Bilevel Transmission Expansion Planning with Joint Chance-Constrained Dispatch

TL;DR

This work develops a bilevel transmission expansion planning model that integrates a lower-level joint chance-constrained dispatch under wind uncertainty with upper-level investment and tariff decisions. To solve the challenging RHS-WDRJCC, it introduces Strengthened Linear Approximation (SLA), an inner convex approximation that remains non-conservative and numerically stable, enabling reformulation to a single-level MPEC via KKT conditions. The approach yields significant computational gains (up to 26x speedups) while preserving out-of-sample reliability; case-study results confirm robust investment strategies under varying risk levels and Wasserstein radii. By explicitly modelling network tariffs and revenue adequacy, the framework provides a practical mechanism for robust grid expansion planning in renewable-rich systems, with implications for tariff design and market-clearing practices.

Abstract

In transmission expansion planning (TEP), network planners make long-term investment decisions while anticipating market clearing outcomes that are increasingly affected by renewable generation uncertainty. Additionally, market participants' sensitivity to network charges and the requirement for cost recovery by the network planner introduce further complexity. Since the day-ahead market clears before uncertainty realizes, explicitly modelling these uncertainties at the lower-level market clearing becomes important in bilevel TEP problems. In this paper, we introduce a novel bilevel TEP framework with lower-level joint chance-constrained market clearing that manages line flow constraints under wind uncertainty and accounts for the effect of network tariffs on participants' actual marginal costs and utility. To solve this complex problem, we propose a Strengthened Linear Approximation (SLA) technique for handling Wasserstein distributionally robust joint chance constraints with right-hand-side uncertainties (RHS-WDRJCC). The proposed method offers more efficient approximations without additional conservativeness and avoids the numerical issues encountered in existing approaches by introducing valid inequalities. The case study demonstrates that the proposed model achieves the desired out-of-sample constraint satisfaction probability. Moreover, the numerical results highlight the significant computational advantage of SLA, achieving up to a 26x speedup compared to existing methods such as worst-case conditional value-at-risk, while maintaining high solution quality.

Figures (7)

• Figure 1: Bilevel structure of the proposed formulation for optimal network planning. The upper-level problem represents the network planner, who determines investment decisions on reconductoring lines ($b^R_{t,l,j}$, $z^R_{t,l}$), parallel lines ($z^P_{t,l,m}$), and sets network charges—both volumetric-based ($\tau^V_l$) and capacity-based ($\tau^C$). Investment decisions affect the transmission capacity and the system susceptance matrix as the network topology changes. The volumetric-based charges $\tau^V_l$ also influence market participants’ marginal costs and willingness to pay, thereby affecting the outcomes of the lower-level market clearing problem. The lower-level problem is formulated as a RHS-WDRJCC problem to account for wind uncertainty. It determines cleared quantities ($g_{t,s,k,b}$, $d_{t,s,k,b}$, $p^{w,sch}_{t,s,k,b}$) and market prices ($\pi_{t,s,b}$). These outcomes are then used by the network planner to compute revenues, which consist of merchandising surplus and network charges, and are compared against investment costs to ensure revenue adequacy for the network planner.
• Figure 2: Impact of risk level $\epsilon$ and Wasserstein radius $\theta$ on out-of-sample reliability analysis. The x-axis represents the desired reliability $1-\epsilon$$\{70\%,80\%,90\%,95\%,97.5\%,99\%\}$, while the y‐axis shows the out‐of‐sample probability that constraints are jointly satisfied. (a)--(d) shows the the out-of-sample reliability results for year 1 to year 4. The black dashed lines represent the desired reliability levels $1-\epsilon$ and the black solid lines represent the maximum $100\%$ reliability levels.
• Figure 3: Computational performance comparison for the proposed problem under $(\epsilon,\theta) = (0.025,0.05)$, $T\in\{2,4\}$. For Time (s) and TimeF (s) plots, dots represent the mean values of the $30$ random runs, with error bars representing the $95\%$ percentile interval (from 2.5th to 97.5th) of 30 randomly generated instances. In contrast, error bars in the Obj. Diff. (M£) plot represent $95\%$ percentile interval of runs for which the proposed SLA and benchmark are both solvable/feasible. The black horizontal line in the Time (s) and TimeF (s) plots represents the $14,400$s TimeLimit. The black horizontal line in the Obj. Diff. (M£) plots indicates zero difference, while negative values denote a lower objective value (social welfare) achieved by the benchmark method compared to the proposed SLA within the TimeLimit.
• Figure 4: The derivation flow and the formulation comparisons of the $\bm{x}$-feasible region defined by different reformulation of RHS-WDRJCC \ref{['constr:wdrjcc']}.
• Figure 5: Topology of the modified Garver’s 6-node transmission network.

Theorems & Definitions (10)

• Theorem 1
• Lemma 1: Lemma 1 in exact_milp_strengthened
• Lemma 2: Lemma 3 in zhou2024strengthenedfasterlinearapproximation
• Theorem 2
• Proposition 1
• Corollary 1
• Corollary 2
• Corollary 3
• Corollary 4
• Proposition 2