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Gradient Methods for Scalable Multi-value Electricity Network Expansion Planning

Anthony Degleris, Abbas El Gamal, Ram Rajagopal

TL;DR

This paper addresses scalable grid expansion planning when the planner’s objectives extend beyond cost minimization to emissions and other market outcomes. It casts expansion planning as a bilevel problem (planner vs. market operator) and introduces multi-value expansion planning (MEP), which optimizes arbitrary functions of dispatch outcomes via the implicit form of the lower-level problem. A fast stochastic implicit-gradient descent algorithm is developed, leveraging strong duality and McCormick relaxations to bound performance and provide good initialization; the method scales linearly with network size and benefits from parallelized scenario computations. Empirical results on a large Western Interconnect model show gradient descent achieving substantial speedups over interior-point methods and producing meaningful trade-offs, such as a 40% reduction in carbon intensity at modest extra cost when emissions penalization is applied. The framework is flexible, extendable to dynamic, stochastic, and multi-resource settings, and supported by open-source software and realistic datasets.

Abstract

We consider multi-value expansion planning (MEP), a general bilevel optimization model in which a planner optimizes arbitrary functions of the dispatch outcome in the presence of a partially controllable, competitive electricity market. The MEP problem can be used to jointly plan various grid assets, such as transmission, generation, and battery storage capacities; examples include identifying grid investments that minimize emissions in the absence of a carbon tax, maximizing the profit of a portfolio of renewable investments and long-term energy contracts, or reducing price inequities between different grid stakeholders. The MEP problem, however, is in general nonconvex, making it difficult to solve exactly for large real-world systems. Therefore, we propose a fast stochastic implicit gradient-based heuristic method that scales well to large networks with many scenarios. We use a strong duality reformulation and the McCormick envelope to provide a lower bound on the performance of our algorithm via convex relaxation. We test the performance of our method on a large model of the U.S. Western Interconnect and demonstrate that it scales linearly with network size and number of scenarios and can be efficiently parallelized on large machines. We find that for medium-sized 16 hour cases, gradient descent on average finds a 5.3x lower objective value in 16.5x less time compared to a traditional reformulation-based approach solved with an interior point method. We conclude with a large example in which we jointly plan transmission, generation, and storage for a 768 hour case on 100 node system, showing that emissions penalization leads to additional 40.0% reduction in carbon intensity at an additional cost of $17.1/MWh.

Gradient Methods for Scalable Multi-value Electricity Network Expansion Planning

TL;DR

This paper addresses scalable grid expansion planning when the planner’s objectives extend beyond cost minimization to emissions and other market outcomes. It casts expansion planning as a bilevel problem (planner vs. market operator) and introduces multi-value expansion planning (MEP), which optimizes arbitrary functions of dispatch outcomes via the implicit form of the lower-level problem. A fast stochastic implicit-gradient descent algorithm is developed, leveraging strong duality and McCormick relaxations to bound performance and provide good initialization; the method scales linearly with network size and benefits from parallelized scenario computations. Empirical results on a large Western Interconnect model show gradient descent achieving substantial speedups over interior-point methods and producing meaningful trade-offs, such as a 40% reduction in carbon intensity at modest extra cost when emissions penalization is applied. The framework is flexible, extendable to dynamic, stochastic, and multi-resource settings, and supported by open-source software and realistic datasets.

Abstract

We consider multi-value expansion planning (MEP), a general bilevel optimization model in which a planner optimizes arbitrary functions of the dispatch outcome in the presence of a partially controllable, competitive electricity market. The MEP problem can be used to jointly plan various grid assets, such as transmission, generation, and battery storage capacities; examples include identifying grid investments that minimize emissions in the absence of a carbon tax, maximizing the profit of a portfolio of renewable investments and long-term energy contracts, or reducing price inequities between different grid stakeholders. The MEP problem, however, is in general nonconvex, making it difficult to solve exactly for large real-world systems. Therefore, we propose a fast stochastic implicit gradient-based heuristic method that scales well to large networks with many scenarios. We use a strong duality reformulation and the McCormick envelope to provide a lower bound on the performance of our algorithm via convex relaxation. We test the performance of our method on a large model of the U.S. Western Interconnect and demonstrate that it scales linearly with network size and number of scenarios and can be efficiently parallelized on large machines. We find that for medium-sized 16 hour cases, gradient descent on average finds a 5.3x lower objective value in 16.5x less time compared to a traditional reformulation-based approach solved with an interior point method. We conclude with a large example in which we jointly plan transmission, generation, and storage for a 768 hour case on 100 node system, showing that emissions penalization leads to additional 40.0% reduction in carbon intensity at an additional cost of $17.1/MWh.
Paper Structure (52 sections, 19 theorems, 80 equations, 11 figures, 2 algorithms)

This paper contains 52 sections, 19 theorems, 80 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Bilevel (market-based) expansion planning.
  4. Solution algorithms.
  5. Multi-value expansion planning
  6. Dispatch model
  7. Dispatch map.
  8. Dual problem.
  9. Discussion of assumptions.
  10. Expansion planning
  11. Bilevel form.
  12. Multi-value expansion planning
  13. Stochastic multi-value expansion planning
  14. Illustrative examples
  15. Dispatch model.
  16. ...and 37 more sections

Key Result

Theorem 1

Consider a continuously differentiable function $\kappa : \mathbf{R}^D \times \mathbf{R}^K \rightarrow \mathbf{R}^D$. Fix $\eta_0 \in \mathbf{R}^K$ and suppose there exists a unique point $z_0 \in \mathbf{R}^D$ such that $\kappa(z_0, \eta_0) = 0$. Then there is a function $z^*(\eta) : \Omega \righta

Figures (11)

  • Figure 1: Illustrative examples of multi-value expansion planning. (A) Initial network before transmission expansion. (B) Transmission expansion (blue lines) that minimizes total cost. (C) Expansion that minimizes emissions (i.e., emissions weight $w = \infty$). (D) Expansion that maximizes the price at node 6.
  • Figure 2: Trade-off between total cost (investment cost plus dispatch cost) and emissions for Problem \ref{['eq:example-emissions']} using the 6-node network in Figure \ref{['fig:sixbus-nets']}. (Black dots) Solutions generated by varying the emissions weight $w \in [0, 10]$ and solving the resulting multi-value expansion planning problem using Algorithm \ref{['alg:gradient-descent']}. (Green line) Pareto frontier for the cost-emissions trade-off. (Red dots) The cost-emissions profile of the networks in Figure \ref{['fig:sixbus-nets']}, Panels B and C.
  • Figure 3: McCormick envelope of the constraint $\phi = \alpha \beta$ over the box $(\alpha, \beta) \in [1, 3] \times [-2, 2]$. projected onto the plane $\{ (\alpha, \beta, \phi) : \phi = 1.5 \}$. (Black line) The nonconvex feasible set. (Gray dashed lines) Linear inequalities defined by convex and concave (McCormick) envelopes of $f(\alpha, \beta) = \alpha \beta$ over the boxed region. (Gray shaded region) Convexified feasible set.
  • Figure 4: Convergence results for gradient descent with varying step sizes. (Dashed black line) Lower bound produced by the relaxation from Section \ref{['sec:relax']}. (Solid black line) Initial choice of the network parameter, produced by the same relaxation. The width of the line is the time to solve the relaxation. (Solid colored lines) Loss curves for gradient descent. Vertical dashed lines mark every 250th iteration.
  • Figure 5: Performance comparison between a local interior point solver (IP), gradient descent (GD), and the solution to the relaxation in Section \ref{['sec:relax']} (RL). (Left side) Box plots of final objective values (divided by the lower bound) for different cases with 1, 4, or 16 scenarios. Box bottom and top lines are the 25th and 75th percentiles, respectively, and midlines are medians. (Right side) Box plot of algorithm runtimes for different cases. For visibility, box bottom and top lines are set to be 10 minutes apart if the IQR is less than 10 minutes.
  • ...and 6 more figures

Theorems & Definitions (41)

  • Theorem 1: Implicit Function Theorem, Dontchev2014-ck, Chapter 1B
  • Theorem 2: Gradient descent convergence
  • proof
  • Definition : $\epsilon$-approximation
  • Lemma 3
  • proof
  • Definition : Convex envelope
  • Lemma 4: McCormick envelope, Al-Khayyal1983-io
  • Remark 1
  • Theorem 5: Tight integer relaxation
  • ...and 31 more