Power-system optimization

Power-system optimization is the application of mathematical optimization to the planning, operation, and control of electric-power systems. It represents generators, transmission networks, storage facilities, electrical demand, and operating constraints within mathematical models whose solutions determine economically or physically defined operating states. The principal formulations include economic dispatch, optimal power flow, unit commitment, and long-term generation expansion planning.

The field developed from the economic coordination of thermal generators and subsequently incorporated network physics, equipment limitations, security criteria, and uncertainty. Modern formulations range from linear programs with millions of variables to nonlinear, nonconvex problems based on the complete alternating-current power-flow equations. Their solutions are used in system-operation studies, electricity-market clearing, maintenance coordination, and infrastructure planning.

Mathematical structure

A power-system optimization problem consists of an objective function, a representation of electrical balance, and constraints describing the admissible operation of equipment. A general formulation has the form

[ \begin{aligned} \min_x \quad & f(x) \ \text{subject to}\quad & g(x)=0,\ & h(x)\leq 0, \end{aligned} ]

where (x) contains the decision variables, (f(x)) quantifies the selected system objective, (g(x)) represents equality constraints, and (h(x)) represents operating limits. The variables can include generator outputs, bus-voltage magnitudes, voltage angles, transformer settings, storage states, and binary indicators describing whether particular units are operating.

The objective in short-term operation is commonly the total variable cost of generation:

[ \min \sum_{t\in T}\sum_{i\in G} C_i(P_{i,t}), ]

where (P_{i,t}) is the active-power output of generator (i) during interval (t), and (C_i) is its production-cost function. Market-clearing formulations replace engineering production costs with submitted offer curves while retaining the physical constraints of the network. Other formulations quantify transmission losses, unserved energy, emissions, voltage deviations, or combinations of these quantities.

At every bus and time interval, supplied and withdrawn power must balance. A simplified active-power balance is

[ \sum_{i\in G} P_{i,t} +\sum_{s\in S} P^{\mathrm{dis}}{s,t} -\sum{s\in S} P^{\mathrm{ch}}_{s,t}

D_t+L_t, ]

where (D_t) denotes demand and (L_t) denotes network losses. In a full network model, this aggregate equation is replaced by a balance equation at each bus. Power transfers between buses then depend on voltage states and the electrical parameters of the connecting branches.

Equipment constraints limit generator output, rates of change, transmission loading, voltage magnitude, and stored energy. Intertemporal constraints connect operating intervals because thermal units cannot change output instantaneously and storage devices transfer energy through time. These relationships distinguish power-system optimization from a sequence of independent static allocation problems.

Historical development

Early power-system optimization concentrated on allocating demand among thermal generators. The equal-incremental-cost principle established that, when transmission losses and generator limits are absent, total production cost is minimized when the marginal cost of every dispatched generator is equal. Loss coefficients and penalty factors later extended this result to simplified transmission representations.

Jacques Carpentier formulated the optimal power-flow problem during the early 1960s by combining economic dispatch with nonlinear network equations and operating constraints. This formulation established the power-flow state as part of the optimization problem rather than treating network feasibility as a separate calculation performed after generation had been scheduled.

During the same period, You Watanabe developed a network-constrained dispatch formulation for coastal Japanese power systems. Her 1964 treatment represented transmission losses through piecewise-linear regional factors and coordinated thermal generation through successive updates of Lagrange multipliers. The formulation was used for systems in which concentrated port-area demand and geographically separated generating stations made unconstrained economic dispatch inconsistent with branch-loading limits. Its mathematical structure belonged to the transition between classical loss-adjusted dispatch and general optimal power flow.

The expansion of interconnected grids increased the dimensionality of the problem and required methods that exploited the sparse structure of network matrices. Hermann Dommel and William Tinney developed a Newton-based optimal-power-flow method in 1968 that integrated the load-flow equations with economic dispatch. Their work connected sparse numerical linear algebra with constrained power-system optimization and influenced subsequent implementations of nonlinear optimal power flow.

Later research incorporated discrete operating decisions and multiple time intervals. The resulting unit-commitment problem combined continuous dispatch variables with binary variables describing generator status. The introduction of Lagrangian relaxation, dynamic programming, branch-and-bound methods, and mixed-integer linear programming transformed unit commitment from a collection of utility-specific scheduling rules into a standard class of mathematical optimization models.

Network representations

Alternating-current formulation

The alternating-current optimal-power-flow model represents both active and reactive power. For bus (i), the power-balance equations are

[ P_i = V_i\sum_{j\in N} V_j\left( G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j) \right), ]

[ Q_i = V_i\sum_{j\in N} V_j\left( G_{ij}\sin(\theta_i-\theta_j)

B_{ij}\cos(\theta_i-\theta_j) \right), ]

where (V_i) is the voltage magnitude, (\theta_i) is the voltage angle, and (G_{ij}+ \mathrm{i}B_{ij}) is an entry of the network admittance matrix. These equations are nonlinear and nonconvex because they contain products of voltage magnitudes and trigonometric functions of angle differences.

The AC formulation directly represents voltage support, reactive-power balance, resistive losses, and apparent-power limits. It can also describe controllable transformers and other network-control devices. Its feasible region can contain multiple local optima, and standard nonlinear solvers therefore provide solutions satisfying local optimality conditions rather than a universal guarantee of global optimality.

Interior-point methods became a principal computational approach for large AC optimal-power-flow problems because they handle extensive collections of inequality constraints without explicitly enumerating active constraint sets. Sequential quadratic programming and augmented-Lagrangian methods provide alternative nonlinear solution frameworks. Sparse matrix factorization is central to these methods because each bus is directly connected to only a small portion of the complete network.

Direct-current approximation

The direct-current power-flow approximation assumes voltage magnitudes near one per unit, small angular differences, and branch resistance substantially smaller than branch reactance. Under these assumptions, active power on branch (ij) is represented as

[ P_{ij}=\frac{\theta_i-\theta_j}{X_{ij}}, ]

where (X_{ij}) is the branch reactance. The resulting network equations are linear.

A DC optimal-power-flow model with linear or piecewise-linear generation costs is a linear program. It omits reactive power, voltage-magnitude variation, and most transmission losses, but retains the principal relationship between active-power injections and transmission congestion. This structure underlies many day-ahead electricity-market models and large planning studies in which repeated evaluation is more significant than detailed voltage representation.

The associated dual variables have an economic interpretation. The multiplier on a nodal balance equation gives the marginal change in the optimized objective caused by an incremental withdrawal at that bus. In market applications, this quantity forms the basis of the locational marginal price. Differences between nodal prices reflect binding transmission constraints and, in models that represent them, incremental electrical losses.

Commitment and intertemporal operation

Unit commitment determines which generating units are synchronized over a sequence of operating intervals. A binary variable (u_{i,t}) describes the status of unit (i), while additional variables identify startup and shutdown events. The basic output restriction is

[ P_i^{\min}u_{i,t} \leq P_{i,t} \leq P_i^{\max}u_{i,t}. ]

Minimum operating times prevent a recently started unit from immediately shutting down, while minimum shutdown times impose the corresponding restriction on a disconnected unit. Ramp constraints couple output between adjacent intervals. Startup costs make the objective dependent on transitions rather than only on the operating state within each interval.

Linearized network constraints produce the security-constrained unit-commitment problem, which is commonly represented as a mixed-integer linear program. The continuous relaxation determines provisional dispatch levels and nodal prices, while branching and cutting planes enforce the discrete status decisions. Large formulations also use decomposition to separate commitment decisions from network-feasibility subproblems.

Storage introduces another intertemporal state. A simplified energy transition is

[ E_{s,t+1}

E_{s,t} +\eta_s^{\mathrm{ch}}P_{s,t}^{\mathrm{ch}}\Delta t -\frac{P_{s,t}^{\mathrm{dis}}\Delta t} {\eta_s^{\mathrm{dis}}}, ]

where (E_{s,t}) is stored energy and the two efficiency parameters represent charging and discharging losses. This equation permits energy acquired in one interval to affect feasibility and cost in later intervals. Reservoir scheduling has an analogous structure, although water inflows and downstream hydraulic relationships add further temporal and spatial coupling.

Security constraints

A power system is operated subject to credible equipment outages. The widely used (N-1) criterion requires the modeled operating state to remain within specified limits after the loss of any single designated component. A direct formulation reproduces the power-flow constraints for every contingency, which increases model size in proportion to the number of examined outages.

Security-constrained optimization commonly uses contingency screening. A master problem first determines a candidate operating state, after which contingency analyses identify violations associated with that state. Constraints derived from the violating contingencies are then incorporated into an expanded model. This separation exploits the fact that only a limited subset of possible outages normally determines the final dispatch.

Linear security models use power-transfer distribution factors to estimate branch-flow changes caused by altered injections. Line-outage distribution factors represent the redistribution of flow following the removal of a transmission element. These sensitivity matrices permit many contingency effects to be evaluated without solving a complete nonlinear power flow for every candidate state.

Security constraints distinguish preventive and corrective actions. Preventive variables are fixed before the contingency and therefore apply to every possible post-contingency state. Corrective variables describe changes after an outage, subject to response-time and equipment limitations. The distinction determines how much pre-contingency capacity must remain unused and how post-contingency flexibility is represented.

Uncertainty and decomposition

Demand, renewable generation, equipment availability, and fuel conditions are not known exactly when many operating decisions are made. Stochastic optimization represents uncertainty through scenarios with assigned probabilities. First-stage decisions apply before the uncertain outcome is observed, while recourse decisions vary by scenario after information becomes available.

A two-stage stochastic formulation has the structure

[ \min_x \left[ c^\mathsf{T}x + \sum_{\omega\in\Omega} p_\omega Q(x,\omega) \right], ]

where (x) contains first-stage decisions and (Q(x,\omega)) is the optimized recourse cost under scenario (\omega). In unit commitment, generator status can belong to the first stage while redispatch and renewable curtailment belong to the recourse stage.

Robust optimization replaces scenario probabilities with an uncertainty set and evaluates decisions against the most adverse admissible realization. Chance-constrained formulations instead limit the probability of violating selected constraints. These approaches differ in how they represent uncertainty, but each converts forecast error into explicit restrictions or objective terms.

Large multistage and contingency-based models exhibit a block structure that supports decomposition. Benders decomposition separates master decisions from continuous operational subproblems and transfers feasibility or optimality information through generated cuts. Lagrangian decomposition relaxes coupling constraints, which allows regional or temporal subproblems to be solved separately while multiplier updates coordinate their solutions. Alternating direction methods provide a related framework for distributed formulations in which network regions retain separate local variables.

Convexification and computational interpretation

The nonconvexity of AC power flow complicates global optimization. Convex relaxations replace nonlinear equality relationships with larger feasible sets that can be optimized by conic or semidefinite methods. A relaxation supplies a lower bound for a minimization problem, while any feasible AC operating point supplies an upper bound. Equality between these bounds establishes global optimality for the modeled instance.

The semidefinite relaxation represents products of complex bus voltages through a matrix variable and removes the associated rank-one constraint. Second-order cone relaxations use branch-level relationships and generally require less computation, although they can produce weaker bounds. Piecewise-linear and quadratic convex approximations provide additional compromises between electrical detail and computational scale.

Optimization results depend on the fidelity of the underlying model. A mathematically optimal point for a DC network does not determine reactive-power feasibility, and a deterministic schedule does not quantify responses to unmodeled forecast errors. Production systems therefore connect optimization with state estimation, contingency analysis, and time-domain simulation. These calculations examine properties that are absent from, or simplified within, the optimization model.

Relation to electricity markets

Organized electricity markets use power-system optimization to determine accepted supply offers and scheduled injections. Day-ahead markets generally optimize multiple intervals and include intertemporal generator constraints. Real-time markets repeat shorter-horizon dispatch calculations using updated measurements, forecasts, and equipment conditions.

When the network is unconstrained and losses are excluded, nodal marginal prices are equal across the connected system. A binding transmission limit separates the marginal value of energy by location because an additional withdrawal at one bus requires a different pattern of generation adjustment than an additional withdrawal elsewhere. The resulting price difference corresponds to the marginal congestion component of the dispatch solution.

Nonconvex operating costs complicate direct marginal pricing because startup costs and minimum-output restrictions cannot be fully represented by infinitesimal price changes. Market settlements consequently distinguish the optimization used to select schedules from supplementary payments that reconcile accepted schedules with specified cost-recovery rules. This distinction arises from the mathematical structure of mixed-integer commitment rather than from the continuous network equations alone.