Economic Operation and Optimal Power Flow

Introduction and context

Power-system operation must satisfy demand continuously while maintaining frequency, voltage, security, and acceptable environmental performance at minimum total cost. The operational problem is hierarchical. Unit commitment (UC) decides which generating units should be on over a scheduling horizon. Economic dispatch (ED) allocates generation among committed units. Optimal power flow (OPF) simultaneously determines generation, network flows, voltages, and sometimes reserves and emissions. Loss allocation assigns transmission losses to users or generators for settlement and planning.

These decisions are coupled. A cheap generator may be far from load and cause congestion; a technically flexible unit may be retained for reserve even when its energy cost is higher; and a dispatch that is economical without losses may be infeasible after network losses are included.

Learning objectives

After studying this chapter, you should be able to:

1. Formulate and solve lossless and loss-including economic dispatch.
2. Explain incremental cost and the equal-incremental-cost condition.
3. Formulate unit commitment with minimum up/down times, start-up and shut-down costs, ramping, and reserves.
4. Distinguish DC and AC optimal power flow and apply the DC approximation with correct units.
5. Calculate and interpret a practical loss-allocation result.
6. Incorporate congestion, uncertainty, demand response, emissions, and security constraints.
7. Evaluate operational decisions using technical, economic, environmental, and regulatory criteria.

1. Economic dispatch

Economic dispatch determines generator outputs for a specified operating period, assuming the committed units are known. For generator i, a common quadratic fuel-cost model is

where C_i is cost in currency per hour, P_i is output in MW, and a_i, b_i, and c_i are plant-specific coefficients. The incremental cost is

The basic lossless ED problem is

where P_D is demand in MW. The equality is valid for a lossless model. With transmission losses P_L(P), the balance becomes

Equal incremental cost

For an interior optimum without losses, the Lagrangian is

Stationarity gives IC_i=λ for every generator not at a limit. Thus, shifting one additional MW from a higher-incremental-cost unit to a lower-incremental-cost unit reduces cost. The condition applies only to unconstrained units; a unit at its minimum or maximum need not have incremental cost equal to λ.

With losses, the stationarity condition becomes

or, equivalently, IC_i/(1-∂P_L/∂P_i)=λ. The term 1-∂P_L/∂P_i is the incremental transmission-loss coefficient under the adopted loss model.

Worked example: two-unit economic dispatch

Two committed generators supply 300 MW. Their costs are

Assume no losses and no active output limits. The balance is P₁+P₂=300. Incremental costs are

Set them equal and use the balance:

With P₂=300-P₁,

so 0.06P₁=15, giving

The common incremental cost is λ=10 currency/MWh. The result satisfies balance and equal incremental costs. If either value violated a generator limit, the limited unit would be fixed at that limit and the remaining units re-dispatched.

Practical ED algorithm

A practical procedure is: (1) specify demand, losses, limits, and cost curves; (2) solve the unconstrained stationarity equations; (3) test all limits; (4) fix violating units at the appropriate limit; (5) re-solve for the remaining units; and (6) verify balance, costs, and feasibility. Numerical solvers generally use active-set, Newton, quadratic-programming, or nonlinear-programming methods.

2. Unit commitment

ED is short-term and continuous; UC is a mixed-integer scheduling problem. Let u_it equal 1 if unit i is on in period t, y_it equal 1 for a start-up at the beginning of period t, and z_it equal 1 for a shut-down. Let P_it be output, R_it upward reserve, and D_t demand.

A representative deterministic UC objective is

subject to the following constraints.

Logical transition:

with u_i0 specified from the initial condition. This equation is essential: it links the start-up and shut-down indicators to changes in commitment status. Additional inequalities such as y_it+z_it≤1 prevent simultaneous start-up and shut-down.

Generation and balance:

Reserve requirement:

The last inequality ensures that scheduled headroom can actually provide upward reserve.

Ramping:

Start-up and shut-down ramp limits may require separate formulations. Minimum up time requires a unit that starts to remain on for at least UT_i periods; one compact form is

for periods where the full interval exists. A corresponding minimum-down-time constraint is

Boundary conditions must incorporate how long each unit has already been on or off before the horizon. Start-up costs can depend on downtime, for example through hot, warm, and cold start categories.

UC is usually solved as a mixed-integer linear program when costs and network constraints are linearized, or as a mixed-integer nonlinear program when AC power flow and nonlinear costs are retained. Rolling-horizon operation updates forecasts, unit states, and actual measurements.

3. Network-constrained optimal power flow

OPF extends dispatch by representing the transmission network. An AC power-flow model uses complex bus voltage V_i=|V_i|∠θ_i, conductance G_ij, and susceptance B_ij. One common sign convention gives

The exact signs depend on the adopted Y-bus and injection convention; consistency is mandatory. AC OPF minimizes cost subject to these equations, generator P and Q limits, voltage limits, transformer taps, line thermal limits, and reference-angle conditions.

DC OPF and its dimensional limitation

The DC approximation assumes approximately 1.0 per-unit voltage magnitudes, small angle differences, negligible resistance, and a network represented by series reactances. In per unit, branch flow is commonly written

where F_ij and X_ij are both expressed consistently in per unit, and angles are in radians. In physical MW units, the scaling cannot be omitted. If X_ij^pu is per-unit reactance and S_base is the MVA base,

under the same DC assumptions. Alternatively, using ohms, the relation requires voltage and base-power factors through the chosen network base. Therefore the unscaled equation must not be interpreted as a general dimensional MW equation.

DC OPF is computationally efficient and is widely used for market clearing, congestion analysis, and security screening. It does not accurately represent voltage magnitude, reactive power, voltage stability, resistance losses, or heavily loaded lines. AC OPF is required when those effects matter.

Figure: A simplified three-bus network. OPF determines injections and flows subject to nodal balance and branch constraints.

Congestion and locational marginal price

In a lossless unconstrained network, the marginal energy price is uniform. When a line limit binds, an additional MW at one bus may require more expensive generation because of its effect on the constrained interface. A locational marginal price (LMP) can be conceptually decomposed into energy, congestion, and loss components:

This is an interpretation rather than a universal accounting identity; exact decomposition depends on market formulation and sign conventions. Security-constrained OPF adds contingency constraints, commonly requiring the system to remain feasible after specified single outages (the N−1 criterion), subject to approved emergency ratings and corrective actions.

4. Loss allocation

Total real-power loss is the difference between total injections and delivered demand:

Loss allocation is not uniquely determined by physics alone. It is a settlement rule, and alternatives include proportional sharing, marginal participation, and power-flow tracing. A transparent method must specify the network model, sign convention, treatment of loads and generators, and whether negative allocations are allowed.

Worked proportional allocation

Suppose three generators inject 120, 80, and 50 MW, while total load is 240 MW. Total generation is 250 MW, hence losses are 10 MW. A simple proportional allocation assigns loss responsibility to generators in proportion to their injections:

Therefore,

The allocated amounts sum exactly to the measured loss. This method is easy to audit but ignores electrical distance and direction. A marginal method instead uses the sensitivity of losses to an incremental injection, ∂P_L/∂P_i, and may produce negative or greater-than-one marginal responsibility under some network conditions. Such results are mathematically possible but require careful regulatory treatment. Flow-tracing methods allocate losses according to traced network usage and are more location-sensitive.

5. Reserves, uncertainty, and demand response

A deterministic schedule based only on expected demand can be inadequate when renewable output, demand, or outages are uncertain. Reserve categories should be defined operationally: regulation responds rapidly to small deviations; spinning reserve is synchronized and rapidly deployable; non-spinning reserve is available after a specified start time; replacement reserve restores readiness over a longer interval.

A probabilistic reserve requirement can be based on forecast-error quantiles. If net-load error has standard deviation σ_N and a normal approximation is acceptable, a one-sided reserve target may be written

where z_α is the selected quantile. This is only a statistical approximation; correlation, non-normal tails, common-mode outages, and delivery time must be considered. Scenario-based stochastic UC minimizes expected cost across scenarios, while robust UC protects against all errors in a specified uncertainty set, often at higher cost.

Demand response can be modeled as a controllable resource. For a shiftable load, energy conservation over a period may be imposed by

with comfort, rebound, maximum-shift, and availability constraints. Curtailment must include customer compensation and reliability limits, not merely a negative load with no cost.

6. Emissions and multi-objective operation

For generator i, a simple emissions model is

An emissions-constrained dispatch may impose

Alternatively, a carbon price τ adds τE_i(P_i) to operating cost. Weighted objectives such as C+ωE require careful selection of the weight ω; they do not automatically reveal the full cost-emissions trade-off. A practical study should report a Pareto frontier or compare explicit policy cases.

7. Illustrative integrated case

This is a fictional instructional case. A 300-MW system has two thermal generators and a 100-MW wind farm. The forecast demand is 320 MW, and 10 MW of transmission loss is expected. Thermal units are:

Wind is dispatched at 90 MW because of a forecasted 10-MW availability margin. Thermal generation must therefore equal 320+10-90=240 MW. If both units are online, a lossless linear-cost ED would use G1 first, subject to its 220-MW maximum, and assign the remaining 20 MW to G2. Thus the schedule is G1 = 220 MW and G2 = 20 MW, but G2 violates its 40-MW minimum. The feasible commitment is instead G1 = 200 MW and G2 = 40 MW, giving 240 MW thermal output. This illustrates why commitment and dispatch cannot be separated blindly.

Suppose the next period demand falls to 250 MW, wind remains 90 MW, and losses are 8 MW. Thermal requirement is 168 MW. If G2 is shut down, G1 alone can produce 168 MW and meet the energy balance. Whether shutting down is economic depends on G2’s shut-down cost, minimum-down time, future demand, reserve requirement, and the cost of retaining flexibility. If 30 MW upward reserve is required, G1 has 52 MW headroom and can provide it, subject to ramp capability; the one-unit schedule may therefore remain feasible. If a contingency requires the largest online unit to be covered, however, the system operator may retain G2 despite its energy cost.

An OPF then checks whether the 240 MW thermal schedule can reach the load without violating line limits. If the G1-to-load interface is limited, the model may increase G2 output and reduce G1 output. The resulting increase is a network-constrained economic decision, not an ED error. A loss-allocation rule can then assign the computed network loss for settlement; the selected rule should be published and applied consistently.

Learning points: minimum generation can make a nominally cheap schedule infeasible; reserves and contingencies have opportunity costs; congestion can reverse the unconstrained merit order; and loss allocation is a policy choice grounded in a physical model.

8. Solution methods and professional practice

Small ED problems can be solved analytically. UC and OPF are generally solved with linear, quadratic, nonlinear, or mixed-integer optimization. Decomposition separates commitment from dispatch; Benders decomposition is common when master commitment decisions and network subproblems can be distinguished. Interior-point methods are effective for large continuous OPF. Commercial and open-source implementations must be validated against independent power-flow calculations and operating limits.

Good practice includes:

• validating generator curves, minimum times, ramp rates, and start costs;
• using a consistent MW/MVA and per-unit base;
• checking angle, voltage, thermal, and reactive-power limits;
• modeling initial conditions and horizon-end effects;
• testing infeasibility with slack variables that carry explicit penalty costs;
• comparing forecast, stressed, and contingency scenarios;
• recording solver tolerances, binding constraints, and dual prices;
• separating physical loss calculations from settlement allocation rules.

9. Limitations, governance, and contemporary developments

Optimization results are only as credible as their data and assumptions. Quadratic costs may not represent valve-point effects, prohibited operating zones, maintenance, fuel contracts, or start-up trajectories. DC models omit reactive power and voltage phenomena. Forecast distributions may be biased, and reserve formulas based on normal errors may understate extreme events.

Market operators must guard against discriminatory bidding, manipulation of congestion, and opaque uplift payments. LMPs and loss-allocation rules should be explainable, auditable, and consistent with approved tariffs. Environmental constraints must account for local pollutants, emissions measurement, and legal limits rather than treating carbon alone as the complete environmental objective.

Modern operations increasingly use high-resolution telemetry, probabilistic forecasting, distributed energy-resource models, and machine learning. AI can improve forecasts, anomaly detection, and warm starts, but safety-critical dispatch should retain physical feasibility checks, cybersecurity controls, human accountability, and a fall-back optimization process. Digital twins and synchronized phasor measurements support model validation, while flexible loads, batteries, and inverter-based resources require updated ramping, reserve, and voltage models.

Chapter Summary

Economic dispatch allocates output among committed units by minimizing operating cost subject to balance and operating limits. Without losses and binding limits, unconstrained units have equal incremental costs. Unit commitment adds binary on/off decisions, start-up and shut-down indicators, minimum up/down times, ramping, reserves, and intertemporal logic. OPF embeds dispatch in the electrical network: AC OPF represents active and reactive power and voltage, whereas DC OPF is a carefully stated per-unit approximation whose physical-MW form requires base scaling. Congestion creates location-dependent marginal prices. Loss allocation is a settlement convention, not a uniquely determined physical fact; proportional, marginal, and tracing methods have different properties. Reliable practice incorporates uncertainty, reserves, demand response, emissions, contingencies, data governance, and validation.

Key Terms

Economic dispatch: Allocation of generation among committed units at minimum operating cost.

Incremental cost: Cost of producing one additional MW at a specified output.

Unit commitment: Scheduling the on/off status of generating units over time.

Start-up cost: Cost incurred when a unit changes from off to on.

Shut-down cost: Cost incurred when a unit changes from on to off.

Minimum up time: Required time a unit must remain on after starting.

Minimum down time: Required time a unit must remain off after shutting down.

Ramping: Limit on output change between successive periods.

Spinning reserve: Synchronized capacity capable of rapid response.

Optimal power flow: Network-constrained optimization of generation and system states.

AC OPF: OPF using nonlinear active- and reactive-power equations.

DC OPF: Linearized OPF based on stated lossless, small-angle assumptions.

Congestion: A binding network transfer or equipment limit.

Locational marginal price: Marginal cost of serving an additional MW at a location.

Transmission loss: Real power dissipated in network elements.

Loss allocation: Rule assigning total losses to market participants.

Security-constrained OPF: OPF enforcing selected outage or contingency requirements.

Demand response: Deliberate modification of customer demand in response to system conditions.

Review Questions

Short Answer Questions

1. Why are incremental costs equal at an unconstrained lossless ED optimum?
2. What role does the equation u_it-u_i,t-1=y_it-z_it play in UC?
3. State two assumptions behind the DC power-flow approximation.
4. Why is loss allocation not uniquely determined by measured total loss?
5. Distinguish spinning reserve from non-spinning reserve.

Long Answer Questions

1. Derive the equal-incremental-cost condition using a Lagrangian and explain how generator limits alter it.
2. Formulate a UC model containing balance, start-up logic, minimum up/down times, ramping, and reserve constraints.
3. Compare AC OPF and DC OPF in terms of equations, accuracy, computational burden, and applications.
4. Compare proportional, marginal, and flow-tracing loss-allocation methods.
5. Explain how congestion, emissions constraints, and demand response can change the least-cost schedule.

Analytical / Application Questions

1. For two generators with costs C₁=400+6P₁+0.01P₁² and C₂=300+8P₂+0.02P₂², find the lossless dispatch for 300 MW if no limits bind.
2. A system has 500 MW generation and 480 MW load. Allocate the loss proportionally among generators producing 250 MW, 150 MW, and 100 MW, and discuss one limitation of the result.
3. A line becomes congested in a three-bus system. Explain why the OPF may dispatch a higher-cost generator and identify two additional checks required before accepting the schedule.

Multiple Choice Questions with Answers

Question 1

In unconstrained lossless economic dispatch, which condition holds for all interior generators?

A. Equal incremental costs

B. Equal maximum outputs

C. Equal fixed costs

D. Equal ramp rates

Correct Answer: A. Equal incremental costs

Question 2

For the worked two-unit dispatch with 300 MW demand, what are the outputs?

A. P₁ = 166.67 MW and P₂ = 133.33 MW

B. P₁ = 250 MW and P₂ = 50 MW

C. P₁ = 150 MW and P₂ = 150 MW

D. P₁ = 200 MW and P₂ = 100 MW

Correct Answer: B. P₁ = 250 MW and P₂ = 50 MW

Question 3

Which UC equation links commitment changes to start-up and shut-down indicators?

A. u_it + u_i,t−1 = y_it + z_it

B. u_it − u_i,t−1 = y_it − z_it

C. u_it = y_it + z_it + 1

D. y_it − z_it = 0 always

Correct Answer: B. u_it − u_i,t−1 = y_it − z_it

Question 4

In physical MW units, the per-unit DC-flow relation requires which factor?

A. Only generator fuel cost

B. Base apparent power, when using per-unit reactance

C. Only reactive load

D. The start-up cost

Correct Answer: B. Base apparent power, when using per-unit reactance

Question 5

In the loss-allocation example, total generation is 250 MW and load is 240 MW. What is total loss?

A. 5 MW

B. 8 MW

C. 10 MW

D. 20 MW

Correct Answer: C. 10 MW

Question 6

Which constraint prevents upward reserve from exceeding available headroom?

A. R_it ≤ P_i^max u_it − P_it

B. R_it ≥ P_it

C. R_it = D_t

D. R_it ≤ P_i^min

Correct Answer: A. R_it ≤ P_i^max u_it − P_it

Question 7

What does AC OPF represent that basic DC OPF does not?

A. Only fuel prices

B. Reactive power and voltage magnitude

C. Only start-up costs

D. Only minimum down time

Correct Answer: B. Reactive power and voltage magnitude

Question 8

Why can congestion produce different locational marginal prices?

A. It changes fuel composition automatically

B. It makes all generators identical

C. It restricts transfers and changes the marginal resource by location

D. It eliminates transmission losses

Correct Answer: C. It restricts transfers and changes the marginal resource by location

Question 9

If three generators produce 120, 80, and 50 MW and 10 MW is allocated proportionally, how much is assigned to the 80-MW generator?

A. 2.0 MW

B. 3.2 MW

C. 4.0 MW

D. 8.0 MW

Correct Answer: B. 3.2 MW

Question 10

Which statement about loss allocation is correct?

A. Physics uniquely determines one settlement allocation

B. Every method must assign identical losses to every generator

C. Allocation is a defined settlement rule that may use proportional, marginal, or tracing methods

D. Losses can be ignored in OPF and settlement

Correct Answer: C. Allocation is a defined settlement rule that may use proportional, marginal, or tracing methods