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Corrective Control to Handle Forecast Uncertainty: A Chance Constrained Optimal Power Flow

Line Roald, Sidhant Misra, Thilo Krause, Goran Andersson

arXiv:1609.02194v1math.OC

TL;DR

Planning corrective control for forecast uncertainty is difficult because it involves many possible scenarios alongside contingencies. The paper combines uncertainty-responsive affine control policies with contingency control in a chance-constrained OPF and solves the resulting problem using a sequential SOCP algorithm. It reports reduced operating cost while maintaining security and demonstrates solutions for larger test systems.

  • Problem

    Planning corrective actions for forecast uncertainty is difficult because it requires considering many uncertainty scenarios as well as contingencies.

    Source: §I · p. 1

  • Method

    The method combines corrective control for uncertainty and contingencies in a chance-constrained OPF and solves the resulting SOC-constrained problem with a sequential SOCP algorithm.

    Source: §VIII · p. 11, B. Results · p. 11

  • Results

    The case studies demonstrate solutions for the IEEE 118-bus, IEEE 300-bus, and 2383-bus Polish systems, with the IEEE 118-bus case including 99 uncertain loads solved within minutes.

    Source: §VIII · p. 11

  • Takeaways & Limitations

    The case studies indicate that corrective control for uncertainty can reduce operating cost while maintaining system security.

    Source: 1) Comparison of operation cost · p. 8, §VIII · p. 11

    The conclusion describes extension to full AC power flow with approximate chance constraints as future work.

    Source: §VIII · p. 11

Abstract

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Higher shares of electricity generation from renewable energy sources and market liberalization is increasing uncertainty in power systems operation. At the same time, operation is becoming more flexible with improved control systems and new technology such as phase shifting transformers (PSTs) and high voltage direct current connections (HVDC). Previous studies have shown that the use of corrective control in response to outages contributes to a reduction in operating cost, while maintaining N-1 security. In this work, we propose a method to extend the use of corrective control of PSTs and HVDCs to react to uncertainty. We characterize the uncertainty as continuous random variables, and define the corrective control actions through affine control policies. This allows us to efficiently model control reactions to a large number of uncertainty sources. The control policies are then included in a chance constrained optimal power flow formulation, which guarantees that the system constraints are enforced with a desired probability. By applying an analytical reformulation of the chance constraints, we obtain a second-order cone problem for which we develop an efficient solution algorithm. In a case study for the IEEE 118 bus system, we show that corrective control for uncertainty leads to a decrease in operational cost, while maintaining system security. Further, we demonstrate the scalability of the method by solving the problem for the IEEE 300 bus and the Polish system test cases.

I. INTRODUCTION

Growing uncertainty and available flow-control devices make corrective control increasingly relevant, but planning reactions to many uncertainty scenarios remains difficult. The paper extends chance-constrained OPF to combine corrective control for uncertainty and contingencies, using affine policies and a scalable solution approach.

I. INTRODUCTION

  • Planning uncertainty-responsive corrective control is difficult because it must handle many possible uncertainty scenarios alongside contingencies.Although corrective control can reduce operational cost, HVDC and PST set-point changes are typically chosen ad hoc in real-time operation.Source: §I · p. 1

II. MODELLING FRAMEWORK FOR CORRECTIVE CONTROL

  • The framework treats outages and uncertainty realizations as distinct triggers for corrective control, with uncertainty occurring more continuously than discrete contingencies.The distinction affects the time available for implementation and the type and modelling of control reactions.Source: §I · p. 1

I. INTRODUCTION

  • The paper combines corrective control for uncertainty and contingencies in a chance-constrained OPF, using an affine policy and a sequential SOCP algorithm.The IEEE 118-bus case study reports reduced operational cost while maintaining system security; IEEE 300-bus and Polish test cases demonstrate scalability.Source: §I · pp. 1–2

A. Modelling Corrective Control for Contingencies

  • Following a line outage, a power-flow controller's set-point changes through an additional optimization variable defined for that outage.The post-contingency set-point is based on its pre-contingency value plus the outage-specific change.Source: A. Modelling Corrective Control for… · p. 2

B. Modelling Corrective Control for Forecast Uncertainty

For forecast uncertainty, controllers adjust their set-points through affine policies proportional to forecast error, with response parameters optimized by the model. The paper combines these policies with contingency control, describes their real-time use, and specifies the system and uncertainty modelling assumptions.

B. Modelling Corrective Control for Forecast Uncertainty

  • Affine policies adjust controller set-points in proportion to forecast error, and their response parameters are optimized.The policy trades off response flexibility against tractability: it can model continuous fluctuations, support optimization, and provide a policy readily implemented by the TSO.Source: B. Modelling Corrective Control for… · p. 2

C. Combined Corrective Control

  • The controller responses to uncertainty are decoupled from whether the system is in a pre- or post-contingency state.This avoids a very large number of additional control variables that would be difficult to handle in optimization and real-time implementation.Source: C. Combined Corrective Control · p. 3

D. Use of Corrective Control in Real-Time Operation

  • Real-time uncertainty-responsive control can be implemented as automatic adjustments or as manual set-point changes when a constraint violation occurs.Once uncertainty is known, the operator can rerun SCOPF deterministically with less restrictive constraints; affine-policy planning guarantees real-time feasibility.Source: D. Use of Corrective Control in… · p. 3

III. POWER SYSTEM MODELLING

  • The power-system model uses a DC power-flow approximation and considers single-line outages for N-1 security, excluding lines that cause islanding.Generation outages are handled through a predetermined reserve requirement.Source: §III · p. 3

A. Uncertainty Sources

  • Uncertain demand or production is represented by expected active-power production plus a zero-mean fluctuating component.The model describes fluctuations with a covariance matrix and represents corrective responses through controller matrices.Source: A. Uncertainty Sources · p. 3

B. HVDC and PST Set-Points With Corrective Control

HVDC and PST set-points are modeled with outage-specific corrective actions and uncertainty-responsive policies. Generator AGC balances fluctuations, while corrective generator control is excluded.

B. HVDC and PST Set-Points With Corrective Control

  • HVDC and PST post-contingency set-point changes are represented by outage-specific variables, while uncertainty-responsive changes use control matrices.The model assumes point-to-point HVDC connections and treats PST angles as continuous because tap positions are sufficiently close.Source: B. HVDC and PST Set-Points With… · p. 3

C. Power Balance and Generation Control

  • Generators adjust in-feeds according to an AGC signal based on total mismatch Ω, with optimized αG distributing that mismatch among generators.The constraints require αG to maintain balance and ensure positive mismatch decreases other generation.Source: C. Power Balance and Generation Control · p. 4
  • The model excludes corrective generator control for both contingencies and uncertainties, although the formulation could be extended with redispatch variables.The authors note that generation control is costly and requires careful pricing of redispatch and balancing.Source: C. Power Balance and Generation Control · p. 4

D. Power flow modelling

Power-flow changes combine uncertainty-driven injection changes and corrective controls; outage flows additionally account for topology changes using line outage distribution factors.

D. Power flow modelling

  • Pre-contingency line flows combine the base-case flow with changes caused by fluctuations.The base-case flow depends on line reactance and the PST angles mapped to lines with PSTs.Source: D. Power flow modelling · p. 4
  • Uncertainty-driven flow changes reflect fluctuations, HVDC and PST corrective control, and generator balancing, with policies able to reduce line congestion.The PTDF matrix maps effective injection changes to line flows, while PST set-point changes also directly affect their connected lines.Source: D. Power flow modelling · p. 4
  • For outages, the flow change includes both corrective actions and topology changes represented by line outage distribution factors.The LODF gives the fraction of flow on an outaged line shifted to another line.Source: D. Power flow modelling · p. 4

IV. CHANCE-CONSTRAINED OPTIMAL POWER FLOW

The CC-OPF minimizes energy and reserve costs subject to power-balance and generator constraints. Reserve activation constraints account for mismatch uncertainty and jointly limit the probability of insufficient reserves.

1) Objective:

  • The CC-OPF objective minimizes the total cost of energy and reserves.Generator bids supply the cost coefficients for energy and reserve provision.Source: 1) Objective · p. 4

2) Power Balance and Generator Constraints:

  • Power-balance constraints cover normal operation and fluctuations, while generator limits, reserve requirements, ramping, and reserve coverage constrain schedules.Reserve activation constraints are chance constraints based on the scalar total mismatch Ω; the generator constraints are jointly enforced at probability 1 − εG.Source: 2) Power Balance and Generator… · p. 5
  • The violation probability εG directly represents the probability of insufficient reserves.The reserve constraints share this interpretation because they depend on the same total mismatch Ω.Source: 2) Power Balance and Generator… · p. 5

3) HVDC and PST Constraints:

Chance constraints bound the risk of HVDC and PST limit violations and transmission overloads. The formulation excludes the intermediate post-contingency, pre-corrective-control state.

3) HVDC and PST Constraints:

  • Chance-constrained HVDC and PST limits allow saturation, but bound the probability that saturation requires additional corrective action below ε.The formulation also permits limits on the amount of post-contingency control.Source: 3) HVDC and PST Constraints · p. 5

4) Power Flow Constraints:

  • Transmission-flow limits are soft constraints, and εl represents the probability of an overload or N-1 violation.Such violations may be accepted or handled through additional real-time measures.Source: 4) Power Flow Constraints · p. 5
  • The formulation omits the intermediate post-contingency, pre-corrective-control state, though constraints for it can be added without conceptual change.Source: 4) Power Flow Constraints · p. 5

B. Reformulation of Chance Constraints

The chance constraints are analytically reformulated under a normality assumption into tractable constraints, including second-order cone constraints. Symmetry reduces the SOC count, while alternative distributional reformulations remain available when normality is unjustified.

B. Reformulation of Chance Constraints

  • Assuming normally distributed forecast uncertainty makes the marginal chance constraints tractable because each is one-dimensional and depends on a weighted sum of the uncertainty vector.The paper motivates the assumption by projection of a high-dimensional vector onto each constraint.Source: B. Reformulation of Chance Constraints · p. 6

1) Reformulated Constraints:

  • The reformulated generator constraints are linear, while the remaining constraints are convex SOC constraints for ϵ ≤ 0.5.The generator constraints depend on total power mismatch and its standard deviation.Source: 1) Reformulated Constraints · p. 6

2) Exploiting Symmetry of SOCs:

  • Exploiting normal-distribution symmetry cuts the number of SOC constraints in half, improving efficiency.Each upper/lower SOC pair becomes two linear constraints and one SOC constraint; the SOC terms tighten the original deterministic constraints.Source: 2) Exploiting Symmetry of SOCs · p. 6

3) Non-Normal Uncertainty:

  • When normality is unjustified, analytical reformulations using limited distributional knowledge are possible, but may be overly conservative and cause high cost or infeasibility.The Chebyshev-based alternative requires only the mean and covariance; the authors instead suggest a normal assumption with out-of-sample testing.Source: 3) Non-Normal Uncertainty · p. 6

V. SOLUTION ALGORITHM

The paper develops a sequential SOCP constraint-generation algorithm to solve the CC-SCOPF efficiently and reach a globally optimal solution. It adds violated full SOC constraints rather than approximating them with cutting planes.

Challenges for cutting planes

  • The prior cutting-plane approach is unsuitable because the problem has many SOCs to evaluate, and its iterations are slow and memory-intensive.Large decision-variable dependencies also require many iterations to obtain a good approximation and feasible solution.Source: §V · pp. 6–7

Sequential SOCP algorithm

  • The sequential SOCP algorithm adds full SOC terms for the most violated constraints to a linear relaxation, repeating until all constraints are satisfied and the original problem's global optimum is reached.The order in which violated constraints are added significantly affects solving time.Source: §V · p. 7

Solving the CC-OPF without security constraints

  • The algorithm preprocesses with an SCOPF without uncertainty, then solves the CC-OPF without security constraints, adding violated base-case SOC constraints iteratively.The base problem includes full PST and HVDC constraints and adds their SOC terms upfront.Source: §V · p. 7

Solving the full CC-SCOPF with security constraints

  • For security constraints, the algorithm starts with constraints active in the SCOPF, then adds the most violated linear constraints and corresponding SOCs before checking remaining violated SOCs.It screens post-contingency SOC evaluations using the LODF matrix, retaining pairs whose LODF exceeds a threshold while maintaining acceptable accuracy.Source: §V · p. 7

Termination

  • After security constraints are satisfied, pre-contingency violations trigger a restart; in all encountered cases, one pass found a feasible global optimum.The algorithm terminates when no pre-contingency constraints are violated.Source: §V · p. 7

VI. CASE STUDY - IEEE 118 BUS SYSTEM

The IEEE 118-bus case study evaluates corrective control under modeled and empirical uncertainty, using Monte Carlo simulations and comparisons among five OPF formulations. The test system includes zonal load fluctuations and installed PST and HVDC controls.

A. IEEE 118 Bus Test System

  • The IEEE 118-bus case uses loads scaled by 1.25, with all 99 loads uncertain at 10% forecast standard deviation and correlation ρ = 0.3 within three zones.Fluctuations across zones are assumed uncorrelated; line, HVDC, and PST chance constraints use ϵ = 0.01.Source: A. IEEE 118 Bus Test System · p. 7
  • The system assumes three PSTs and three HVDC connections, while reserve requirements are dimensioned to cover total fluctuation with probability 1 − ϵG.Generator reserve provision is capped at 20% of total generation capacity.Source: A. IEEE 118 Bus Test System · pp. 7–8
  • Two 2,000-sample Monte Carlo simulations test modeled multivariate-normal uncertainty and out-of-sample Austrian grid data scaled to the assumed mean and covariance.The empirical data cover 8,492 points across 28 buses; samples are constructed for each of 99 buses.Source: A. IEEE 118 Bus Test System · p. 8

B. Numerical Results

  • Five formulations compare standard OPF, security-constrained and corrective variants, and chance-constrained cases with or without corrective response to fluctuations.The study compares dispatch and reserve costs, empirical violations, and corrective-control effects across violation probabilities and uncertainty levels.Source: B. Numerical Results · p. 8

1) Comparison of operation cost:

Corrective control reduces both security-related and uncertainty-related operating costs in the IEEE 118-bus case. For line 119 after an outage, it also reduces flow variability and shifts corrective responses toward nearby fluctuations.

1) Comparison of operation cost:

  • 0.35% lower cost of uncertainty results from adding uncertainty corrective control, while post-contingency corrective control reduces cost of security by 0.9%.The comparisons are formulation (d) versus (e) for uncertainty control and (b) versus (c) for post-contingency control.Source: 1) Comparison of operation cost · p. 8
  • 35.2 MW to 19.7 MW: corrective HVDC and PST control reduces the standard deviation of line 119 flow after line 126's outage.The reduced variance permits a higher nominal power flow, which leads to lower cost.Source: 1) Comparison of operation cost · pp. 8–9
  • HVDC and PST controls respond strongly to nearby fluctuations and have nearly zero response coefficients for most other fluctuations.The control parameters shown are αDC and αγ.Source: 1) Comparison of operation cost · p. 9, Figure 4 · p. 9

2) Number of Violations and Out-of-Sample Testing:

Monte Carlo tests show that chance-constrained formulations sharply reduce violations, and corrective control for uncertainty lowers dispatch cost without materially worsening the violation rate. The cost advantage grows as uncertainty rises or the accepted violation probability falls.

2) Number of Violations and Out-of-Sample Testing:

  • About 85% of samples violate constraints under SCOPF, versus about 5% under either CC-SCOPF formulation.The table reports around 1,700 violating samples for SCOPF cases (b) and (c), compared with around 100 for CC-SCOPF cases (d) and (e).Source: Table II · p. 9, 2) Number of Violations and… · pp. 9–10
  • For the 19 constraints with observed violations, APG samples exceed the 1% limit on four tight constraints, but every constraint remains below 2%.With normal samples, the four tight constraints are approximately at 1%; most constraints remain below the accepted 1% level.Source: Figure 5 · p. 9, 2) Number of Violations and… · p. 9

3) Influence of the confidence level:

  • The cost reduction from corrective control grows as the accepted violation probability falls or uncertainty increases; at 14.25% standard deviation, only formulation (e) is feasible.Table III compares dispatch costs for formulations (d) and (e) across violation-probability levels and uncertainty standard deviations.Source: Table III · p. 10, 3) Influence of the confidence level · p. 10

2) Number of Violations and Out-of-Sample Testing:

  • APG samples increase violated scenarios by less than 1% versus normal samples for CC-SCOPF cases (a) and (e), while corrective control (e) retains similarly low violations as (d) at lower cost.The comparison is an out-of-sample test using APG data.Source: 2) Number of Violations and… · p. 10

VII. CASE STUDY - SCALABILITY

2) IEEE 300 bus test system:

The authors test the method on modified IEEE 300-bus and Polish systems, then report problem sizes and computational results. These cases demonstrate scalability, while SOC-constraint evaluation remains a runtime bottleneck.

2) IEEE 300 bus test system:

  • The modified IEEE 300-bus and Polish cases include uncertain loads, three HVDC connections, and PSTs; neither includes post-contingency corrective control.The Polish case uses 157 uncertain loads, while the IEEE 300-bus case uses 109; both use zero load correlation.Source: 3) Polish Test System · p. 10, 2) IEEE 300 bus test system · p. 10

B. Results

  • The Polish grid case is two orders of magnitude larger than IEEE 118-bus by constraint count, featuring more than 8 million variables and 8 million SOC constraints.The IEEE 300-bus case is one order of magnitude larger than IEEE 118-bus by constraint count.Source: B. Results · p. 10
  • The current implementation spends significant time checking SOC constraints; LODF prescreening cuts the checks threefold, but the Polish case still requires close to 1.4 million evaluations.Table VII reports run times and SOC-evaluation counts for the test cases.Source: B. Results · p. 10, Table VII · p. 11
  • The sequential SOCP algorithm needs only 2–3 SOC evaluations, whereas the cutting-plane algorithm requires many more and would lead to prohibitively long run times in this implementation.More efficient coding and parallelization could reduce SOC-check time, which the paper leaves for future work.Source: B. Results · p. 11

VIII. CONCLUSIONS

  • PSTs and HVDCs respond similarly to nearby fluctuations, suggesting that geographically close uncertainty sources could be aggregated.The authors connect this aggregation possibility with the small number of active constraints in large transmission systems when discussing scalability.Source: §VIII · p. 11
  • Extending the method to full AC power flow with approximate chance constraints linearized around the forecast operating point remains future work.The authors also call for further scalability tests and implementation improvements.Source: §VIII · p. 11
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