Learning to Optimize meets Neural-ODE: Real-Time, Stability-Constrained AC OPF

The AC Optimal Power Flow (AC-OPF) problem plays a fundamental role in power systems as it determines the optimal generator dispatch to meet demands while adhering to physical and engineering constraints. Traditionally, OPF problem instances are solved on timescales spanning 5 minutes to 1 hour, and often utilize linear approximations which do not satisfy steady-state AC power flow [1]. Additionally, the stochastic nature of renewable energy sources has increased uncertainty, necessitating more accurate power flow representations, and more frequent adjustments by system operators to preserve system stability [2]. Currently, these operations typically rely on heuristics for maintaining stable frequencies and voltages of the electrical generators, which lead to inefficiencies and higher losses. Rapid fluctuations of intermittent renewable generation, for example, can lead to curtailments as well as increasing emissions [3]. Similarly, governor droop control, which uses proportional control rules to modify the power generator based on frequency imbalances, introduces additional inefficiencies, losses, and emissions [4].

There are two key challenges within this setting. The first is the complexity of solving the full AC-OPF problem, which limits the frequency of these operations, and the second is the dynamic nature of the system, which impacts global system stability [5]. Indeed, relying solely on steady-state models for power dispatching operations is insufficient to guarantee compliance with system dynamic requirements. Nonetheless, adhering to these dynamics is not only computationally demanding on its own, but fully incorporating them within the AC-OPF model would introduce complex interdependencies among the problem variables, significantly increasing the computational complexity of an already challenging problem.

In an effort to address the first challenge, there has been recent interest in applying machine learning (ML) based models to learn solutions mappings from load conditions to AC-OPF solutions [6]. These approaches have demonstrated enormous potential for replacing traditional numerical solvers to approximate complex supply/demand balance problems in near real-time. Central to these advancements is the concept of Learning to Optimize (LtO), [7] where neural network-based models serve as proxies for traditional numerical solvers to produce near-optimal solutions for parametric constrained optimization problems. However, while LtO approaches have shown promising results in solving the AC-OPF problem with high fidelity in real-time [8, 9], they address the steady-state problem, and may not be suitable to capture the dynamics of the system. In Section 6 the paper will show that, on the benchmark systems analyzed, many of the solutions generated by state-of-the-art LtO methods for AC-OPF violate the dynamic requirements of a synchronous electrical generator, which is unacceptable in practice.

To overcome these challenges, this paper proposes Dynamic OPF-Net (DynOPF-Net), a learning-based model that combines Learning to Optimize with ML-based dynamic predictors to solve real-time AC OPF problems while also addressing the dynamic stability of the generators. A key component of DynOPF-Net is the integration of neural ODEs (NODEs) [10] within the Learning to Optimize framework. NODEs approximate the continuous dynamics of systems through neural networks, allowing efficient modeling of system behaviors that evolve over time. The paper shows how this integration is able to produce near-optimal and stable solutions for a variety of operational conditions.

Contributions. This paper offers the following key contributions: (1) We introduce Dynamic OPF-Net (DynOPF-Net), a novel framework that seamlessly integrates machine learning with optimization techniques to incorporate generator dynamics directly into the AC-OPF model. (2) We provide empirical evidences which highlight the critical need to model the system dynamics within the OPF framework. Specifically, we demonstrate that existing LtO methods, which overlook these dynamics, systematically fail to satisfy stability requirements. In contrast, DynOPF-Net consistently generates solutions that adhere to these dynamic constraints, helping address stability. (3) We show that DynOPF-Net not only achieves a decision quality comparable to that of state-of-the-art LtO methods that address only steady-state constraints but also uniquely ensures compliance with dynamic requirements. This is achieved without sacrificing computational efficiency, offering the first solution, to our knowledge, for real-time power system operations under dynamic conditions.

This section introduces the stability-constrained AC OPF problem, starting from two key components: the (steady state) AC OPF problem and the synchronous generator dynamics. The paper adopts the following notation: lowercase symbols are used for scalars, bold symbols represent vectors, and uppercase symbols represent complex variables, denoted either in rectangular or polar form. Sets are denoted with standard calligraphic symbols (e.g., $\pazocal{X}=\{x_{1},\dots,x_{n}\}$), and special calligraphic symbols are reserved for models, such as a deep neural network parameterized by vector $\bm{\phi}$, denoted as $\mathscr{X}_{\bm{\phi}}$.

AC Optimal Power Flow. The AC OPF problem determines the most cost-effective generator dispatch that satisfies demand within a power network subject to various physical constraints. Typically, the setting focuses on a snapshot of the OPF problem in time. In a power network, represented as a graph $(\pazocal{N,L})$, the node set $\pazocal{N}$ consists of $n$ buses, and the edge set $\pazocal{L}$ comprises $l$ lines. Additionally, $\pazocal{G}$ is used to represent the set of synchronous generators in the system. The AC power flow equations use complex numbers for current $I$, voltage $V$, admittance $Y$, and power $S$, interconnected through various constraints. The power generation and demand at a bus $i\in\pazocal{N}$ are represented by $S^{r}_{i}=p_{i}^{r}+jq_{i}^{r}$ and $S^{d}_{i}=p_{i}^{d}+jq_{i}^{d}$, respectively, while the power flow across line $ij$ is denoted by $S_{ij}$, and $\theta_{i}$ describes the phase angles at bus $i\in\pazocal{N}$. The Kirchhoff’s Current Law (KCL) is represented by $I^{r}_{i}-{I^{d}_{i}}=\sum_{(i,j)\in\pazocal{L}}I_{ij}$, Ohm’s Law by $I_{ij}={Y}_{ij}(V_{i}-V_{j})$, and AC power flow denoted $S_{ij}=V_{i}I_{ij}^{\ast}$. These principles form the AC Power Flow equations, described by  equation 1f and equation 1g in Model 1. The goal is to minimize a function equation 1a representing dispatch costs for each generator. Constraints equation 1b-equation 1c represents voltage operational limits to bound voltage magnitudes and phase angle differences, while equation 1d-equation 1e set boundaries for generator output and line flow. Constraint equation 1h sets the reference phase angle. Finally, constraints equation 1f and equation 1g enforce KCL and Ohm’s Law, respectively.

Synchronous Generator Model. To capture the dynamic behavior of synchronous generators with high fidelity, this study considers the classical “machine model” [25], a simplified version of the “two-axis model”. The dynamics of a synchronous generator $g\in\pazocal{G}$ are defined as:

where stator currents $i_{d}^{g}(t)$ and $i_{q}^{g}(t)$ in the reference frame of machine $g\in\pazocal{G}$, are computed as:

For a generator $g\in\pazocal{G}$, the state vector $\begin{bmatrix}e^{\prime g}_{q}&e^{\prime g}_{d}&\delta^{g}&\omega^{g}\end{bmatrix}^{\top}$ describes its voltage components, $e^{\prime g}_{q}$ and $e^{\prime g}_{d}$, associated with the ‘d’ and ‘q’ axis (dq-axes of a reference frame for $g$), its rotor angle $\delta^{g}$, and its angular frequency $\omega^{g}$. The synchronous angular frequency is denoted as $\omega_{s}$ in Equation equation 2. The mechanical power $p_{m}^{g}$ from the shaft turbine serves as the control input. Constitutive machine parameters are given by $\begin{bmatrix}x^{\prime g}_{d}&m^{g}&d^{g}\end{bmatrix}^{T}$, where $x^{\prime g}_{d}$ denotes the the transient reactance, $m^{g}$ the machine’s inertia constant and $d^{g}$ the damping coefficient. Importantly, stator currents are connected to the terminal voltage magnitude $|V_{g}|$ and phase angle $\theta_{g}$ via equation 3. In this model, voltage components $e^{\prime g}_{d}(t)$ and $e^{\prime g}_{q}(t)$ are held constant, meaning the update function for these states is zero. The interested reader is referred to [25] for a more in-depth discussion of the classical machine model. Given the assumptions on the voltage $e^{\prime g}_{d}$ and $e^{\prime g}_{q}$, the dynamic model of synchronous generators results in:

Initial Values of Rotor Angles and EMF Magnitudes. For each generator $g\in\pazocal{G}$, the initial values of rotor angle $\delta^{g}(0)$ and electromotive force (EMF) $e^{\prime g}_{q}(0)$ are derived from the active and reactive power equations, assuming the dynamical system equation 4 initially being in a steady state condition, $\frac{d}{dt}\begin{bmatrix}\delta^{g}(t)&\omega^{g}(t)\end{bmatrix}_{t=0}^{T}=\begin{bmatrix}0&0\end{bmatrix}^{T}$, from which:

Following the same assumption, it follows that

Stability Limit. To guarantee stability of a synchronous generator $g\in\pazocal{G}$, the rotor angle $\delta^{g}(t)$ is required to remain below an instability threshold $\delta^{\max}$, as defined by SIngle Machine Equivalent (SIME):

Unstable conditions arise when violating equation 8, which is the principal binding constraint that necessitates re-dispatching.

Stability-Constrained OPF. The stability-constrained OPF problem is thus formulated in Model 2. This problem determines the optimal power dispatch subject to physical, engineering, and dynamic stability constraints equation 9b-equation 9h. Given a load demand $\bm{S}^{d}$ , the objective is to find the OPF variables $\bm{S}^{r},\bm{V}$ that minimize the cost function equation 9a while satisfying the constraints equation 9b-equation 9h. The inclusion of the dynamic behavior of synchronous generators complicates the problem due to the coupling between the OPF variables $S_{g}^{r},V_{g}$ and the generator state variables $\delta^{g}(t),\omega^{g}(t)$, as shown in equation 9d-equation 9f. Additionally, the non-linear nature of the generator model and the time dependencies of the state variables further increase the complexity. As a result, traditional numerical optimization algorithms are unable to handle, in a computationally viable way, the stability-constrained AC-OPF. To address this challenge, we develop a learning-based dual framework integrating NODEs with LtO models.

Given a load profile $\bm{S}^{d}=(\bm{p}^{d},\bm{q}^{d})$, the goal is to predict the generators set-points that minimize the objective equation 9a while simultaneously satisfying the operational and dynamic constraints equation 9b-equation 9h of Problem 2. A significant challenge in this learning task is ensuring the satisfaction of both steady-state equation 9b and dynamic constraints equation 9h. To address this challenge, we propose Dynamic OPF-Net (DynOPF-Net), a dual neural network architecture consisting of a learning-to-optimize model (cyan network in Fig. 1) and a collection of NODE models, one for each generator, as illustrated in the orange networks of Fig. 1. The LtO model, denoted as $\mathscr{D}_{\psi}$, where $\psi$ represents its parameters, predicts the OPF variables $\bm{S}^{r}=(\bm{p}^{r},\bm{q}^{r})$ and $\bm{V}$. Each NODE model $\mathscr{N}^{g}_{\phi}$, parametrized by $\phi$, captures the system dynamics of the corresponding generator $g\in\pazocal{G}$. Both the steady-state and dynamic constraints are integrated into the learning task using a Lagrangian Dual Learning framework [8] as reviewed in Section 4.2.

The effectiveness of DynOPF is tested on two power networks of different sizes and complexity: the WSCC 9-bus system, and the IEEE 57-bus system [26]. Our approach is benchmarked against $3$ widely adopted LtO methods approximating the AC-OPF problem 1 solutions $\bm{y}=(\bm{S}^{r},\bm{V})$ given $\bm{S^{d}}$. With reference to equation 13, equation 14, equation 15, they use the following loss functions for model training.

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  Zamzam et. al [17] uses a loss function:

  $$\begin{cases}\pazocal{L}_{p}(\bm{y},\hat{\bm{y}})=\|\hat{\bm{y}}-\bm{y}^{\star}\|^{2}\\ \pazocal{L}_{c}(\hat{\bm{y}},\hat{\bm{x}}(t))=0,\end{cases}$$

  which minimizes the mean squared error (MSE) between the predicted solution and its corresponding label.

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  Lagrangian-Dual [8] uses a Lagrangian loss function:

  $$\begin{cases}\pazocal{L}_{p}(\bm{y},\hat{\bm{y}})=\|\hat{\bm{y}}-\bm{y}^{\star}\|^{2}\\ \begin{aligned} \pazocal{L}_{c}(\hat{\bm{y}},\hat{\bm{x}}(t))=\sum_{j=1}^{n^{\prime}_{eq}}{\lambda}_{h_{j}}\nu\left(h_{j}(\hat{\bm{y}},\hat{\bm{x}}(t))\right)+\sum_{l=1}^{n^{\prime}_{ineq}}{\lambda}_{u_{l}}\nu\left(u_{l}(\hat{\bm{y}},\hat{\bm{x}}(t))\right),\end{aligned}\end{cases}$$

  where $n^{\prime}_{eq}$, $n^{\prime}_{ineq}$ denotes the number of static equality and inequality constraints of model 2, specified by functions $h_{j},\;j=1,\ldots,n^{\prime}_{eq}$ and $u_{l},\;l=1,\ldots,n^{\prime}_{ineq}$ equation 9b.

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  DC3 [9] uses the loss function:

  $$\begin{cases}\pazocal{L}_{p}(\bm{y},\hat{\bm{y}})=f(\hat{\bm{y}})\\ \begin{aligned} \pazocal{L}_{c}(\hat{\bm{y}},\hat{\bm{x}}(t))={\lambda}_{h}\sum_{j=1}^{n^{\prime}_{eq}}\|h_{j}(\hat{\bm{y}},\hat{\bm{x}}(t))\|^{2}+{\lambda}_{u}\sum_{l=1}^{n^{\prime}_{ineq}}\max(0,u_{l}(\hat{\bm{y}},\hat{\bm{x}}(t))^{2}),\end{aligned}\end{cases}$$

  where $f$ is objective function equation 9a. This method relies on a completion-correction technique to enforce constraints equation 9b satisfaction of Model 2, in self-supervised training.

The rest of this section describes the training setting.

This section presents the results of each benchmark system of DynOPF-Net and each baseline LtO method. The experiments focus on two main aspects: (1) assessing the effectiveness of NODEs and PINNs in capturing generator dynamics and comparing their precision and computational efficiency to a numerical ODE solver; (2) comparing DynOPF-Net with LtO methods that capture only the steady-state AC-OPF problem, focusing on the stability constraint violations. Unless otherwise stated, results are reported as an average of 40 independent runs on a subset of dataset $\pazocal{D}$ where DynOPF-Net and each LtO model is not trained on, which we refer to as the test set. Specifically, we report:

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  The inference time (measured in seconds) of NODEs which we compare to the computational time of a traditional numerical ODE solver and the precision of state variables’ estimate (measured as the percentage relative $\ell^{2}$ error) of NODEs and PINNs, a different learning based approach discussed in Section 6.1.

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  The average constraint violations (measured as $\frac{1}{n_{test}}\sum_{i=1}^{n_{test}}|h_{j}(\hat{\bm{y}}^{i},\hat{\bm{x}}^{i}(t))|$ for the $j$-th equality and $\frac{1}{n_{test}}\sum_{i=1}^{n_{test}}\max(0,u_{l}(\hat{\bm{y}}^{i},\hat{\bm{x}}^{i}(t)))$ for the $l$-th inequality, where $n_{test}$ is the test set size), incurred by DynOPF-Net and each baseline method approximation of ${\bm{y}}$.

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  The average percentage steady-state gap incurred by DynOPF-Net and the baselines LtO predictions $\hat{\bm{y}}$, and measured as $\frac{|f(\bm{y}^{\star}(\bm{S}^{d})-f(\hat{\bm{y}}(\bm{S}^{d}))|}{|f(\bm{y}^{\star}(\bm{S}^{d}))|}*100$, where $f$ is objective equation 9a and prediction error (MSE) $\|\bm{y}^{\star}-\hat{\bm{y}}\|^{2}$ of the OPF variables with respect to optimal (steady-state) generators set-points values. The steady-state assumption is crucial for evaluating how closely each solution approximates the AC-OPF optimal results, though it does not reflect the stability-constrained AC-OPF problem results which is the main focus of this paper. Given the non-linearity of both the dynamics and optimization in the stability-constrained AC-OPF, computing exact optimal decisions $\bm{y}^{\star}$ with traditional methods is unfeasible. Consequently, while our method may achieve slightly higher steady-state gaps or prediction errors, these should not be interpreted in the context of the stability-constrained AC-OPF problem and the key desiderata and goal is that of producing solutions with low stability violations.

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  The inference time (measured in seconds) of DynOPF-Net and each LtO model to generate $\hat{\bm{y}}$.