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Policy-based reserves for power systems

Joseph Warrington, Paul Goulart, Sebastien Mariethoz, Manfred Morari

arXiv:1212.2119v2math.OC

TL;DR

Power systems must maintain system integrity while incorporating highly variable renewable energy. This paper uses robust optimization to set time-coupled responses to prediction errors; a case study reports lower costs when reserve policies are enabled.

  • Problem

    Power systems need to maintain integrity while accommodating highly variable renewable energy sources.

    Source: §1 · p. 1

  • Method

    Robust optimization is used to find efficient, time-coupled responses to prediction errors in power-system operation.

    Source: §1 · p. 2, §6 · p. 20

  • Results

    For a given finite-horizon optimization with coupled generation costs, costs were reduced when reserve policies were enabled; a case study demonstrated operating benefits.

    Source: §6 · p. 20

  • Takeaways & Limitations

    Policy-based reserves are proposed as products that could be bought or sold on power markets, with associated prices.

    Source: §6 · p. 20

    It may be impossible to find causal policies that provide the required robustness at costs approaching the stated value, so the reported savings are limited by that boundary.

    Source: §6 · p. 21

Abstract

from arXiv · show

This paper introduces the concept of affine reserve policies for accommodating large, fluctuating renewable infeeds in power systems. The approach uses robust optimization with recourse to determine operating rules for power system entities such as generators and storage units. These rules, or policies, establish several hours in advance how these entities are to respond to errors in the prediction of loads and renewable infeeds once their values are discovered. Affine policies consist of a nominal power schedule plus a series of planned linear modifications that depend on the prediction errors that will become known at future times. We describe how to choose optimal affine policies that respect the power network constraints, namely matching supply and demand, respecting transmission line ratings, and the local operating limits of power system entities, for all realizations of the prediction errors. Crucially, these policies are time-coupled, exploiting the spatial and temporal correlation of these prediction errors. Affine policies are compared with existing reserve operation under standard modelling assumptions, and operating cost reductions are reported for a multi-day benchmark study featuring a poorly-predicted wind infeed. Efficient prices for such "policy-based reserves" are derived, and we propose new reserve products that could be traded on electricity markets.

1 Introduction

The introduction proposes time-coupled reserve policies that let power-system entities respond to forecast errors, addressing the challenge of maintaining system integrity while using highly variable renewable energy. It reports lower reserve operating costs than rules that do not use policies.

1 Introduction

  • The challenge is maintaining system integrity while making best use of energy from highly variable intermittent renewables.Source: §1 · p. 1
  • The paper proposes optimal time-coupled policies: advance-agreed rules for how power-system entities respond to prediction errors under bounded uncertainty.The uncertainty bounds are chosen by probabilistic criteria or arise inherently, such as from wind-farm capacities.Source: §1 · p. 2
  • Reserve operating costs fall with policies under large uncertainty, compared with reserve rules that do not use them.The comparison is stated for operating reserves in the presence of large uncertainty.Source: §1 · p. 2
  • These policies use future wind prediction-error measurements systematically to reduce average reserve-provision costs.Source: §1 · p. 3

Summary of notation by theme

2 Power system model

This section models power-system operation over a finite planning horizon, representing uncertain renewable infeeds and fluctuating loads alongside participant dynamics, costs, operating constraints, and a linearized transmission network. It specifies feasibility through supply-demand balance, participant limits, and transmission constraints expressed as phase-angle-difference limits between adjacent buses.

2 Power system model

  • The planning horizon has T discrete trading steps and extends up to 24 hours, beyond which renewable predictions are assumed too poor for sophisticated decisions.The steps correspond to intra-day market intervals of 5 minutes to 1 hour.Source: §2 · p. 5

2.1 Participant model

  • Participants are generators, loads, or storage units connected at fixed network locations, with inelastic flows and controllable elastic flows.Each participant may have either or both flow types.Source: §2.1 · p. 5

Inelastic power flows

  • Inelastic power flows combine a nominal prediction with a linear mapping of future forecast errors, which lie in a compact set with known first and second moments.No other restrictions are imposed on the forecast-error probability distribution.Source: Inelastic power flows · p. 5

Elastic power flows

  • Elastic flows follow linear time-invariant participant dynamics, with control inputs determining future states and network-visible power injections.Participant state can also encode internal dynamics or memory of previous states.Source: Elastic power flows · p. 6

Costs

  • Participant costs are modeled as quadratic functions of state and input, with positive semidefinite Hessians so the resulting optimization problem is convex.Coupling costs across time steps can be represented by augmenting the state with memory of prior states.Source: Costs · p. 6

Constraints

  • Participant operating limits are compact polyhedral constraints that can couple state and input trajectories, including generator ramp rates and storage empty/full limits.Generator start-up and shut-down decisions are excluded because unit commitment is assumed to have been decided earlier.Source: Constraints · p. 7

2.2 Network model

  • The network uses a linearized high-voltage grid model and imposes zero net injection at each time and line-flow limits in both directions.The model assumes lossless lines, constant voltage magnitudes, and small phase differences.Source: §2.2 · pp. 7–8
  • Transmission-line constraints limit phase-angle differences between adjacent buses.This modeling choice is used in the derivation of equation (III.4) in.Source: §2.2 · p. 8

3 Choosing optimal reserve policies

Existing reserves respond to prediction errors as they unfold, without systematically using expected future conditions. The section formulates predictive reserve policies and tractable optimization methods, while noting their computational growth with the planning horizon.

3 Choosing optimal reserve policies

  • Predictive reserve policies account for short-term uncertainties to reduce expected running costs over the horizon, addressing the time-coupled costs and constraints overlooked by existing reserve actions.Existing primary, secondary, and tertiary mechanisms respond to errors as they unfold and do not systematically use what is expected to happen in the future.Source: §3 · p. 8

3.1 Finite horizon optimization

  • Each participant’s schedule combines a nominal plan with linear adjustments to prediction errors, and its lower-triangular response matrix prevents dependence on errors not yet known.The policy is chosen before errors are known; the optimization seeks causal responses subject to local and network constraints.Source: §3.1 · pp. 9–10
  • With positive-semidefinite quadratic costs, expected policy costs can be represented convexly using only the first two moments of prediction errors.The stated assumption allows the expectation to be minimized using E[δ] and E[δδ′].Source: §3.1 · p. 10

3.2 Equivalent tractable reformulation

  • The robust constraints, required to hold for every error in the uncertainty set, are reformulated into a finite tractable optimization using auxiliary variables and strong duality.The resulting constraints encode nominal schedule tracking, response to error vectors, and transmission-line current limits.Source: §3.2 · pp. 11–12

3.3 Computational requirements

  • The optimization’s decision-variable count grows quadratically with the horizon through the response matrices, whereas the compared existing-mechanism formulation grows linearly.The paper notes that thousands of variables and constraints arise even for modest parameter choices.Source: §3.3 · p. 12

4 Implementation of policy-based reserves

The section derives efficient payments for policy-based reserves and describes possible traded products and real-time implementation. It also distinguishes the opportunities and trade-offs of batchwise and receding-horizon policy updates.

4.1 Efficient prices for policies

  • Efficient market prices comprise a nodal power price and a matrix of reserve-policy prices, whose entries give the marginal value of each response-matrix element.The nodal price has a global component and a local component induced by line congestion; reserve-policy prices exhibit the same locational variation as LMPs.Source: §4.1 · pp. 13–14

4.2 Policy-based reserve products

  • A participant commits to a nominal schedule for accurate predictions and a policy specifying how its output responds to realized errors.The policy matrix can be read row-wise as the rule for power at a given step or column-wise as an a posteriori response to an error.Source: §4.2 · p. 14

Example product

  • A reserve product can sell the agreed row-wise response with payments based on each response coefficient; a column-wise product would instead sell an a posteriori response to errors.Scheduled power is paid separately using the nominal-power price.Source: Example product · p. 14

4.3 Real time operation

  • Closed-loop operation can update policies batchwise or in a receding horizon, with only the first step played before an immediate update in the latter scheme.A rolling adjustment is motivated by prediction-error bounds changing over the horizon as new forecast information arrives.Source: §4.3 · pp. 15–16
  • Shifting prior policies into each new optimization can preserve earlier commitments, provided uncertainty bounds grow monotonically along the prediction horizon.The feasibility statement is given without proof in the section.Source: §4.3 · p. 15
  • Renewing policies at every step allows updates using new information, but only the first-row policy entries are used, leaving later contracted responses uncalled upon.The text notes that later-step costs and constraints can still affect the chosen first-step schedule and response.Source: §4.3 · p. 15

5 Numerical case study

The numerical case study evaluates policy-based reserves under wind uncertainty, comparing reserve costs and examining sensitivity to wind capacity. It reports savings from two-step and time-coupled policies as wind share grows, as well as infeasibility above a specified scaling factor when wind is not curtailed.

5 Numerical case study

  • A modified 39-bus network with 7 thermal generators, 2 storage units, 19 loads, and 3 wind farms was simulated for three days across 50 wind realizations.The network replaced three original generators with wind farms.Source: §5 · p. 16

5.1 Uncertainty model

  • Uncertainty came from random wind availability while loads were assumed perfectly predicted; the three wind farms were driven by a temporally and spatially correlated process with lower dimension than the affected nodes.The lower-dimensional process can reduce computational effort or reflect uncertainty that is difficult to model.Source: §5.1 · pp. 17–18

5.2 Comparison of reserve costs

  • 38.4%: full policies (Scheme 3) reduced reserve costs versus the best possible non-recourse reserve scheduler (Scheme 2); additional total power generation costs reached up to 1.10% under Scheme 2 and 0.73% under Scheme 3.Average costs under Scheme 1 were 4.474 × 10^7 for the three-day test.Source: §5.2 · p. 19
  • 32.4%: a two-step policy saved this average share of reserve costs, compared with 38.4% for full policies, so most full-policy savings came from the two-step policy.The two-step policy allowed only diagonal and immediate subdiagonal entries in the D-matrices to differ from zero, reducing optimization variables relative to Scheme 3.Source: §5.2 · p. 19

5.3 Sensitivity analysis

  • Wind uncertainty was scaled by φ from 0.2 to 1.2, with variance scaled by φ^2; for φ > 1.2 the model became infeasible because wind power was not curtailed.The expected proportion of load energy supplied by wind was φ · 29.0%.Source: §5.3 · p. 19
  • The additional percentage cost of accommodating uncertainty increased sharply, apparently more than quadratically, as wind capacity increased.Source: §5.3 · p. 20
  • Time-coupled policies offset around 40% of this additional cost as the wind share grew.Source: §5.3 · p. 20
  • Although percentage savings decreased slightly as φ increased, absolute savings still grew quadratically; the large saving at φ = 0.2 was measured relative to a tiny number.Source: §5.3 · p. 20

6 Conclusions

The paper formulates policy-based reserves as robust, time-coupled responses to uncertain supply or load over a finite horizon, and develops their market interpretation. Its case study reports cost reductions, while the authors note that reported savings are lower bounds and practical AC-grid feasibility requires additional checks.

6 Conclusions

  • The approach uses robust optimization to choose time-coupled responses to uncertain load or supply over a finite horizon.The paper develops policy-based reserves as products that could be bought or sold with associated prices, and reports a case study of their operating principles and benefits.Source: §6 · p. 20
  • Potential savings depend on the time coupling of participant costs, constraints, and prediction errors, including how correlated errors affect policy-matrix contributions to expected costs.The paper identifies spatial and temporal prediction-error correlation as relevant to the expected-cost contributions of entries in Di.Source: §6 · p. 20
  • For a given finite-horizon optimization, costs were reduced when reserve policies were enabled.The authors relate this result to prediction-error correlation and time-coupled generation costs.Source: §6 · p. 20
  • Cost savings remain under receding-horizon operation, and much of the reduction is available without using all subdiagonal entries in policy matrices Di.The authors report savings when optimization is repeated in receding-horizon fashion.Source: §6 · p. 21
  • Reported savings are lower bounds because causal policies that provide the required robustness may not attain the lower-bound cost.The paper states that Scheme 1 is a lower bound on costs achievable under receding-horizon schemes.Source: §6 · p. 21
  • The optimality of reserve policies depends on the current power-system state, but the relationship is not straightforward.Source: §6 · p. 21
  • The tractable formulation uses standard power-system simplifications, so real implementation needs additional checks to confirm AC-grid flow feasibility.The paper notes that these checks may require iteration and resemble current system-operator practice.Source: §6 · p. 21
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