A Multiobjective Model for Distribution System Planning based on Tabu Search

Benvindo Rodrigues Pereira Junior José Roberto Sanches Mantovani

Electrical Engineering Departament São Paulo State University, UNESP

Ilha Solteira, Brazil brpjunior@gmail.com,

mant@dee.feis.unesp.br

Antonio Marcos Cossi Mathematical Departament

São Paulo State University, UNESP Ilha Solteira, Brazil

cossi@dm.feis.unesp.br

Javier Contreras E.T.S Ingenieros Industriales

University of Castilla - La Mancha Ciudad Real, Spain

Javier.Contreras@uclm.es

Abstract—This paper presents a multiobjective model for the multistage planning problem of a distribution system. The problem is formulated as a multiobjective mixed integer nonlinear programming model. The actions deemed in this model for each period of the planning horizon are: increase in the capacity of existing substations (or construction of new ones), exchange of cables in existing lines (and construction of new feeders), reconfiguration of the network, allocation of sectionalizing switches and construction of tie lines. Taking into account these actions, the objective functions used in this model are the investment and operational costs as well as the system’s reliability under contingencies using an n-1 criterion. Reliability is calculated by means of the non-supplied energy. To solve the proposed multiobjective model a multiobjective tabu search method is used. Numerical results are presented for a 54-bus system.

Index Terms—Distribution system planning, multiobjective optimization, reliability, tabu search.

NOMENCLATURE

Sets and IndexesSet of available cables to be installed in thenetwork;Set of existing branches in period p;Set of proposed branches in period p;, Set of feeders of substation n in period p;Set of network buses in period p;Set of buses connected to substation n in period p;Number of load levels;Set of lines connected to substation n in period p;Maximum number of switches allowed in the network;Years of the planning horizon;Number of periods of the planning horizon;Set of existing substations in period p;Set of proposed substations in period p;

Functions, ( , , ) Active power injection at bus i with load level kin period p;, ( , , ) Reactive power injection at bus i with load level k in period p;

CostsCosts to build a new branch with a cable of type k; in $/km;Costs to replace a branch with a cable of type aby a cable of type k; in $/km;Costs to expand substation n; in $;Costs to build a new substation n; in $;Costs to install a switch in line ij ; in $;, KW/h costs, in $, with load level k in period p.

Binary variables, , Decision to build (1) or not (0) branch ij with acable of type k in period p;, , Decision to replace (1) or not (0) the cable of branch ij by a cable of type k in period p;, Decision to build (1) or not (0) substation n inperiod p;, Decision variable to expand (1) or not (0) substation n in period p;, Decision to operate (1) or not (0) substation nin period p;, Decision to use (1) or not (0) branch ij in period p;, Decision to install (1) or not (0) a switch in line ij in period p.

Continuous variables and their limits, , Voltage at bus i with load level k in period p;Minimum voltage level of the feeder;Maximum voltage level of the feeder;, , Current of branch ij in load level k in period p;, Maximum current allowed through branch ij inperiod p;

, Active power generated at bus i with load level k in period p;, Active power demand at bus i with load level kin period p;, Reactive power generated at bus i with load level k in period p;, Reactive power demand at bus i with load level k in period p;

, Capacity of existing substation n in period p;, , Load at bus i connected to substation n withload level k in period p;, , Losses in branch ij connected to substation nwith load level k in period p.

Parameters and DefinitionsLength of branch ij;Investment discount factor in period p;Operation discount factor in period p;Annual interest rate;Duration in hours of each load level;, Resistance of branch ij in period p.

I. INTRODUCTION

Distribution system planning (DSP) is based on the technical and economic analysis of future operation conditions. Traditional planning models seek to minimize the required investments in new distribution equipment to supply energy for customers, taking into account demand forecasts, voltage drop, equipment capacity, etc., i.e., aiming at quality and reliability [1-5].

The DSP problem solution indicates how and when the conductors’ branches and substations must be reinforced, and when and where to build new substations and circuits. These actions improve energy supply quality, but not reliability.

System reliability can be improved with tie lines and the allocation of sectionalizing switches to control energy supply interruptions, which can occur due to permanent faults or preventive maintenance of distribution system components,through restoration procedures.

The DSP models can be classified in terms of the planning horizon as: static, pseudo-dynamic and dynamic; and mono or multiobjective in relation to the number of objectives.Modeling and solving the exact DSP problem can be a difficult job, because of its complexity and combinatorial nature. Over the years many works describing these models and proposing solution techniques have been presented to solve this problem and these techniques can be divided into two categories: heuristcs and metaheuristics (Genetic algorithms, Tabu Search, Simulated Annealing, etc.) [1-3]; and classical mathematical programming methods (Simplex, Branch and Bound, etc.) [4, 5].

In this paper, DSP is formulated with a multiobjective and multistage dynamic model. The objective functions proposed are: i) the investment and operation costs (to increase the capacity of existing substations or to build new ones, to exchange cables of existing branches and to build new circuits, to sectionalize switches allocation and to build tie

lines), and ii) reliability, expressed as the non-supplied energy costs (NSEC). The proposed model is different from [1] because we use a multistage dynamic model.

The multiobjective approach is very important in solving these kinds of problems, because it prevents decision-making elements to be implicitly incorporated into the model, leaving the decision making responsibility for planners, since the multiobjective models do not have a specific solution, but a set of solutions that allows to observe clearly the compromise relationships (trade-off) between the analyzed objectives.

To solve this model a multiobjective tabu search algorithm is proposed [6], which uses dominance concepts to do the optimization. Results for a 54-bus system are presented.

II. MATHEMATICAL MODEL

The distribution system planning problem is formulated in this paper as a multiobjective mixed integer nonlinear programming problem, and the proposed mathematical formulation is as follows:

= . . , ,∈∈∈+ . . , ,∈∈ + ( ). ,∈+ ( ). ,∈ + . ,∈ .+ . , . , . , , . ,∈∈ .∈

(1)

= , ,∈ ,∈∈ (2)

subject to:

, ( , , ) − , + , = 0 ∀: ∈ , ∈ , ∈ (3)

, ( , , ) − , + , = 0 ∀: ∈ ,∈ , ∈ (4)

< , , < ∀: ∈ , ∈ , ∈ (5)

, , ≤ , . , ∀: ∈ , ∈ , ∈ (6)0 ≤ , , +∈ , ,∈ ≤ , . , ∀: ∈ , ∈ , ∈ (7)

, = −∈ ,∈ ∀ ∈ (8)

, , ≤ 1 ∈ ∀: ∈ , ∈ (9)

, , ≤ 1 ∀: ∈ , ∈∈ (10)

, ≤∈ ∀ ∈ (11)

, , , , , , , , , , , , , , , ∈ {0,1} (12)

The considered are the investment and the operational costs. The investment costs are determined at the beginning of each period and represent the costs involved in the installation of new branches ( ), the exchange of conductors sizes of existing branches ( ) (in both cases the costs of the physical infrastructure are included), the increase of the capacity of an existing substation ( ) or the construction of a new one ( ), and the costs of allocationsectionalizing switches ( ). The operational costs for each period correspond to the annual energy losses for each load level.

In this paper, it is assumed that the investment in lines, sectionalizing switches and substations are performed in the first year of each period. The costs of the annual energy losses are calculated year by year, so the net present value over the horizon planning is calculated using and .

= 1(1 + ) .( ) (13)

= 1(1 + ) .( )/

(14)

is related with the distribution system reliability in terms of the non-supplied energy, based on the topology and sectionalizing switches allocated in the system. The , , formulation details are presented in [1].

The problem constraints are the active and reactive node balance equations (3) and (4), upper and lower voltage limits (5), branch conductors capacity (6), substation operational capacity (7), radiality conditions (3)(4)(8)[7], the choice of just one type of cable to replace the cable in an existing branch (9) or to be installed in a new branch (10) the maximum number of sectionalizing switches allocated (11), and the setting the decision variables as binary (12). In addition to the constraints mathematically modeled above, the following ones are also considered:

every element of the system (branch, substation and sectionalizing switch) installed or replaced in period p is present in period p +1; investment in new elements is considered only once; replacement branches or new branches may only be used after the corresponding investment has been made.

III. SOLUTION TECHNIQUE

The proposed model is solved by multiobjective tabu search [6]. To codify the problem a decimal base codification is used, which allows including important design aspects in an easy and practical way, that would be hardly been considered with a binary codification [2]. The codification is based on [1, 2], so for each period the codification structure is illustrated in Figure 1.

Branches {Lep U Lpp} Subs {SSexp U SSprp} Branches {Lep U Lpp}

3 3 0 1102b1 b2 bn bk-1 bk

1 02s1 sn sk

1 0 0 11b1 b2 bn bk-1 bk

Part I Part II Part II

Figure 1. Codification system for period p.

In part I, encoding is as follows:

0 – Branch not constructed; 0 < number < 100 – Branches in operation. The number represents the type of cable installed inthe branch; number > 100 – Tie lines. The number represents the type of cable installed in the branch added 100. For example, is a tie line with cable type 2.

Part II:

0 – Substation not constructed; 0 < number – Type of substation. If is 1, is an existing substation; 1 represents the initial substation capacity, and 2 that the substation has to be expanded. On the other hand, if ∈2 represents that the substation has been constructed.

Part III:

0 – There are no sectionalizing switches installed; 1 – There are sectionalizing switches installed.Sectionalizing switches are installed only inconstructed branches.

For n periods the structure of Figure 1 is repeated n times. It should be taken into account that, when working with multi-periods, the elements built in period p must be present in period + 1, whether in operation or not.

From the system characteristics provided by the encoding structure (topology, location of substations, type of cables that make up the branches, location of sectionalizing switches and tie lines), a radial power flow algorithm [8] is used to calculate the system operation state, making possible to evaluate

.

Despite the planned network is meshed, its operation is radial. The tie lines along with the sectionalizing switches improve network reliability. So with this information provided by the encoding, the can be evaluated. To

calculate is a difficult task, because several factors need to be taken into account in the restored sections (group of branches between adjacent sectionalizing switches) affected by one permanent fault. Some of these sections can be restored by switching loads blocks through energized circuits and tie lines. This way to evaluate the real value of of the proposed system needs to solve a restoration problem considering the system operation constraints (3-8) [9-10].

The neighborhood is defined through the simplest movements, which should consider two important conditions: connectivity of busses and radiality of the distribution network. The neighborhood movements are the following:

i. system reconfiguration using a branch exchange technique [11];

ii. eliminate or add branches (or tie lines) that supply power to a system node. If the branch added or removed isn’t a tie line radiality and connectivity must be verified;

iii. change the size of a substation in a system node; change the size of branches (or tie lines) that supply power to a system node;

iv. eliminate or add a substation in a system node. Eliminating a substation entails the elimination of all feeders that leave the substation, and adding new branches to transfer the load of these feeders to other substation feeders. Adding a substation entails the creation of feeders to the new substation maintaining the system radiality;

v. eliminate or add sectionalizing switches in the system; vi. change the positions of existing sectionalizing switches

in the system.

In addition to the tabu list (TL) present in the classical tabu search, two new lists are incorporated in the multiobjective tabu search used. The first one is the Pareto list (PL) that stores the non-dominated solutions found by the algorithm during the search process. The second one is the Candidate List (CL) which stores all other non-dominated solutions that have not been stored in the LP and that were not explored until the current iteration. These solutions can be selected to be the new seed solution, i.e., their neighborhoods will be explored if they keep their status as non-dominated solutions during the iterative process of the tabu search multiobjective algorithm [6].

In the same way as the classical tabu search, aneighborhood is generated from an initial seed solution, which is sorted in dominated and non-dominated solutions. One of the non-dominated solutions is selected to be the new seed solution, from which the search will continue. It’s an intensification process that performs the search in a specific region of the search space. On the other hand, a diversification process is applied when non-dominated solutions are not found in the current neighborhood, selecting the new seed solution from the CL. The search restarts from this solution to search a new region of the Pareto frontier, thus avoiding the premature termination of the search process due to regions of local optima.

The convergence criteria applied in the multiobjective tabu search implemented in this paper is the maximum number of iterations or until no more non-dominated solutions are found and = ∅.

IV. RESULTS

The implemented algorithm is tested on a 54-bus system based on [12]. This system is composed of 2 existing substations which can be expanded, 2 proposed substation, 16 existing branches and 45 proposed ones, operating at 13.8 kV. The initial system is illustrated in Figure 2. Six types of cables are considered for the new and existing lines expansions. Data regarding cables is given in Tables 1 and 2. Substation data are presented in Table 3.

4056

5711

61

55

60

59

14

16

15

13

12

3338

43

3435

37

36

42

41

39

5453

5250

4

3

5

7

3230

31

2946

4445

47

48

49 51

9

2

17

20 18

18

19

24 1021

22 23

2526 27

286

28

101 3

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5

6

103

7

27

26

825

2

1

2423

917

18

19

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2221

104

36

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10

31

37

43

30

29

13

12

11 102

15

40

14

16

41

46

47

42

48

49

5058

Figure 2. Initial system.

TABLE I. CABLE DATA

Type Current Capacity (A) Impedance (Ω/km)1 150 0.3655 + j0.25202 250 0.2359 + j0.24026 350 0.1827 + j0.12604 400 0.1460 +j0.12335 500 0.1180 + j0.12016 600 0.0966 + j0.1201

TABLE II. COST TO BUILD AND REPLACE BRANCHES (103 $/KM)

Type of cables Y X 1 2 3 4 5 6

0 20.0 30.0 42.0 50.0 67.0 85.01 - 23.0 37.0 40.0 58.0 76.02 - - 30.0 35.0 50.0 66.03 - - - 29.0 43.0 53.04 - - - - 35.0 45.05 - - - - - 38.0

In relation to cable costs, positions (0, Y), Y = 1,⋯, 6 in Table 2, indicate the branch construction costs with a cable of

type Y, and positions (X, Y), X = 1,⋯, 5 and Y = 1,⋯, 6, indicate cable replacement costs of a branch with cable X by cable Y.

TABLE III. SUBSTATION DATA

BusInitial

capacity (MVA)

New capacity (MVA)

Construction costs

(103 $/km)

Expansioncosts

(103 $/km)Existing

101 16.7 33.4 - 1400.0102 16.7 30.0 - 1200.0

New103 - 22.2 3000.0 -104 - 22.2 3000.0 -

Three load levels are used in the tests: Light – a loading factor of 0.65 with duration of 1000 hours/year; Nominal – aloading factor of 0.80 and duration of 6760 hours/year; and Peak – with a loading factor of 1.0 and duration of 1000 hours/year. Upper and lower voltage limits are 0.95 and 1.05 p.u., and the voltage at the substation is fixed in 1.0 p.u. The price of energy to evaluate the losses is 100 $/MWh for all load levels. The costs of sectionalizing switches are $1.0x103.The planning horizon adopted is 15 years subdivided into 3 periods of 5 years.

The annual interest rate is 10%. The rate of permanent faults used is 0.4 faults/km/year. The values of interruption costs for each customer category and the interruption duration can be seen in Table 4.

TABLE IV. INTERRUPTION COSTS

Customer category

Interruption costs ($/MW/year)Restoration (15 min) Repair (120 min)

Residential 40.0 2000.0Commercial 1200.0 12800.0

industrial 1600.0 10400.0

The Pareto frontier obtained with the implemented algorithm is shown in Figure 3.

Figure 3. Pareto frontier solutions.

To verify the quality of the solutions found by the algorithm a detailed analysis is presented for the highlighted

solution in Figure 3. This solution has investment and operation costs of $4836.97x103 and NSEC of $4207.24x103.Figures 4 to 6 show the proposed topology for the highlighted solution.

61

6

4

4-sw

4

4-sw

6

2

6

2

2

1

61

4-sw

2

6-sw

6

4

2-sw2

1

1

2-sw

6

2

1

21

1

1 1

2

6

1

2 66-sw

1

2

1 21

4 4-sw 4 4

28

101 3

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825

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47

42

48

2-sw2-sw

x-sw

x

Tie lines with x cable typex

operation lines withx cable type

operation lines with xcable type and switches

103

Figure 4. Proposed solution for period 1.

61-sw

16

4

4-sw

4

4-sw

6

102

6-sw

4-sw

5-sw

61

61-sw

4-sw2-sw

2-sw

6-sw

6

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2-sw2

1

1-sw

2-sw

6

2

1-sw1

1

1

2-sw101

1

1 101

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6

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2 66-sw

1-sw

102

1 1021

4 4

4 4-sw 4 4

28

101 3

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825

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2423

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11 102

15

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48

49

1022-sw

103

Figure 5. Proposed solution for period 2.

1900

2900

3900

4900

5900

6900

7900

8900

4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000

NSE

C (1

03 $)

Costs (103 $)

1

1

61-sw

16

4

4-sw

4-sw

4-sw

6

102

6-sw

4-sw

5-sw

61

61-sw

4-sw2-sw

2-sw

6-sw

6

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1

1

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6

1-sw1

1

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1

1 1

2

6

1

2 66-sw

1-sw

2-sw

1 2-sw1

4 4

4 4-sw 4 4

28

101 3

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1022-sw

2-sw

50

103

Figure 6. Proposed solution for period 3.

IV. CONCLUSION

This paper presents a computational evaluation of a multistage and multiobjective optimization model for distribution system planning. The proposed model takes into account investment and operational costs. The problem is formulated as a multiobjective mixed integer non-linear problem and solved by a multiobjective tabu search algorithm.

The developed algorithm is efficient to treat the multiobjective nature of the DSP problem. Through the set of solutions found by the algorithm is possible check the compromise (trade-off) between costs (investment and operational) and reliability of the planned network.

In problems with a multiobjective nature a very important role is assigned to humans deciding what is the best option.Thus the results obtained from the model and the technique proposed in this paper can aid a system planner (decision-maker) to select the best decision.

ACKNOWLEDGMENT

The authors gratefully acknowledge FAPESP (grant 2009/08428-4) and CAPES (grant BEX 0144/12-6) for the economic support.

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