addressing uncertainty in power system operation and ... · probabilistic methods for energy...

Probabilistic Methods for Energy Networks, King’s College London, March 15, 2017

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Addressing uncertainty in power system operation and planning via two-stage robust

optimization

José M. Arroyo

E-mail: [email protected] Departamento de Ingeniería Eléctrica, Electrónica, Automática y Comunicaciones

Universidad de Castilla – La Mancha


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Contents

Introduction

Dealing with uncertainty

Two-stage robust optimization

Numerical results

Conclusions


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Introduction

Power balance must be kept under continuously changing conditions

Reaction to changes constrained by:

Physical limits of system components

Non-instantaneous deployment of corrective actions

Need for decision making ahead of time Consideration of uncertainty


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Introduction

Tractability Hierarchical decision-making process according to time horizon

Operation Short-term decisions related to generation scheduling and dispatch

Planning Long-term decisions related to infrastructure investment

Decisions under different levels of uncertainty


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Two-stage decision framework

First-stage decisions anticipate uncertainty

Generation scheduling, base-case dispatch, investment decisions

Second-stage decisions model the reaction against uncertainty

System operation under each uncertainty

realization


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Two-stage decision framework Operation-related example

Day-ahead market clearing, look-ahead unit

commitment

Prior to day d Day d

Generation schedule and

base-case dispatch

Actual generation dispatch


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Two-stage decision framework Planning-related example

Static transmission network expansion

planning

Planning horizon

Investment

decisions

System operation


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Contents

Introduction



Numerical results

Conclusions


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Uncertainty sources

Availability of system components

Fluctuations in nodal injections

Demand

Renewable-based generation levels

Injections/withdrawals from electric vehicles


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Addressing component outages

Two types of contingencies:

Random failures Existence of a probability distribution

Non-random deliberate outages Absence of a probability distribution

Industry practice driven by criticality of power system infrastucture


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Addressing component outages

Notion of deterministic security (worst-case

setting) Capability to withstand outages

Implemented via reserves (operation) and redundancy (planning)

Deterministic security criteria: n – 1, n – 2, n – K, n – K

G – K

L

Contingency-constrained models


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Addressing component outages Contingency-constrained models

Power balance guaranteed for:

Pre-contingency state (normal state)

A set of credible contingencies (security criterion)

Replication of operational constraints


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Large-scale mixed-integer program

Economic goal

Second-stage constraints

First-stage constraints


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Contingency-constrained models Each event represents a combination of

unavailable assets

Asset availability is characterized by binary

parameters (generator outages) and

(line outages)

1 for available asset

0 for unavailable asset


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Contingency-constrained models

Contingency set determined by the security criterion being adopted

({ }

{

}

)

Objective function does not explicitly depend on second-stage variables

Computational issue Problem dimension depends on the size of the contingency set


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Contingency 1

First-stage decisions Second-stage decisions

Contingency 2

Contingency |C|

Feasible reaction under

contingency 1


contingency 2


contingency |C|


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Equivalent penalized model

Nodal power imbalance is allowed Slack variables

Penalty term in the objective function associated with the worst-case system power imbalance

Equivalence guaranteed by a sufficiently large

penalty coefficient

Useful to understand the relationship with robust optimization


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Equivalent penalized model

Also a large-scale mixed-integer program


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Addressing nodal fluctuations Traditional context

Nodal fluctuations were typically associated

with demand, which was characterized by: Short-term fluctuation (typically low)

Long-term demand growth (typically steady)

Use of deterministic models relying on forecast values


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Addressing nodal fluctuations today

Increasing penetration of renewable-based generation and future electric vehicles

Larger short-term fluctuations due to variability and intermittency

Unknown long-term penetration level

Worldwide economic context Unforeseen impact on demand growth

Need to consider associated uncertainty


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Two-stage stochastic programming

Probabilistic approach suitable for two-stage decision making

Uncertainty is modeled based on:

Known probability distributions

Scenario-based discretizations

Best performance “on average” driven by a measure of risk aversion


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Two-stage stochastic programming

Uncertainty

realization 1 with

probability 1

First-stage decisions Second-stage decisions

Uncertainty

realization 2 with

probability 2

Uncertainty

realization || with

probability ||

Not necessarily feasible

reaction under

uncertainty realization 1


reaction under

uncertainty realization 2


reaction under

uncertainty realization ||


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Two-stage stochastic programming Scenario-based models

Large-scale mixed-integer program

Scenario-dependent economic goal

Second-stage constraints

First-stage constraints


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Two-stage stochastic programming Scenario-based models

Somewhat similar to contingency-constrained models:

Replication of second-stage constraints

Each scenario represents a combination of nodal injections through parameters

Salient aspect Objective function explicitly depends on second-stage variables


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Two-stage stochastic programming Practical issues

Accurate probabilistic information may be either difficult to obtain or simply unavailable

Scenario-based models Non-trivial tradeoff between accuracy (discretization of uncertainty) and tractability (problem size dependent on scenario set size)

Potential infeasibility in some scenarios Crucial in planning


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Addressing component outages and nodal injection uncertainty

Contingency-constrained two-stage stochastic

programs Increased intractability issues

( )


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Contents

Introduction



Numerical results

Conclusions


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Modeling framework

Worst-case rather than probabilistic setting:

Optimization process considers the impact of uncertainty realizations on first-stage decisions while disregarding their likelihood

Objective function:


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Modeling framework

Suitable for:

Contingency-constrained models relying on deterministic security criteria

Models with inaccurate probabilistic

characterizations of uncertainty Non-random events, penetration of renewables

Planning models where criticality of power

system infrastructure prevails


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Key aspects Implicit second-stage model

Uncertain parameters ( ,

, ) replaced

with decision variables ( ,

, )

Contingency- or scenario-dependent variables ( , ) replaced with variables ( )

Second-stage constraints replaced with a

max-min problem Trilevel counterpart

Size independent of cardinalities of and


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Key aspects Uncertainty-modeling variables

Driven by the maximization of the “damage” (infeasibility, operation cost) of the reaction against uncertainty

Feasibility characterized by a pre-specified,

uncertainty set, i.e., Y

Y encompasses the security criterion and the

limited availability of probabilistic information


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Key aspects Uncertainty set

Discrete and/or interval based:

{ }

[

]

Bounds based on practical aspects and/or historical data


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Key aspects Uncertainty set

Uncertainty budgets to limit conservativeness:

Security criterion ({ }

{

} )

Fluctuation limits for based on

engineering judgement

Polyhedral versus ellipsoidal uncertainty sets


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Key aspects Variables under uncertainty

Driven by the minimization of the “damage” (infeasibility, operation cost) of the reaction against uncertainty

Feasibility characterized by second-stage constraints parameterized in terms of first-stage and uncertainty-modeling variables:


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Robust counterpart

subject to:

su

First stage

Second stage


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Robust counterpart

Large-scale mixed-integer nonlinear trilevel program

Binary variables may be present both in the objective function and the constraint set at any level

No general exact and computationally-efficient solution approach available


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Existing solution techniques

Heuristics or metaheuristics not based on

sound mathematical programming Greedy randomized adaptive search procedures (GRASP)

Decomposition-based methods Finite convergence to optimality under specific conditions and assumptions

All available methods rely on the solution of the max-min second-stage problem


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Decomposition-based methods

Iterative solution of a master problem and a subproblem:

Accelerated Benders decomposition

Column-and-constraint generation

algorithms

Very similar approaches


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Decomposition-based methods Master problem

Relaxation of the trilevel counterpart First-

stage decisions

Iteratively tightened with information from the subproblem

Large-scale mixed-integer program of growing dimension along the iterative process


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Decomposition-based methods Subproblem

Two lowermost levels for given first-stage

decisions from the master problem Second-stage decisions:

Worst-case values for uncertainty-modeling

variables

Best reaction

Max-min problem


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Decomposition-based methods

MASTER PROBLEM

First-stage decisions

Tightening constraints

(Benders cuts, lower-level constraints) SUBPROBLEM

Second-stage decisions

Upper bound for optimal value of upper-level objective function

Lower bound for optimal value of upper-level objective function

x


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Decomposition-based methods Techniques for the subproblem

Duality- or KKT-based bilinear single-level equivalent and iterative heuristic (outer

approximation, separation procedure) Fast heuristic suitable for large-scale problems

Duality- or KKT-based linearized single-level

equivalent Exact for correct bounding parameters for dual variables, scalability issues


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Decomposition-based methods Techniques for the subproblem

Block coordinate descent method (no dual information, no linearization schemes, bilevel

structure kept) Fast iterative heuristic suitable for large-scale problems

Column-and-constraint generation algorithm (in the presence of lower-level binary

variables) Exact, scalability issues


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Contents

Introduction



Numerical results

Conclusions


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Models

Contingency-constrained generation scheduling (CC-GS)

Contingency-constrained generation scheduling under wind uncertainty (CC-U-GS)

Contingency-constrained transmission network expansion planning (CC-TE)

Transmission network expansion planning with uncertain generation and demand (U-TE)


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CC-GS

Accelerated Benders decomposition and linearized duality-based single-level conversion

Master problem and subproblem are MILP solved with CPLEX under GAMS

IEEE RTS, IEEE 118-bus system, 1 period

n – K and n – KG – K

L security criteria

Optimality gaps well below 1%


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CC-GS. IEEE RTS case

K Contingency-

dependent model Robust model

System cost ($)

Time (s) System cost ($)

Time (s)

0 2068020 00000.30 2068020 00.34

1 2688760 00003.60 2688760 00.75

2 3751680 00334.78 3751680 01.79

3 - 14400.00 5232740 16.77

4 Out of memory Infeasible 01.31

5 Out of memory Infeasible 02.45


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CC-GS. IEEE 118-bus case


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CC-U-GS

Column-and-constraint generation algorithm and linearized duality-based single-level conversion


IEEE 118-bus system, 24 periods

Single-component outages, 10 wind farms



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CC-U-GS. IEEE 118-bus case Performance of solution approach


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CC-U-GS. IEEE 118-bus case Solution quality

118-BUS SYSTEM–RESULTS FROM THE OUT-OF-SAMPLE ASSESSMENT

Imbalance (%)

ARO model Conventional model

Number of

samples

0 8668 8706 8779 9140 5606 (0.0, 0.5] 1267 1271 1199 0847 4256 (0.5, 1.0] 0064 0023 0021 0013 0133

>1.0 0001 0000 0001 0000 0005

Expected imbalance (%) 00.028 00.022 00.021 00.014 00.053 CVaR of imbalance (%) 00.435 00.367 00.366 00.271 00.500

Cost increases between 0.1% and 6.8%


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CC-TE

Accelerated Benders decomposition and linearized duality-based single-level conversion


n – K security criterion

IEEE 118-bus system, IEEE 300-bus system



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CC-TE. IEEE 118-bus case


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CC-TE. IEEE 300-bus case


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U-TE

Column-and-constraint generation algorithm

Primal block coordinate descent method (P-CC) versus linearized duality-based single-level conversion (D-CC)

MILP and LP solved with CPLEX under GAMS

IEEE 118-bus system, 2383-bus system


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U-TE. IEEE 118-bus case Quality and performance


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U-TE. 2383-bus system Practical applicability


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Contents

Introduction



Numerical results

Conclusions


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Conclusions

Robust optimization is suitable for operational and planning models under uncertainty

Implicit consideration of uncertainty realizations yields significant computational advantages over contingency- and scenario-dependent models

Tradeoff between accuracy of the uncertainty characterization, conservativeness and computational tractability


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Further research

Incorporation of impact of correlated uncertainty sources

Consideration of more accurate/sophisticated operational models (ramping, nonlinear load flow, line switching, quick-start units)

Analysis of alternative solution approaches to avoid the duality-based transformation


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References

A. Street, A. Moreira, J. M. Arroyo. “Energy and reserve scheduling under a joint generation and transmission security criterion: An adjustable robust optimization approach”. IEEE Transactions on Power Systems, vol. 29, no. 1, pp. 3-14, January 2014

A. Moreira, A. Street, J. M. Arroyo. “An adjustable robust optimization approach for contingency-constrained transmission expansion planning”. IEEE Transactions on Power Systems, vol. 30, no. 4, pp. 2013-2022, July 2015


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References

A. Moreira, A. Street, J. M. Arroyo. “Energy and reserve scheduling under correlated nodal demand uncertainty: An adjustable robust optimization approach”. International Journal of Electrical Power and Energy Systems, vol. 72, pp. 91-98, November 2015

N. G. Cobos, J. M. Arroyo, A. Street. “Least-cost reserve offer deliverability in day-ahead generation scheduling under wind uncertainty and generation and network outages”. IEEE Transactions on Smart Grid, in press, 2017


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References

A. Street, A. Moreira, J. M. Arroyo. “On the solution of large-scale robust transmission network expansion planning under uncertain demand and generation capacity”. https://arxiv.org/abs/1609.07902


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