Dynamic Economic Dispatch Problem

• Dynamic Economic Dispatch is an optimization framework that minimizes electricity generation costs by accounting for time-coupled constraints, ramping, and uncertainties.
• Key methodologies include convex relaxations, mixed-integer programming, hybrid metaheuristics, and distributed algorithms to enhance scalability and reliability in power systems.
• Emerging approaches integrate data-driven techniques, machine learning models, and robust optimization to improve real-time adaptability and support renewable integration.

The dynamic economic dispatch (DED) problem is a central optimization challenge in power system operations, focusing on minimizing the cost of electricity generation over a time horizon while meeting demand, respecting operational constraints, and adapting to system dynamics and uncertainties. DED generalizes static economic dispatch by considering time-coupled constraints such as generator ramping, storage dynamics, non-deferrable and elastic (controllable) loads, and various network-wide operational limits (e.g., voltage, power factor, and security constraints). Formulations and solution methods for DED span a wide range of mathematical programming, distributed algorithms, robust and stochastic optimization, and, more recently, data-driven and machine learning paradigms.

1. Mathematical Formulation and Problem Structure

The DED problem is typically formulated as a multi-period convex or nonconvex program, depending on the network and generation models. The canonical objective is to minimize the total or average generation and procurement cost over a scheduling horizon, represented for instance by

$$\min_{\text{control variables}} \ \sum_{t=1}^T \sum_{i\in \mathcal{G}} C_{i,t}(P_{i,t})$$

subject to physics-based network constraints and operational constraints. Examples:

• Network (power flow) constraints: Real and reactive power balance at each node and time: For unbalanced three-phase systems, e.g., $V_{n,t}^\phi (I_{n,t}^\phi)^* = \text{[net power injection at node$n$, phase$\phi$, time$t$]}$, with per-phase indices $\phi$.
• Generator operating constraints: $P_{i,t}^{\min} \leq P_{i,t} \leq P_{i,t}^{\max}$, ramp rate limits $|P_{i,t+1} - P_{i,t}| \leq R_i$, reserve requirements, and, for units with storage, storage capacity and SoC evolution constraints.
• Elastic (controllable) and non-deferrable (fixed) loads: With time/energy requirements for each elastic load $d$ at node $n$ and phase $\phi$, as $$\sum_{t} \widehat{P}_{d,n,t}^\phi \Delta_t = E_{d,n}^\phi$$, $$0 \leq \widehat{P}_{d,n,t}^\phi \leq P_{d,n}^{\max}$$.
• Voltage regulation: $$V_{n,t}^{(\min)} \le |V_{n,t}^\phi| \le V_{n,t}^{(\max)}$$.
• Additional constraints: Power factor at substations, thermal limits, prohibited operating zones (POZ), and spinning reserves.

When incorporating network nonlinearities (especially power flow in unbalanced systems), the problem is nonconvex. In DED with storage, SoC trajectory of storage units is added, with degradation costs (e.g., using Rainflow cycle counting) considered in the objective (Bansal et al., 2020).

2. Algorithmic Approaches: Centralized, Convexification, and Relaxations

Several major classes of methods have been developed:

• Semidefinite Programming (SDP) Relaxation: For unbalanced networks, nonconvexities in power flow and cost terms are "lifted" using outer-product voltage matrices $V_t = v_t v_t^*$, so quadratic constraints become linear in $V_t$ (Dall'Anese et al., 2012). The SDP relaxation drops the nonconvex rank-1 constraint; if the solution is rank-1, a global optimum is directly recoverable.
• Piecewise Linearization and Mixed-Integer Programming: Nonconvex, non-smooth cost terms, e.g., valve-point effects, are approximated via piecewise linearization, yielding mixed-integer linear programs (MILPs) (Pan et al., 2017, Pan et al., 2017), which can be solved globally to a set tolerance with modern solvers. MILP is also employed for DED with prohibited operating zones and transmission losses, via perspective cuts and Taylor expansion (Pan et al., 2017).
• Metaheuristics and Hybridization: Heuristic approaches such as Particle Swarm Optimization (PSO), often hybridized with Simulated Annealing (SA) or Sequential Quadratic Programming (SQP), are prevalent for DED when the cost function or constraints are highly nonconvex or non-differentiable (Karthikeyan et al., 2013, Alam, 2018). Hybrid PSO-SQP and chaotic/self-adaptive PSO variants are specifically developed to address dynamic constraints (e.g., ramping and reserves), non-smooth costs, and security/contingency constraints.
• Dynamic Programming and Approximate Dynamic Programming: For problems with explicit process or storage dynamics (e.g., CCGT plants with thermal inertia (Lin et al., 2021)), the DED is cast as a Markov decision process (MDP). Approximate dynamic programming with value function approximation, e.g., convex piecewise linear fits, is used to retain tractability.
• Robust Optimization: DED under wind or demand uncertainty is modeled using two-stage adaptive robust optimization, where dynamic uncertainty sets capture temporal and spatial correlations of renewables (Lorca et al., 2014, Sharf et al., 2021). Dynamic uncertainty is handled via autoregressive characterizations, and robust rolling-horizon optimization is used, often yielding Pareto-efficient tradeoffs between cost and system risk metrics.

3. Distributed and Decentralized Methods

Scalability, privacy, and resilience concerns motivate distributed DED algorithms:

• Consensus+Optimization and Dynamic Average Consensus: Distributed Laplacian (nonsmooth) gradient flows coupled with dynamic average consensus blocks enable initialization-free DED under varying load and generator commitment (Cherukuri et al., 2014). Robustness is analytically established for generator dropout/join and time-varying demand, using a refined invariance principle for differential inclusions.
• Dual Decomposition and Privacy: Dual gradient-based, continuous-time distributed coordination is developed where only local dual variables (or incremental costs) are exchanged, ensuring privacy and adaptation to grid changes (Yun et al., 2018). Infeasibility is transparently detected via divergent dual variable trajectories.
• ADMM and Local Consensus: A fully distributed Alternating Direction Method of Multipliers (ADMM) variant, using dynamic average consensus for local estimation of network-level dual variables and power mismatches, provides decentralized DED updates requiring only neighbor communications (Wasti et al., 2020), enabling real-time adaptability under renewables.
• Market-Based and Primal-Dual Dynamics: Co-optimization across transmission and distribution networks with high DER penetration is addressed using primal-dual gradient algorithms, augmented by distributed market-like signals to preserve privacy and enable local agent (DER) participation (Zhou et al., 2020, Wang et al., 2023). Aggregate equivalence and projection-based spatial/temporal decomposition support rapid convergence and scalability for large integrated networks.

4. Treatment of Uncertainty, Flexibility, and Storage

Contemporary DED frameworks explicitly address uncertainty and system flexibility:

• Adaptive Robust Optimization: Wind power uncertainty is captured by dynamically constructed (autoregressive) uncertainty sets, forming the basis for robust ED with rolling-horizon dispatch (Lorca et al., 2014). The framework provides cost-reliability Pareto frontiers and guidance on tuning conservativeness.
• Flexibility Metrics and AGC Integration: Novel robust optimization models interlink DED and automatic generation control (AGC), resulting in quantifiable flexibility indices (e.g., upward/downward ED/AGC flexibility), solved via Benders decomposition with problem-specific linearizations (Fan et al., 2020).
• Storage Degradation: Multi-period DED formulations co-optimize generator and storage dispatch, where the storage degradation is modeled by convex stress functions over half-cycle depths (Rainflow counting). Incentive compatible market clearing prices are shown to align individual profit-maximizing decisions with system cost minimization, and uniqueness of storage response is established under appropriate incidence matrix rank conditions (Bansal et al., 2020).

5. Data-Driven and Machine Learning Methods

Recent research introduces data-driven and learning-based paradigms to the DED problem:

• Differentiable Predictive Control (DPC) with Koopman Surrogates: DED that incorporates generator swing dynamics is made computationally feasible by (i) learning a Koopman operator-based linear surrogate for the nonlinear system and (ii) applying differentiable predictive control to train explicit neural control policies offline (King et al., 2022). At runtime, dispatch decisions are computed via neural policy evaluation—orders of magnitude faster than online optimization—with small performance degradation.
• LLMs for Economic Dispatch: Off-the-shelf LLMs can solve the classic ED problem via few-shot prompting, enabling feasible and near-optimal dispatch allocations without explicit mathematical modeling or labeled data (Mohammadi et al., 28 May 2025). LLMs can leverage few-shot examples to emulate reasoning in both non-evolutionary (simple reasoning) and evolutionary (mutation, crossover) styles, achieving cost and power balance errors within acceptable margins in standard test cases (e.g., IEEE 118-bus benchmarks).

6. Applications, Benchmarks, and Comparative Analysis

DED formulations have been validated on a wide range of benchmarks:

Approach	Key Test Cases	Strengths/Limitations
SDP Relaxation	IEEE 13-node feeder	Global optimality for unbalanced, small systems
MILP for DED-VPE/POZ	10–500 unit, 6–180 unit ED systems	Guarantees on optimality, scalable with advanced solvers
Hybrid PSO/GA/metaheuristic	3-unit, 10/30-unit, IEEE 118-bus	Robust to nonconvexities, slower convergence
Distributed (Consensus/ADMM)	IEEE 118-, 30- ,300-bus, microgrid	Robust, initialization-free, high scalability
Data-driven (Koopman, LLMs)	9-bus, IEEE 118-bus	Ultra-fast, promising accuracy, scalability ongoing

Convergence guarantees, tradeoffs between accuracy and computational effort, scalability to large-scale systems, as well as privacy and practical implementability are major axes of comparison (Dall'Anese et al., 2012, Pan et al., 2017, Wasti et al., 2020, Mohammadi et al., 28 May 2025).

7. Significance, Open Challenges, and Future Directions

The DED problem is evolving to encompass high renewable penetration, storage integration, distributed architectures, and uncertainty modeling:

• Robust optimization and decomposition techniques offer resilience and computational tractability, essential for real-time operations in complex, large-scale, and decentralized environments.
• SDP and convexification approaches yield tractable solutions in previously intractable unbalanced or nonlinear settings, especially when relaxation is tight.
• Metaheuristic and hybrid methods remain valuable for tackling nonconvex, discontinuous, or combinatorial constraints not amenable to convex relaxations.
• Data-driven and learning-based methods (including neural control and LLMs) hint at transformative potential for operator decision support, provided their scalability and constraint-satisfaction guarantees can be enhanced.
• Integration of storage degradation, AGC/frequency regulation, and market mechanisms into DED formulations reflects the increasing sophistication of operational requirements.
• Open challenges include ensuring exactness of relaxations for general network topologies, handling large-scale practical constraints (POZ, N-1 security), achieving real-time scalability, fully capturing stochasticity, and reliably bridging model-based and data-driven paradigms.