to clear faults before high

42

dc currents start posing a threat to the MTdc network [63]. This is the reason why several efforts are

currently focusing on research and development of HVdc switch breakers [64, 65].

Dynamic Behaviour

The stability of MTdc networks has also been thoroughly analyzed in the literature [61, 66]. It has been

shown that MTdc networks formed by VSCs are inherently unstable systems which need feedback control

to compensate the unstable system poles [67].

Power Flow Control

Inside a MTdc transmission system, controlling the network direct voltage is one of the most important

tasks given to VSC-HVDC stations. A well-controlled direct voltage on a HVDC link is a guarantee of

power balance between all the interconnected nodes [47].

The control of point-to-point HVDC transmission systems is typically arranged so that one terminal

controls the dc network voltage, whereas the other operates in current or power regulation mode. This

control philosophy – of having only one converter controlling the direct voltage – can be extended to

MTdc networks. However, as the MTdc network grows, it will be increasingly difficult to assure power

balance by having only one terminal responsible for voltage regulation. Hence, for large MTdc networks,

controlling the voltage at a single terminal is not desirable. For its successful development and operation,

MTdc networks will require a control strategy capable of sharing the dc system voltage control amongst

more than one network node.

Different control strategies have been proposed in the literature to operate MTdc networks. Four different

direct voltage control methods were analyzed and compared according to their capability of performing

different market dispatch schemes in [33, 68]. A brief introduction to these control methods is given next.

• Voltage droop method: it uses a droop mechanism to control the dc voltage. Although this method

was initially thought for controlling MTdc networks using HVdc classic technology, it can also be

applied for VSC-based MTdc networks [69];

• Ratio control: a power ratio between onshore stations may be established. The power transmission

ratio between the stations can be set and varied by the system operator from time to time [70];

• Priority control: one onshore terminal has priority in terms of receiving the energy generated

offshore over the others. With this control strategy, the remaining onshore terminals will not

receive any energy until the capability or the active power limit of the first terminal has been

reached [71];

• Voltage margin method: in this method, each converter station inside the network is given a

dc voltage reference marginally offset, generating the voltage margin. The method was initially

developed to control a three-terminal back-to-back HVDC system in Japan [72].

The aforementioned control methods have drawbacks regarding their expandability, dynamic response

and flexibility, i.e. the capacity of steering the power flow inside large MTdc networks [68]. A nre

method of controlling VSC-MTdc networks called distributed voltage control (DVC) is proposed [73]. In

the DVC method the dc system voltage references are obtained through means of a dc Optimal Power

Flow.

43

A.2 Market Schemes

A restructuring process in the power industry, which started at the end of the seventies, gave birth to the

world’s first competitive market for electricity generation in Chile, in 1982 [74]. The order to privatize

the United Kingdom electricity industry in 1988, concluded with the creation, in 1992, of the England

and Wales Power Pool [75]. In the same year, the approval of the Electricity Policy Act (EPAct) ordered

an open access to transmission networks in the USA. Ever since, a variety of electricity pool markets have

been created; among them, the Pennsylvania-New Jersey-Maryland (PJM) Pool (1997), the Californian

market, and the New York Power Pool (1998) [75].

Several other countries have also implemented electricity markets, or are in the process of doing so; e.g.

Australia, Brazil, Germany, Norway and Spain. The world-wide restructuring process of the electricity

industry begun more than twenty years ago and has since deeply accelerated, creating a political and

technical turmoil [76]. The operation of large investors or state-owned utilities is being transformed from

an rate-of-return basis to a competitive basis by the creation of electricity markets.

Future offshore transnational offshore networks will most probably employ a centralized model imple-

menting a pool market scheme to regulate the wind farms output power [35]. Moreover, the possibility of

setting up an international independent system operator (ISO) to regulate a future offshore transnational

DC network in the North Sea is currently under discussion [77].

A.2.1 Pool Market

A Pool Market is defined as a centralized marketplace which clears the market for buyers and sellers. In

the case of an offshore MTdc network, the sellers will be represented by the offshore wind farms whereas

the buyers will be constituted by the onshore networks, somehow interconnected via the MTdc network.

The market participants submit their bids to the pool, stating the amount of power that they are willing

to trade. The sellers in a pool market compete for the right to supply energy to the grid, not to specific

customers. If an electricity seller would bid too high, it might not be included for dispatch. On the other

hand, buyers compete for purchasing power, therefore, if their bids would be too low, they may be left

out of the market. This market scheme is shown in Figure A.1.

POOL MARKET

Bids MCP

Bids MCP

Onshore Stations

Offshore Wind Farms

Figure A.1: Diagram of a pool market applied to a future offshore MTdc network.

ISO

A competitive electricity market needs an independent operational grid control. Such task cannot be

guaranteed without establishing an ISO [76]. The ISO administers transmission tariffs, maintains the

system security, coordinates maintenance scheduling, and has a decisive role in coordinating the long-term

planning. It functions independently of any market participants, such as transmission owners, generators,

distribution companies and end-users. Moreover, it should also provide non-discriminatory open access

to all transmission system users. The ISO has the authority to commit and dispatch some or all the

44

system resources; and to shed loads if necessary to maintain system security, e.g. remove transmission

violations or balance supply and demand.

By running an auction for the electricity trade, the ISO ensures a competitive marketplace. The spot

price, also designated market clearing price (MCP), is obtained based on the lowest bid, which is the bid

of the last demand entity which was admitted to enter the market. In order to obtain the spot price, the

ISO makes use of an optimal power flow (OPF) dispatch model, taking into consideration the constraints

of the network. Market participants must provide extensive data, such as cost data for every generator

and daily demand for every consumer or load. Utilizing these extensive data, the ISO obtains the unit

commitment and the correspondent dispatch, which maximizes the social welfare and sets transmission

congestion prices. The social welfare can be mathematically described as:

Social welfare =(CT

DPD − CTS PS

)(A.1)

where CD and CS are, respectively, demand and supply bids vectors in e/MWh, whereas PD and PS

represent the demand and supply power bids vectors in MW.

A.2.2 Transmission Loss Allocation

In a practical implementation of a MTdc network, the question of whom should be the responsible for the

system losses payment naturally arises [78]. Since system losses can be typically estimated as five to ten

percent of the total generation, if capitalized over the course of one year, the value of system losses can be

in the range of tenths of millions of euros [36]. Consequently, a fair power losses allocation among loads

and generators has an important impact on their benefits and needs to be included in the optimization

algorithm.

Although transmission losses are nonlinear functions of the line flows(Plosses ∝ I2line

), several studies

have tried to determine which amount of a line power flow is the responsibility of a given generator or

demand. In [79] and [80] new methods for tracing the flow of electricity in meshed electrical networks

are proposed.

Due to the lack of a perfect algorithm for tracing the power flow, its acceptance will mainly depend on the

perceived fairness as seen by each one of the pool participants [78]. In this work, the chosen transmission

losses allocation procedure is a circuit-based strategy known as Z-bus loss allocation method [36]. A brief

description of the procedure is given next.

Z-bus Loss Allocation Method

The classic Z-bus allocation method used mostly on AC networks [36], is here modified for its application

in MTdc networks. The Z-bus method emphasizes the currents, instead of the active power flow, during

the losses allocation. It also takes into consideration the network topology. Therefore, generators or

loads which are located in distant regions from the network center of gravity tend to proportionally be

allocated higher losses.

The Z-bus allocation method makes use of a previously solved power flow to systematically distributes the

power transmission losses, Plosses, amongst all the nodes present in the MTdc network. This procedure

can be mathematically expressed as:

Plosses =n∑

k=1

Lk (A.2)

45

where Lk stands for the power losses fraction allocated to the k-th DC node, and n is the total number

of nodes inside the MTdc network.

In this way, the responsibility of paying for its losses share is assigned to each individual market partici-

pant. The extra cost due to the loss allocation must then be deducted from the revenue of the generating

entities and added to the load payments in order to maintain the pool revenue null.

If a certain DC node has both generation and demand, its allocated loss component, Lk, may be divided

among the different players in a pro rata manner [36]. On the contrary, if a specific DC node has neither

load nor generation, its loss allocation must be zero.

In order to calculate the allocation fractions, Lk, the Z-bus method makes use of the network impedance

matrix, Z. The latter can be further separated into two matrixes: the resistance matrix, R, and the

reactance matrix, X. However, for the power losses calculation only the resistance matrix is of concern

[36]. Hence, the power losses can be achieved through:

Plosses =

n∑k=1

Ik

⎛⎝ n∑

j=1

RkjIj

⎞⎠ (A.3)

where Ik is the DC current injection at the node k, Rkj is the DC cable resistance between the nodes k

and j, and Ij is the DC current injection at node j.

Finally, the loss allocation terms, Lk, can be expressed as:

Lk = Ik

⎛⎝ n∑

j=1

RkjIj

⎞⎠ (A.4)

As can be seen from (A.4), the loss component Lk, is equivalent to a weighted function where the current

in node k, Ik, serves as weight to the sum of the currents in all the network nodes. With this approach,

no special assumptions or approximations are necessary to derive the loss allocation terms.

A.3 MTdc Network Control Principles

The control of point-to-point HVDC transmission systems is typically arranged so that one terminal

controls the dc network voltage, whereas the other operates in current or power regulation mode. This

control philosophy – of having only one converter controlling the direct voltage – can be extended to

MTdc networks. Mathematically, disregarding transmission losses and considering the magnitude of the

transmitted powers, this control strategy for MTdc networks can be translated as:

PmaxV dc ≥

N∑i=1

P iIdc(t) (A.5)

where PmaxV dc is the maximum rating of the VSC controlling the direct voltage (N-th node) and P i

Idc(t) is

the power production at the i-th VSC terminal operating in current regulation mode.

As it can be seen from (A.5), as the MTdc network grows, it will be increasingly difficult to assure power

balance by having only one terminal responsible for voltage regulation. Hence, for large MTdc networks,

controlling the voltage at a single terminal is not desirable. In addition, if an outage would affect the

only direct voltage controlling station, direct voltage control would be lost until somehow transferred

to another node in the network. Such scenario is, however, also not desirable. The fast nature of dc

46

phenomena could trigger protection equipment, within only a few cycles of the ac network [48].

For its successful development and operation, MTdc networks will require a control strategy capable of

sharing the dc system voltage control amongst more than one network node. A more suitable control

strategy for large multi-terminal dc networks, would be then to assign each dc-voltage-controlling VSC

terminal with a specific voltage set-point.

In this way, any predefined load flow scenario can be achieved while no single converter is left alone

with the responsibility of balancing the power inside the transmission system, i.e. the control of the dc

system voltage is distributed between several nodes inside the MTdc network. This method of controlling

VSC-MTdc networks is called distributed voltage control (DVC) [25] and it is depicted in Fig. A.2.

Distributed DC Load Flow

Algortihm

OPF Algorithm

No

Generation at OWFs

Voltage setpoints

to GS-VSCs

Optimum?

MTDC Secure?

Check for N-1 Security

No

Yes

Yes

DVC

SCADAsignal

SCADAsignal

Figure A.2: Flow chart of the distributed voltage control method.

A.3.1 Information flow

The information flow inside the system is as follows: firstly, according to the wind speeds, a certain power

is produced at the OWFs. The information about the generation is given to the MOOA, which, knowing

the MTdc network topology, will obtain optimal power flows according to the several constraints. Finally,

according to the DVC method, the dc system voltage reference values are transmitted to the onshore

VSC stations to generate the optimal power flow inside the MTdc grid. In Fig. A.3 it is shown how the

entire system is interconnected.

MOOA

MTDCnetwork

OnshoreVSCs

ACgrids

OffshoreVSCsOWFsWind

*1DCV

*DCjVWFi

DCP

1WFDCP

... ...

Figure A.3: Information flow between the OWFs, the GA and the onshore VSC stations.

It is important to point out that the MOOA is provided with the OWFs averaged power production over

a certain period. In this way, it is expected that the actual and the desired load flows will somehow

differ. However, since several OWFs will be connected to the MTdc network, it is anticipated that the

variability in the total power production will be smoothed out, due to the effect of integrating the wind

energy production over a large area [81].

47

A.3.2 Social Welfare

Social welfare is widely used as an efficiency indicator to evaluate liberalized power markets and may be

mathematically described as:

SW =

qλ∫0

(B − S) dq (A.6)

where B and S are, respectively, the buyers and sellers aggregated bid curves as a function of the power

quantity, and qλ is the traded power quantity.

The aggregated curves for sellers and buyers for two areas, 1 and 2, may be described as [82]:

S1,2 = α1,2 + β1,2 · q ; B1,2 = γ1,2 − φ1,2 · q (A.7)

where β and φ are the curves slopes and α, β, γ and φ > 0.

Without trade (Fig. A.4 (a)), the traded quantities and MCPs (λ) of both areas are:

qa1,2 =γ1,2 − α1,2

β1,2 + φ1,2; λa

1,2 =α1,2φ1,2 + β1,2γ1,2

β1,2 + φ1,2(A.8)

(b)

(c)

(a)Price

PriceBuyers WelfareSellers Welfare

S1

B1

S2

B2

Price

1aq 1

bq

1a1b

2b

2a

1a

2a

1aq

2bq 2

aq

2aq

1aq 1

cq 2aq2

cq

q q

qqq

qqq

1a

2a

1 2c c

2 1c c

1 2bq

1 2cq

Figure A.4: Social welfare of two areas, 1 and 2, for three distinctive cases: (a) no trade; (b) limitedtrade; (c) unlimited trade.

In order to maximize social welfare, the ISO establishes power flows from areas with lower MCPs to

areas with higher MCPs (see Figs. A.4 (b) and (c)) . In such way, costly and highly-pollutant units are

48

not dispatched. Moreover, cheaper energy suppliers will see their welfare increased, enhancing market

competition.

The social welfare might not be fully maximized if the transfer capacity between the areas is limited

(Fig. A.4 (b)). Mathematically this can be expressed as:

SWnt︸︷︷︸no

trade

≤ SWltc︸︷︷︸limitedtransfercapacity

≤ SWutc︸︷︷︸unlimitedtransfercapacity

(A.9)

49

Appendix B

Multi-Objective Optimization

Algorithms

B.1 Introduction

In many real-world situations there may be several objectives which must be optimized simultaneously

in order to solve certain problems. The main difficulty when considering multi-objective optimization

problems is that, in such cases, there is no definition of optimum solution. Therefore, it is difficult to

compare solutions and decide which ones are better. The quality of a solution is subjective and depends

on the needs of the designer or decision maker [83].

Although multi-objective genetic optimization algorithms do not guarantee the optimal trade-off between

the criteria taken into consideration, they provide good solutions within acceptable computation times

[84]. These algorithms mimic the behavior of natural evolution presented by Darwin in 1859 [85] and

treat solution candidates as individuals of a population that compete in a virtual environment. Each

solution is also designated as a chromosome, composed by genes.

The mathematical foundations for multi-objective optimization were laid down by Vilfredo Pareto more

than one hundred years ago [86]. Pareto optimality became an important notion in economics, game

theory, engineering, and social sciences. It defines the frontier of solutions that can be reached by

trading-off conflicting objectives in an optimal manner. From the Optimal Pareto Front (OPF) the

decision maker may choose a solution that best suits the problem considered [87].

MOOAs are an important instrument since, among other characteristics, they do not require any infor-

mation about the targets to be optimized besides their own evaluation. Therefore, the goal functions

are not restrict with respect to differentiation, continuity or any other particular behavior in the search

space of the problem. MOOAs are able to work simultaneously with real and integer representation of

variables [41]. A very important feature for the success of such algorithms is the fact that they treat the

multi-objective problem with non-commensurable objectives, i.e. no a priori information is needed [42].

Over the past two decades, several MOOAs have been suggest. The most famous and used algorithms

are, among others, the NSGA-II [88], Pareto-archived evolution strategy (PAES) [89] and the strength-

Pareto EA (SPEA) [90]. A comparison between the standard MOOAs is elaborated in [91]. Different

approaches from the genetic algorithms have been introduced in the literature, being the Particle Swarm

Optimization one of the most famous algorithms [92]. This algorithm, differently from the GAs does not

rely only in the information of the present population to create new solutions, since it keeps track of the

best result of each solution. Its multi-objective optimization form was presented in [93].

50

The purpose of a MOOA is to obtain the optimal Pareto-front (OPF) of a problem. This front gives

the optimal trade-off between the considered fitness functions. A MOOA can be generically described by

[91]:

MOOA

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Optimize F

gi(x) ≤ 0 ; 1 ≤ i ≤ J

hi(x) = 0 ; J + 1 ≤ i ≤ J +K

LBi ≤ xi ≤ UBi ; 1 ≤ i ≤ m

where F consists of n objective functions fi, each representing one criterion to be optimized; gi(x) are

the inequalities of the problem; hi(x) are the equalities that the algorithm solutions need to respect and

LBi and UBi are the variables boundaries.

Optimal Pareto Front

Being X the solution space of the problem, an element x∗ ∈ X is Pareto optimal (and hence, part of

the optimal set X∗) if there is no other element in X that dominates x∗. Mathematically, this can be

expressed as:

x∗ ∈ X∗ ⇔ ∀x /∈ X∗ : x∗ � x (B.1)

In terms of Pareto optimization, X∗ is called the OPF.

An element x1 dominates x2 (x1 � x2) if, x1 is better than x2 in at least one objective function and not

worse with respect to all other objectives [94]:

x1 � x2 ⇔∀i : 1 ≤ i ≤ n ⇒ wifi(x1) ≤ wifi(x2) ∧∃j : 1 ≤ j ≤ n : wjfj(x1) < wjfj(x2)

(B.2)

where wi is 1 if fi is to be minimized and −1 if a maximization of fi is desired.

In Fig. B.1 it is possible to see an example of an OPF if the minimization of two functions is considered.

1 1( ,..., )nf x x

21

(,..

.,)

mfx

x

Optimal Pareto Front (X*)

SolutionSpace (X)

Figure B.1: Pareto front of a Multi-Objective Minimization Problem.

51

B.2 Proposed Multi-Objective Optimization Algorithm

The proposed multi-objective optimization algorithm resembles greatly the already referred standard

MOOAs. However, new ranking and sort strategies for the population are introduced. Moreover, a new

genetic operator, so-called regeneration, is also introduced. In Fig. B.2 it is shown a high-level flow chart

of the proposed MOOA while in (B.3) the pseudo-code is presented.

6. Terminate?

1. Population Initialisation

2. Finess Evaluation

3. RankPopulation

4. SortPopulation

5. ArchiveUpdate 7. Selection

8. Crossover

9. Mutation

10. Regeneration

11. Output Archive

Yes

No

Figure B.2: Flow chart of MOOA implemented.

52

Archive actual size = 0;

N generations = 1;

Population = initialization;

while (1)

Population( : , Aug F it1 · · ·Fitm) = fitness(Population);

Population( : , Aug F it1 · · ·Fitm) = fitness(Population);

Population( : , Rank) = rank(Population);

Population = sort(Population);

/ ∗Archive update ∗ /Archive = cat(Archive, Population( : , Rank == 1));

Archive( : , Rank) = rank(Archive);

Archive = Archive( : , Rank == 1);

Update Archive actual size;

if Archive actual size > Archive max size

Archive = crowding competition(Archive);

end

increment N generations;

if N generation > N generations max

break;

end

Mating Pool = selection(Archive, Population);

Population = crossover(Mating Pool);

Population = mutation(Mating Pool);

Population = regeneration(Mating Pool);

end

(B.3)

In the next subsections each step of the flow chart will be introduced and described.

B.2.1 Population Initialization

There are two ways to initialize the population, P(t=0), of a MOOA [83]. One of the options is to

randomly initialize all the variables of the chromosomes. However, it has to be assure that the random

values respect the boundaries of the respective genes. The initialization can be performed according to

the following formula:

genei = α× (UPi − LBi) + LBi (B.4)

where LPi and UPi represent the boundary limits of the i− th gene, and α is a random number between

0 and 1.

The second approach is to choose the initial variable values for the genes of the chromosomes. This

53

strategy may be mainly used if there is some a priori knowledge about the problem. In Fig. B.3 it

is shown the population initialization procedure implemented in the algorithm. Independently of the

variables initialization, the value 1 is attributed in the origin field of the chromosomes.

PopulationInitialization

User specified values

Bonded random values

Origin = 1

Figure B.3: Population initialization process.

B.2.2 Fitness evaluation

In this step the fitness values of the solutions are determined. In this way, the chromosomes are tested

in order to see how well they perform in the optimization goals that are being considered.

Most of the real world problems have constrains over the variables problems. This way, genetic algorithms

must have a way to handle constrains and, therefore, present feasible solutions at the end, i.e. solutions

that respect all the constrains and boundaries of the considered problem. In Appendix A a description

about constraint handling techniques is presented.

The penalty term is obtained according to the following formula:

pen(x) = φα × λ (B.5)

where φ =[1 · · · J +K

]and λ is the sorted vector with the constraint violations.

The terms λi(x) are calculated as:

λi(x) =

⎧⎪⎨⎪⎩

0 if x is feasible

|gi(x)| if x is infeasible ∧ 1 ≤ i ≤ J

|hi(x)| if x is infeasible ∧ J + 1 ≤ i ≤ J +K

B.2.3 Constraint Handling

There are several constrain handling strategies in the literature (for surveys regarding constraint handling

techniques in a multi-objective genetic algorithm see [95, 96]). In this report the two most common

strategies to handle constrains will be present and described.

Before presenting the strategies it is shown, once again, the constrains that a genetic algorithm can be

submitted to:

Constraints

⎧⎪⎨⎪⎩

gi(x) ≤ 0 ; 1 ≤ i ≤ J

hi(x) = 0 ; J + 1 ≤ i ≤ K

LBi ≤ xi ≤ UBi ; 1 ≤ i ≤ n

where gi(x) are the inequalities of the problem; hi(x) are the equalities that the solutions of the algorithm

need to respect and LBi and UBi are the variables boundaries.

54

In general, the solution space can be divided in two distinctive parts: the feasible and the infeasible areas.

This distinction in areas is depicted in Fig. B.4.

1 1( ,..., )nf x x

21

(,..

.,)

mfx

xOptimal Pareto

Front (X*)

Feasiblespace

SolutionSpace (X)

Figure B.4: Pareto front of a Multi-Objective Minimization Problem.

Preservation of feasible solutions

One of the simplest ways to assure that all the solutions are feasible at all times is to rejected offspring that

are infeasible. With this strategy, all offspring after their generation are tested in order to verify if they

respect all the constrains of the problem. Another approach is to try and change an infeasible solution

into a feasible one via a deterministic procedure - repairing strategy. Although these strategies have the

advantage of never generating infeasible solutions, in a highly constrained problem, infeasible solutions

may have an important role. The movement of solutions through infeasible areas tend to increase the

velocity of the optimization and to produce better solutions [97].

Penalty term

Penalty techniques are most probably the most common strategy to handle constrains in a genetic algo-

rithm [40]. With this approach, a constrained problem becomes unconstrained and the infeasible solutions

are penalized, having, this way, worse fitness values. The penalty term that is added to infeasible solu-

tions traduces their infeasibility. Therefore, if the penalty term is small that means that the solution is

infeasible but it is close to be the feasible area. On the other hand, if the penalty value is high, that

corresponds to a chromosome that is far away from the desired feasible region.

In general, there are two distinctive ways to obtain the fitness value of a chromosome with the penalty

term: addition or multiplication of the penalty with the fitness value. If the addiction approach is chosen,

the augmented value (total evaluation of the score of a chromosome) of the fitness of a chromosome is

obtained through:

Aug Fit = Fit + Penalty (B.6)

Since in a minimization problem a solution with a lower fitness value is a better solution, the penalty

term has to be derived as:

{pen(x) = 0; if x is feasible

pen(x) > 0; if x is infeasible(B.7)

There are several options to generate the penalty term. The strategy implement to obtain the penalty

value has to be carefully chosen since the guide has to be effectively guided towards a promising area

55

of the solution space. The penalty function can be classified in two groups: constant and dynamic

penalties. The approach that uses the constant penalty value is known to have a worse performance than

the strategies that implement a variable penalty scheme and, therefore, a higher attention is being given

to the last one by the researchers [98].

The dynamic penalty approaches can be separated in two groups: the variable penalty ratio and the

penalty value for the violation of the constraints [99]. The last one applies a penalty term that varies

with the “amount” of violation of the constraints by a chromosome. The variable penalty ratio can be

implemented according to the degree of the violation of the constraints or according to the generation

number of the genetic algorithm.

The dynamic penalty approaches can be separated in two groups. In the first group the penalty term is

determined by the degree of the violation of the constrains. In this way, a solution that violates severely

the constraints of the problem is given a higher penalty (if a minimization problem is considered). For

the second group of dynamic penalties, the generation number of the genetic algorithm is the variable

that controls the penalties attributed to the infeasible solutions of the population.

B.2.4 Population Ranking

After the fitness evaluation, the population is ranked. In Table B.1 the possible ranks are introduced and

distinguished, while in (B.8) the pseudo-code for the rank attributions is laid down.

Table B.1: Possible rank values and their description.

Rank Description

1 Non dominated and feasible solution2 Non dominated, feasible and repeated solution3 Non dominated and infeasible solution4 Dominated Solution

Pop(:, Rank) = 4;

for each solution of the Pop

if Pop(i, Rank) == 4

Index(1 : Popsize) = true;

Index2(1 : Popsize) = true;

for each optimization goal fi

Index(Index) = Pop(Index, fi) ≤ Pop(i, fi);

Index2(Index2) = Pop(Index2, fi) == Pop(i, fi);

end

if sum(Index) == sum(Index2)

if Pop(Index) are feasible

Pop(i, Index) = 2;

Pop(i, Rank) = 1;

else

Pop(i, Rank) = 3;

end

end

end

(B.8)

56

B.2.5 Population Sorting

The next step of the MOOA is the population sorting. The first step is to sort the population through the

rank value. Thereafter, each rank has its own sorting, being the maximum constraint violation the second

sort parameter. In this way, each rank gives priority to solutions that disrespect with lower severity the

problem constraints. The attempt of this principle is to reach feasible solutions. In Table B.2 it is shown

the aspect of the population after being sorted.

Table B.2: Description of the population sorting.

Rank Sorting

1 Lowest to Highest Max constraint2 Lowest to Highest Max constraint3 Lowest to Highest Max constraint4 Lowest to Highest Max constraint

B.2.6 Archive Update

The Archive represents what in a single-objective genetic algorithm is known as elitism, which is a

strategy to ensure that the best chromosomes found so far are not lost due to randomized operators such

as crossover and mutation. In the multi-objective context, the elitism is done through the preservation

of the non-dominated solutions found so far [91]. There are two approaches to keep stored the non-

dominated solutions: to keep all found so far or the pruning strategy where only part of the solutions

are kept [100, 101]. If the later is chose it is important to assure that it is achieved a balancing between

proximity and diversity of the OPF [102].

After sorting the population all the solutions that were assigned with Rank 1 are merged with the present

Archive. After an archive ranking is performed, and in this way, the solutions that are dominated will

be erased from the Archive. If the number of archived solutions is higher than the archive maximum

size, a crowding competition is performed in order to choose with chromosomes to keep. In Fig. B.5 the

archiving process is depicted.

Archive

Population Solutions with Rank=1

Crowding Competition

Yes

NoNumber of solutions > Elite size?

Archive

Merge Rank

Figure B.5: Archiving process of the non dominated solutions.

It is important to refer that a crowding competition will always be performed if the actual archive size is

higher than the number of solutions that are sent from the Archive to the mating pool (elite size). In this

way, the chromosomes sent to participate in the creation of new solutions are the ones with the biggest

cuboids. Such feature enhances the search of Pareto front zones that were the least explored.

Crowding competition

The crowding distance, originally from the NSGA-II algorithm [88], is used to estimate the solutions

density surrounding a particular solution i. To do so, the average distance of the two solutions that are

57

next to i is taken for all the objectives. The average distance, di , is an estimation of the perimeter of

the cuboid formed by the closest neighbors. In Fig. B.6 it is shown the cuboid for the solution i.

2f

1f1i 1

1i 1

i

Cuboid

Non dominated solution

Dominated solution

Figure B.6: Crowding competition among the non dominated solutions present in the archive.

In order to calculate the crowding distance of the archived solutions the following steps are used:

1. Obtain the number of solutions in the Archive: l = size(Archive).

2. For all the solutions assign d = 0.

3. For each fitness function, m = 1, 2, ...,M , find the sorted indices vector in worse order of fitness

value, Im = sort(fm), and assign dIm1

= dIml

= ∞.

4. For the other chromosomes, i = 2, ..., l − 1, the crowding distance is calculated as:

dImi

= dImi+

fImi+1

m − fImi−1

m

fmaxm − fmin

m

(B.9)

The elements Im1 and Iml represent the best and the worst fitness values for the fitness function m,

respectively. The last term of the equation shown above represents the difference in fitness function

values between the two neighbors of the solution Imi . The parameters fminm and fmax

m represent the

maximum and the minimum value of fitness for the m-th function.

B.2.7 Ending criteria

In a MOOA the presence of a termination criteria for the algorithm is needed. The most common

terminating conditions found in the literature are [103]:

• Maximum number of generations (iterations);

• Maximum time;

• The fitness value of at least one of the chromosomes is lower than a predefined minimum;

• Maximum number of fitness function evaluations;

• etc..

The conditions above can be used solo or in combinations. The choice for the terminating condition is

somehow problem related and also depends of the desire of the user. In the present work the maximum

number of iterations was used as ending criterion.

58

B.2.8 Selection

The selection operator resembles the process of natural selection and the survival of the fittest of Dar-

winian evolution theory. In this process, an intermediate population, called mating pool. The selected

chromosomes in the mating pool, also designated as parents, will take part in the subsequent genetic

operations [83].

In this step of the algorithm the mating pool is populated with solutions coming from the population

and from the archive. As it is possible to observe in Fig. B.7 the solutions coming from the Population

participate in a 4-th tournament selection [104]. The winner of each tournament goes to the mating pool.

Mating PoolArchive

Population Tournament Selection

Elite size

Figure B.7: Selection of solutions from the current population and archive to population the mating pool.

If the Archive in not empty, the archived solutions with the biggest cuboids are sent to the mating pool.

The ratio between solutions coming from the population and the archive is user specified. If the number

of chromosomes in the archive is not enough to cover its share part in the mating pool, extra solutions are

picked from the Population. However, the number of solutions coming from the Archive to the Mating

Pool is never higher than the user specified value (elite size).

B.2.9 Crossover

In the nature, although it may be much more complicated, crossover basically occurs as follows: chromo-

somes of both parents are randomly divided from the same gene positions into a number of segments and

the corresponding segments are exchanged and copied to the chromosome of the newly created offspring.

Therefore, the offspring inherit traits from both parents.

The idea behind crossover is that the new chromosomes may be better than both the parents if they

take the best characteristics from each of the parents. Crossover occurs during evolution according to a

user-definable crossover probability. Some popular crossover methods, when a binary representation for

the chromosome genes is used, are single-point, two point and uniform crossover [104].

In the single-point crossover a crossover point is randomly chosen and then the two parents, also ran-

domly chosen from the population, are interchanged at this point in order to produce two new offspring.

Variations of this technique such as two-point (can also be generalized to n-point) were already presented

[83].

In real-valued approaches, one of the major challenges is how to use the real-valued genetic information of

both parents to create offspring. However, the techniques to perform crossover for binary representations

can also be used for this case, most of the time that does not occur. This is due to the fact that there is

a large diversity of blending operators, e.g. the BLX-α technique which is able to create offspring out of

the gap formed by the parents [40] and the quadratic crossover where the information of three solutions

is used to produce one offspring [105].

The crossover technique used is the heuristic crossover [106]. This approach tries to create better off-

59

spring via extrapolation. After the random selection of two parents from the mating pool, their genetic

information is blended according to the following formula:

offpringi = α× (x2i − x1i ) + x1i (B.10)

where α is a random value between 0 and 2 and x2i having a better rank than x1

i . If both parents have

the same rank they are randomly assigned.

With this approach it is expected that the offspring will have better fitness values than their progenitors

and in this way tend to the true Pareto front of the problem. Although most of the offspring produced

might represent worse solutions since only two parents are considered at a time, after a large number

of tryouts the predictions will hopefully lead the population towards the Pareto front. In Fig. B.8 it is

depicted the possible result of the heuristic crossover. Differently from other crossover strategies, with

this method only one child is created from the matting of two parents.

Gene valueParent 2

Possible value for the gene value of the child

Gene valueParent 1

Figure B.8: Possible effects of the crossover operator on the genes of the new solution.

B.2.10 Mutation

The mutation in a genetic algorithm is also a process that was inspired in nature. During the transfer of

genetic contents from parents to offspring, something may go wrong, and therefore, random changes may

occur in the new chromosome. As a result, the quality of the individual might degrade considerably. On

the other hand, it can occur that the quality of the new chromosome is higher. Moreover, this process

helps to keep diversity in population, and therefore, it helps the algorithm to abstain from getting stuck

onto a local minimum of the problem search space. The concept of mutation is employed by introducing

a mutation probability parameter. Before inserting an individual to the next generation, random changes

on its chromosome is performed according to this probability. This parameter is most of the times set

low in order to avoid the algorithm of becoming a primitive random search.

In binary chromosome representation, the random change that may occur in the new chromosome is the

inversion of the value of a random gene position from 0 to 1 (and vice-versa). On the other hand, if a

real-valued representation is used, one of the chromosome’s genes may be altered and the new value is

a randomly generated real number between the upper and lower bounds of that specific gene. Another

option is the non-uniform mutation technique which enhances diversity in the first generations of the

population. However, with the increase of the number of generations, the mutation process starts to

converge in a way that there is a high probability of creating a mutated chromosome close to the initial

value focusing in this way the search [40]. Other widely known strategy is to perform mutation in a

real-valued chromosome by adding a normally distributed random value [107]. The desired effect of the

mutation process can be visualized in Fig. B.9 [108].

In the algorithm a point mutation was employed with three different possibilities of mutation with the

same probability of occurrence. The mutated gene might receive the value of its upper or lower bound

and a random value between its boundaries. Such mutations are described as:

60

Global minimum

Local minimum

Fitness value

Search space

Figure B.9: Desired effect of the mutation operator on the genes of the new solution.

genei

⎧⎪⎨⎪⎩

α(UBi − LBi) + LBi

LBi

UBi

where α is a random value between 0 and 1.

The first point random mutation process is not only completely independent of the gene’s value of the

parent but is also able to create a value from the entire interval set by the boundaries of the particular

variable being mutated.

B.2.11 Regeneration

The genetic operator regeneration can be seen as a severe mutation since all the genes of the new

chromosome are altered when compared to the solution from where it was originated. However, it can

also be seen as a crossover with only one parent that alters, regenerates, itself in order to, hopefully,

create a new better solution.

There are two possibilities for the regeneration operator. In the first approach, each gene of a randomly

chosen solution, x1 , from the mating pool is added of itself times a random value that lies in the interval

[−0.05, 0.05]. In this way, the genes with higher values will suffer a higher increase when compared to

the ones with lower gene values. Such approach is benefic in the cases where the distance between the

genes is the key to reach better areas of the search space.

offpring = x1 + α× x1 (B.11)

In the second approach, once again one chromosome is randomly chosen from the mating pool. However,

differently from the first strategy, now a random variable of the gene is chosen and multiplied by α (with

α ∈[−0.05, 0.05

]). After, each gene is summed with this value. In this way, all the genes are

modified with the same amount and, therefore, the difference between the genes is kept constant.

offpring = x1 + α× x1(β) (B.12)

This strategy is benefic when the difference between the genes is already optimal but it is still necessary

to change the absolute values.

It is important to refer one feature of this genetic operator. If the population converges to only one point

the effect of the crossover operator will disappear. In such cases the algorithms rely in the mutation

operator to introduce diversity in the population. The regeneration operator does not share this feature

61

[%]

GeneValue

5-5

ExpansionContraction

0

Figure B.10: Possible effects of the renegeration operator on the genes of the new solution.

with the crossover operator since it is able to change one chromosome into another one without information

of any other solution.

A study on the effectiveness of the regeneration operator was conducted and may be found in Appendix

C.

B.2.12 Chromosome composition

The first n entries of each chromosome are the variables of the problem. They are real-valued and

lower and upper bounded. For each optimization goal there are two entries in the solutions. One is the

augmented fitness, which is composed by the fitness value summed with the penalty term, while the other

entry corresponds to the fitness value itself. The last three entries correspond to the absolute value of

the maximum constraint violation, the rank of the solution and its origin. In Fig. B.11 the encoding of

each solution is shown.

X [var1 var Aug_Fit 1 Aug_Fit Fit 1 Fit Rank Max_constraint Origin]n m mvar Aug_Fit 1 Aug_Fit Fit 1 Fit RaFit 1 Fit Au Fit 1 Au Fit nk Maxk Aug Fit 1 Aug Fit Fit 1 Fit Aug Fit 1 Aug Fit Fit 1 Fit Aug Fit 1 Aug Fit

LB variables UBn nFitness + penalty

termFitness values Highest

constraint violation1 Rank 4

1 Origin 4

Figure B.11: Composition of the chromosomes.

In Table B.3 it is explained the possibilities for the last gene of each chromosome.

Table B.3: Description of the field Origin.

Value Origin

1 Initial Population2 Crossover3 Mutation4 Regeneration

B.2.13 Population composition

After the creation of solutions via the genetic operators, the population is once again tested for the

fitness value of its chromosomes. The amount of solutions coming from each genetic operator, crossover,

mutation and regeneration are user specified. The population after the genetic operators is illustrated in

the next picture.

62

Crossover

Mutation

Regeneration

Population

Figure B.12: Composition of the population after the genetic operators.

In Appendix B, the presented MOOA is tested via several benchmark functions.

63

Appendix C

Benchmark Results

In this appendix the performance of the developed MOOA is tested through well-known benchmark

functions for multi-objective optimization algorithms.

Before introducing the test function it is important to present how the MOOA was set. For all the

benchmarks the population size was set to 100, while the Archive size was defined to 250. The contribution

of each genetic operator was also kept constant throughout the benchmark session and defined as shown

in Fig. C.1. The termination criteria used was the number of fitness evaluations allowed by each test

function. A 4− th tournament selection was implement to select solutions from the Population to enter

the Mating Pool. The size of the Mating Pool was set to 20 and the maximum contribution of the Archive

to the Mating Pool was determined to be 10.

Crossover

Mutation

Regeneration

Population

80%

10%

10%

Figure C.1: Contribution of each genetic operator to the population with new solutions.

Test function 1

The test function 1 is a problem proposed by Kita [109]:

Minimize F = (−f1(x, y), −f2(x, y)), where

f1(x, y) = −x2 + y, f2(x, y) =1

2x+ y + 1

subject to:

• 1

6x+ y − 13

2≤ 0,

1

2x+ y − 15

2≤ 0, 5x+ y − 30 ≤ 0

64

• 0 ≤ x, y ≤ 7

• Maximum number of fitness evaluations: 5000

Test function 2

The test function 2 can be found in [93]:

Minimize F = (f1(x, y), f2(x, y)), where

f1(→x) =

n−1∑i=1

(−10 exp

(−0.2

√x2i + x2

i+1

)), f2(

→x) =

n∑i=1

(|xi|0.8 + 5 sin (xi)

3)

subject to:

• −5 ≤ x1, x2, x3 ≤ 5


Test function 3

The test function 3 was proposed by K. Deb [110]:

Minimize F = (f1(x1, x2), f2(x1, x2)), where

f1(→x) = x1, f2(

→x) = g(

→x) · h(→x)

where

g(→x) = 11 + x2

2 − 10 cos(2πx2)

h(→x) =

⎧⎪⎨⎪⎩ 1−

√f1(

→x)

g(→x)

, if f1(→x) ≤ g(

→x)

0, otherwise

subject to:

• 0 ≤ x1 ≤ 1, −30 ≤ x2 ≤ 30


Test function 4

The test function 4 was also proposed by K. Deb [110]:

Minimize F = (f1(x1, x2), f2(x1, x2)), where

f1(→x) = x1, f2(

→x) =

g(x2)

f1

where

g(x2) = 2.0− exp

{−(x2 − 0.2

0.004

)2}

− 0.8 exp

{−(x2 − 0.6

0.4

)2}

subject to:

65

• 0.1 ≤ x1 ≤ 1, 0.1 ≤ x2 ≤ 1


Test function Kita

This test function was proposed by Kita [109]:

Minimize F = (−f1(x1, x2), −f2(x1, x2)), where

f1(x1, x2) = x2 − x21, f2(x1, x2) = 0.5x1 + x2 + 1

subject to:

• x1, x2 ≥ 0

• −(6.5− x1

6− x2

)≤ 0

• −(7.5− x1

2− x2

)≤ 0

• − (30− 5x1 − x2) ≤ 0


Test function OSY

This test function may be found in [111]:

Minimize F =(f1(

→x), f2(

→x)), where

f1(→x) = −25

[2∑

i=1

(xi − 2)2+ (x3 − 1)

2+ (x4 − 4)

2+ (x5 − 1)

2

], f2(

→x) =

6∑i=1

x2i

subject to:

• 0 ≤ x1, x2, x6 ≤ 10, 1 ≤ x3, x5 ≤ 5, 0 ≤ x4 ≤ 6

• − (x1 + x2 − 2) ≤ 0

• − (6− x1 − x2) ≤ 0

• − (2 + x1 − x2) ≤ 0

• − (2− x1 + 3x2) ≤ 0

• −(4− (x3 − 3)

2 − x4

)≤ 0

• −((x5 − 3)

2+ x6 − 4

)≤ 0


66

Test function SRN

This test function may be found in [111]:

Minimize F =(f1(

→x), f2(

→x)), where

f1(→x) = 2 + (x1 − 2)

2+ (x2 − 1)

2, f2(

→x) = 9x1 − (x2 − 1)

2

subject to:

• −20 ≤ x1, x2 ≤ 20

• − (225− x21 − x2

2

) ≤ 0

• − (10− x1 + 3x2) ≤ 0


Test function TNK

This test function was first proposed in [112]:

Minimize F =(f1(

→x), f2(

→x)), where

f1(x) = x1, f2(x) = x2

subject to:

• 0 ≤ x1, x2 ≤ π

• −(x21 + x2

2 − 1− 0.1 cos

(16 arctan

(x1

x2

)))≤ 0

• −(0.5−

2∑i=1

(xi − 0.5)

2)

≤ 0


Test function DTZL9

The test function DTLZ9 can be found in [113]:

Minimize F =(f1(

→x), f2(

→x), f3(

→x),), where

f1(→x) =

1

10

10∑i=1

xi, f2(→x) =

1

10

20∑i=10

xi, f3(→x) =

1

10

30∑i=20

xi

subject to:

• 0 ≤ →x ≤ 1

• −(f23 (

→x) + f2

1 (→x)− 1

)≤ 0

• −(f23 (

→x) + f2

2 (→x)− 1

)≤ 0


67

The Pareto fronts found by the MOOA, for the test cases during a normal run described above, are shown

in Fig. C.2. It is possible to observe that the algorithm was able to found the true Pareto front of each

problem and, moreover, the entire front was achieved.

-20 -19 -18 -17 -16 -15 -14 -13-12

-10

-8

-6

-4

-2

0

2Test function 2

f1

f2

-6 -4 -2 0 2 4 6 87.4

7.6

7.8

8

8.2

8.4

8.6

8.8Test function 1

f1

f2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4Test function 3

f1

f2

-4 -2 0 2 4 6 87.4

7.6

7.8

8

8.2

8.4

8.6

8.8Test function Kita

f1

f2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8Test function 4

f1

f2

-300 -250 -200 -150 -100 -50 00

10

20

30

40

50

60

70

80Test function OSY

f1

f2

0 50 100 150 200 250-250

-200

-150

-100

-50

0

50Test function SRN

f1

f2

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4Test function TNK

f1

f2

0

0.5

1 0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

f2

Test function DTLZ9

f1

f3

Figure C.2: Non dominated solutions of the final Archive obtained by the developed MOOA.

Due to the Origin filed present in each chromosome it is possible to track which genetic operator generated

them. In Table C.1 the percentages of solutions generated per each genetic operator is shown for the

different test function. The crossover is the genetic operator that most contributes to the non dominated

solutions present in the final Archive in all functions. The regeneration operator, as shown in the table, for

some functions generated up to 33% of the non dominated solutions. However, for two of the benchmark

function it did not contribute with any solution. In this way, it is possible to conclude that the contribution

of the regeneration operator is problem dependent.

Table C.1: Contribution of each genetic operator for the non dominated solutions in the final Archive.

Test Function Crossover [%] Mutation [%] Regeneration [%]

F1 95 0 5F2 64 3 33F3 73 8 19F4 88 12 0Kita 100 0 0OSY 79 15 6TNK 84.42 0.65 14.94DTLZ9 64 19 17

68

optimization of north sea transnational grid solutionsnorthsea transnational grid work package5...

Documents