Dynamic Economic Dispatch (DED)

• Dynamic Economic Dispatch is an optimization framework for scheduling power generation over a time horizon, ensuring cost minimization and adherence to operational limits.
• It integrates advanced cost models, including valve-point effects and prohibited operating zones, to capture nonconvex and nonsmooth cost behaviors.
• Solution strategies range from MILP-based centralized optimization and hybrid MILP-IPM methods to metaheuristics and distributed consensus algorithms, facilitating robust renewable integration.

Dynamic Economic Dispatch (DED) is an optimization framework central to the operational planning of power systems over a finite or rolling time horizon. The DED problem aims to schedule the output of generating units to meet time-varying demand at minimal aggregate cost while strictly enforcing operational, temporal, and network constraints that evolve dynamically with system states, regulation requirements, renewable uncertainty, and device physics.

1. Core Mathematical Formulation and Constraints

DED models a multi-period optimal scheduling problem for a fleet of generation assets, commonly indexed $i=1,\ldots,N$ over discrete intervals $t=0,\ldots,T-1$. For each unit and period, the primary decision variable is the active power output $P_{i,t}$. The canonical DED is:

$$\min_{P_{i,t}} \sum_{t=0}^{T-1} \sum_{i=1}^{N} C_i(P_{i,t})$$

subject to

$$\sum_{i=1}^N P_{i,t} = D_t + P^{\rm loss}_t \qquad (\text{power balance each hour})$$

$P_i^{\min} \leq P_{i,t} \leq P_i^{\max}$

$-DR_i \leq P_{i,t} - P_{i,t-1} \leq UR_i$

where:

• $C_i(\cdot)$ is the unit cost (often quadratic with possible nonsmooth/nonconvex terms),
• $D_t$ is the time-varying demand,
• $P^{\rm loss}_t$ are network losses (e.g. via the B-matrix),
• $t=0,\ldots,T-1$0 and $t=0,\ldots,T-1$1 are per-unit capacity and ramp constraints.

Additional practical constraints may encode spinning reserve requirements, storage/DER participation, forbidden zones, and more (Pan et al., 2017, Cherukuri et al., 2016, Pan et al., 2017).

2. Advanced Cost Modeling: Nonconvexities and Network Effects

Several critical extensions define modern DED:

• Valve-Point Effects (VPE): Non-convex, non-smooth "rippling" cost terms model the effects of valve-point loading in thermal units, often formulated as

$t=0,\ldots,T-1$2

rendering the DED problem nonconvex and numerically challenging (Pan et al., 2017, Pan et al., 2017, Alam, 2018).

• Prohibited Operating Zones (POZ): Output segments where operation is not allowed (e.g. due to mechanical or emissions reasons), leading to disjoint feasible sets, typically using binary assignment variables to encode segment selection (Pan et al., 2017).
• Transmission Losses: Quadratic expressions via Kron/B-matrix or first-order Taylor approximations are included in constraints for accurate system representation (Pan et al., 2017, Pan et al., 2017, Alam, 2018).

3. Deterministic and Stochastic Solution Methodologies

A. Centralized Optimization

Mixed-Integer Linear Programming (MILP):

• Piecewise linearization enables the transformation of nonconvex/nonsmooth cost curves (e.g., with VPE or POZ) into MILP, solvable with global optimality guarantees (to prescribed tolerances) using branch-and-bound (Pan et al., 2017, Pan et al., 2017, Pan et al., 2017).
• Granularity (number of breakpoints or segments) governs the tradeoff between accuracy and computational tractability. For example, dividing each sine period of the VPE term into M intervals with $t=0,\ldots,T-1$3 yields bounded optimality gaps (Pan et al., 2017).

Hybrid MILP–Interior Point Methods (MILP-IPM):

• To overcome local minima in nonconvex NLPs (as in DED-VPE), a two-stage MILP-IPM approach first solves a MILP relaxation (without losses), then initializes a differentiable NLP with this result and refines to local optimality with IPM (Pan et al., 2017).

Approximate Dynamic Programming (ADP):

• For systems with complex physical dynamics such as CCGT units, DED is posed as a finite-horizon Markov Decision Process (MDP). Value function approximation (VFA), post-decision states, and SPAR-based slope monotonicity enforcement yield near-optimal control policies at scale (Lin et al., 2021).

Koopman-based Differentiable Predictive Control (DPC):

• By learning a finite-dimensional Koopman operator for low-level generator dynamics and training an explicit neural policy, DPC policies provide solutions orders of magnitude faster at run-time, relaxing the need for online optimization (King et al., 2022).

B. Metaheuristics

• Particle Swarm Optimization (PSO) and variants are widely utilized for large, highly nonconvex, or mixed-integer DEDs, supporting discrete and continuous variable mixing (e.g., binary POZ variables, ramping, reserves). Modern PSO variants include chaotic, adaptive, penalty-free, SQP-hybrid, and eigen-analysis approaches (Alam, 2018).

C. Robust and Stochastic Optimization

• Adaptive robust DED with dynamic uncertainty sets incorporates the temporal and spatial correlation of uncertain renewables (notably wind). This is realized with autoregressive models on load and wind, generating dynamic polyhedral uncertainty sets, and employing two-stage min-max-min optimization with tractable scenario-based reformulations (Lorca et al., 2014).

Approach	Strengths	Key References
MILP (piecewise linear)	Global optimality (tolerance-bound), scalability for convexifiable cases	(Pan et al., 2017, Pan et al., 2017, Pan et al., 2017)
MILP-IPM hybrid	Nonconvex/nonsmooth (VPE), better local minima it capture	(Pan et al., 2017)
Metaheuristics (PSO)	Large nonconvex search spaces, hybrid constraints	(Alam, 2018)
ADP/VFA/Koopman-DPC	Physically dynamic models, near–real-time closed-loop control	(Lin et al., 2021, King et al., 2022)
Robust optimization	Spatio-temporal uncertainty, rolling-horizon simulation	(Lorca et al., 2014)

4. Distributed and Consensus-Based Algorithms

Distributed DED methodologies address privacy, scalability, renewables integration, and real-time operation by decentralizing optimization among networked agents:

• Distributed Dynamic Average Consensus: Each agent estimates both load-generation mismatch and dual variable via local communication (e.g., "dac+Lap∂+∂" architecture), enabling exact or approximate optimality without a central coordinator (Cherukuri et al., 2016, Cherukuri et al., 2014, Wasti et al., 2020, Pal et al., 14 Mar 2026).
• ADMM Variants: Fully decomposed primal updates where agents share only minimal