Docs/kpg run/Economic Dispatch (ED)

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

Feature	ED	UC	DC-OPF	AC-OPF
Problem Type	LP/QP	MIP	LP/QP	NLP
Network Model	✗	✗	✓ DC (Linearized)	✓ AC
Time Periods	Single	Multiple	Single	Single
Commitment	✗	✓ Binary	✗	✗
Solve Time	Fastest	Slow	Fast	Medium

→ Detailed Comparison

Mathematical Formulation

Problem Statement

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

Sets and Indices

Symbol	Description	Example
	Set of generators
	Generator index	(first generator)

Decision Variables

Variable	Unit	Description
	p.u.	Active power output of generator

Domain: (non-negative power generation)

Parameters

Parameter	Unit	Description
	$/h	Generation 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:

Sort generators by marginal cost (ascending)
Dispatch cheapest first until capacity reached
Continue up the merit order until demand met
Marginal unit sets system marginal 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.

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

Understand network constraints: DC-OPF Formulation →
Learn multi-period scheduling: Unit Commitment →
Compare all models: Solver Comparison →
Try it yourself: KPG Run Getting Started →

← User Interface Guide Unit Commitment (UC) →