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Economic Dispatch (ED)

Economic Dispatch is the simplest optimization model for power system operation, determining the cost-minimizing generator dispatch to meet system load without considering network constraints.

flowchart TB
    A[System Demand] --> B[Economic Dispatch]
    C[Generator Costs] --> B
    D[Capacity Limits] --> B
    
    B --> E{Optimization}
    
    E --> F[Generator Outputs]
    E --> G[System Marginal Price λ]
    E --> H[Total Cost]

Solver Comparison

FeatureEDUCDC-OPFAC-OPF
Problem TypeLP/QPMIPLP/QPNLP
Network Model✗✗✓ DC (Linearized)✓ AC
Time PeriodsSingleMultiple (24+)SingleSingle
Commitment✗✓ Binary✗✗
Solve TimeFastestSlowFastMedium

→ Detailed Comparison

Mathematical Formulation

Problem Statement

Find the cost-minimizing generation dispatch that meets system demand while respecting generator capacity limits.

Sets and Indices

SymbolDescriptionExample
Set of generators
Generator index (first generator)

Decision Variables

VariableUnitDescription
p.u.Active power output of generator

Domain: (non-negative power generation)

Parameters

ParameterUnitDescription
$/hGeneration cost function for generator
p.u.Total system demand (load)
p.u.Minimum generation limit for generator
p.u.Maximum generation limit (capacity) for generator

Cost Function Forms:

Linear:

Quadratic:

Where:

  • : Quadratic coefficient (heat rate increase)
  • : Linear coefficient
  • : Constant (no-load cost)

Formulation Explanation

(1a) Objective Function

Minimize total generation cost:

Sum the individual generator costs to get total system cost.

Economic interpretation: Find the cheapest way to generate power.

Constraint Explanation

(1b) Power Balance Constraint

Supply equals demand:

Total generation must exactly match system load.

Physical interpretation: Conservation of energy (ignoring losses).

Dual variable: = System Marginal Price (SMP)

(1c) Generation Limit Constraints

Capacity bounds:

Each generator operates within its physical limits.

Physical interpretation:

  • : Technical minimum (turbine stability)
  • : Nameplate capacity (maximum output)

Dual variables:

  • : Shadow price of minimum limit
  • : Shadow price of maximum limit

Solution Process

flowchart TB
    A[Start] --> B["Input Data (Costs, Limits, Demand)"]
    
    B --> C[Formulate Optimization]
    
    C --> D[Solve LP/QP]
    
    D --> E{Converged?}
    
    E -->|Yes| F[Extract Solution pi, λ, μ]
    E -->|No| G[Check Data/Reformulate]
    
    G --> D
    
    F --> H[Validate Constraints]
    
    H --> I[Compute Total Cost]
    
    I --> J[Output Results]
    
    J --> K[End]

Merit Order Dispatch

ED solution follows merit order:

  1. Sort generators by marginal cost (ascending)
  2. Dispatch cheapest first until capacity reached
  3. Continue up the merit order until demand met
  4. Marginal unit sets system price
graph LR
    A["Nuclear 5-8 $/MWh (Baseload)"] --> B["Coal 22-32 $/MWh (Mid-merit)"]
    B --> C["LNG 45-65 $/MWh (Peaking)"]

KKT Optimality Conditions

Lagrangian Function

Introduce Lagrange multipliers:

  • : Power balance (equation dual)
  • : Lower bound duals
  • : Upper bound duals

Stationarity Condition

First-order optimality:

Interpretation: Marginal cost equals system price (adjusted for binding constraints).

Complementary Slackness

For lower bounds:

For upper bounds:

Interpretation: Dual variable is non-zero only when constraint is binding.

Three Cases

Case 1: Generator at minimum

  • (binding)
  • Marginal cost > System price

Case 2: Generator in interior

  • ,
  • Marginal cost = System price

Case 3: Generator at maximum

  • (binding)
  • Marginal cost < System price

Interpreting Dual Variables

System Marginal Price (λ)

Definition: The dual variable of the power balance constraint.

Economic meaning:

  • Marginal cost of serving one more MW of load
  • Value of generation at the margin
  • System-wide electricity price (in a perfect market)

Usage:

  • Electricity market clearing price
  • Optimal generator bidding strategy
  • Value of demand response

Shadow Prices (μ)

Lower bound shadow price :

  • Value of relaxing minimum generation by 1 MW
  • Non-zero only when
  • Indicates generator wants to reduce output but can’t

Upper bound shadow price :

  • Value of adding 1 MW more capacity to generator
  • Non-zero only when
  • Indicates generator wants to produce more but can’t

Example interpretation:

If $/MWh:

  • Adding 1 MW capacity to Gen 1 would save $15/hour
  • Over a year: $/year
  • Informs capacity expansion decisions

Visualization

Interactive ED Chart

Below is an interactive visualization showing how generation is dispatched by cost as demand changes.

Economic Dispatch

Demand: 80 MW

Marginal Generator: Gen 2 | SMP: $20/MWh

Observations:

  • Cheapest generator (Gen 1) dispatches first
  • As demand increases, more expensive units come online
  • System price equals marginal generator cost
  • Total cost grows faster as expensive units dispatch

Merit Order Stack

graph TB
    subgraph "Merit Order (Cost per MWh)"
        A[Nuclear: $6]
        B[Coal 1: $24]
        C[Coal 2: $28]
        D[LNG 1: $48]
        E[LNG 2: $55]
        F[LNG 3: $62]
    end
    
    A --> G["Baseload (24/7)"]
    B --> G
    C --> H["Mid-Merit (Most Hours)"]
    D --> H
    E --> I["Peaking (High Load Only)"]
    F --> I

Limitations of ED

What ED Ignores

1. Network Constraints

  • Transmission line limits
  • Voltage constraints
  • Power flow equations
  • Congestion costs

Impact: ED costs are lower bound (optimistic).

Solution: Use DC-OPF or AC-OPF for network-aware dispatch.

2. Temporal Constraints

  • Minimum up/down times
  • Startup costs
  • Ramping limits
  • Multi-period optimization

Impact: ED may suggest infeasible schedules.

Solution: Use Unit Commitment for realistic scheduling.

3. Reactive Power

  • Generator Q limits
  • Voltage stability
  • Reactive compensation

Impact: May not be AC feasible.

Solution: Use AC-OPF for complete power flow.

Next Steps