Economic Operation and Optimization

Introduction and Context

An electric power system must supply demand continuously while satisfying physical, security, environmental, and market constraints. Economic operation is the coordinated scheduling and control of generating resources so that the required electrical load is served at minimum feasible cost. Optimization extends this idea by providing mathematical methods for selecting the best operating point when multiple objectives and constraints exist.

The problem is more complex than simply selecting the generator with the lowest fuel price. A generating unit may have a high fixed cost, transmission losses may depend on dispatch, ramp-rate limits may restrict output changes, and reliability constraints may require reserves. Modern systems also include intermittent renewable generation, storage, demand response, flexible loads, and electricity markets. Consequently, economic operation is a constrained optimization problem spanning several time scales:

• Unit commitment: deciding which generating units should be on or off over a scheduling horizon.
• Economic dispatch: allocating power among committed units at a particular time.
• Security-constrained dispatch: performing dispatch while maintaining acceptable operation after specified contingencies.
• Optimal power flow: simultaneously determining generator outputs, bus voltages, angles, and network flows.
• Real-time balancing: correcting deviations between forecast and actual generation or demand.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain the economic principles underlying power-system operation.
2. Derive and apply the equal-incremental-cost criterion for economic dispatch.
3. Include transmission losses, generator limits, ramp limits, and prohibited operating zones in dispatch models.
4. Formulate unit commitment and security-constrained optimization problems.
5. Distinguish classical optimization from market-based and multi-objective approaches.
6. Interpret locational marginal prices and the role of optimization in modern power systems.
7. Evaluate practical, environmental, regulatory, and computational limitations.

1. Cost Models of Generating Units

The short-run production cost of a thermal generator is commonly represented by a quadratic heat-rate or fuel-cost curve:

where C_i is the cost of unit i in currency per hour, P_i is generator output in MW, a_i is the no-load cost in currency per hour, b_i is the linear cost coefficient in currency per MWh, and c_i is the quadratic coefficient in currency per MW²h. The model is applicable over the unit's permitted operating range and assumes that valve-point effects, start-up costs, and discrete operating states are either negligible or modeled separately.

The incremental cost is the derivative of cost with respect to output:

It is measured in currency per MWh and represents the additional operating cost of producing one more MW for one hour. For a convex cost curve, c_i > 0, incremental cost increases with output.

A more detailed model may include piecewise-linear costs, minimum up and down times, start-up costs, shut-down costs, emissions costs, and nonconvex operating regions. Hydroelectric plants have opportunity costs associated with stored water rather than ordinary fuel costs. Renewable generators generally have low short-run marginal cost, although curtailment and balancing costs may be economically significant.

2. Economic Dispatch Without Transmission Losses

Suppose N committed generators supply a fixed demand P_D. Ignoring losses, the dispatch problem is

subject to

and

Here P_i^min and P_i^max are the minimum and maximum feasible outputs. The balance equation and limits must be satisfied in consistent power units, normally MW.

For an interior optimum, the Lagrangian is

Stationarity gives

Thus, all unconstrained units operate at equal incremental cost. The multiplier λ is the marginal system cost: the increase in minimum operating cost caused by a one-MW increase in demand, under the stated assumptions.

The equal-incremental-cost rule does not mean that all units produce equal power or incur equal total cost. A unit with a lower incremental-cost curve normally supplies more power. If a generator reaches a limit, it is fixed at that limit and the remaining unconstrained units satisfy the equality among themselves.

Worked Example: Two-Unit Dispatch

Two generators supply 300 MW. Their costs are

Assume no losses and output limits that do not bind. The balance equation is P₁ + P₂ = 300. Incremental costs are

At optimum, 8 + 0.008P₁ = 7 + 0.012P₂. Substituting P₂ = 300 − P₁ gives

8 + 0.008P₁ = 7 + 0.012(300 − P₁),

so 0.020P₁ = 2.6 and P₁ = 130 MW. Therefore P₂ = 170 MW. The common incremental cost is 9.04 currency/MWh.

The result should be checked against generator limits and the convexity assumption. If either output violates its limit, the limiting unit is fixed and the calculation is repeated for the remaining units.

3. Economic Dispatch with Transmission Losses

Transmission losses depend on network flows and generator locations. A simplified loss formula is the B-coefficient model:

The coefficients depend on the network, voltage assumptions, reference conditions, and choice of units. They are not universal constants. The power-balance constraint becomes

For an interior optimum, differentiation of the constrained problem produces

where ∂P_L/∂P_i is the incremental transmission-loss coefficient. Equivalently, the incremental cost multiplied by the appropriate penalty factor is equal across unconstrained units:

A generator electrically close to the load may have a lower loss penalty than a remote generator. The loss formula is useful for instructional and approximate studies; detailed operational decisions normally use AC or DC optimal power flow.

4. Network-Constrained Optimal Power Flow

Economic dispatch treats generation and demand as aggregated quantities. Optimal power flow (OPF) includes the electrical network. A standard AC-OPF minimizes total operating cost subject to nonlinear power-flow equations, generator limits, voltage limits, transformer constraints, and line thermal limits.

For bus k, the active and reactive power-balance equations in polar form are commonly written as

Here V_k is voltage magnitude, θ_k is voltage angle, and G_km and B_km are entries of the bus-admittance matrix. Sign conventions must be defined consistently for injections and withdrawals.

Typical constraints include

and line-flow limits such as S_l ≤ S_l^max, where S_l is apparent-power flow on line l. AC-OPF is nonlinear and generally nonconvex. DC-OPF linearizes the network by assuming small angle differences, approximately constant voltage magnitudes, negligible resistance, and simplified reactive-power behavior. It is computationally efficient but may not capture voltage, reactive-power, or loss-related constraints accurately.

5. Unit Commitment

Economic dispatch assumes that the set of available units is already known. Unit commitment determines the binary operating schedule. Let u_it ∈ {0,1} indicate whether unit i is on during period t, and let P_it be its output. A simplified objective is

where SU_it and SD_it are start-up and shut-down costs, and EC_it represents emissions or other operating charges.

The principal constraints are

capacity limits, minimum up and down times, ramp-rate limits, and reserve requirements. A ramp constraint can be expressed as

where RU_i and RD_i are upward and downward ramp limits in MW per scheduling interval.

Unit commitment is a mixed-integer optimization problem. Dynamic programming, priority-list methods, branch-and-bound, mixed-integer programming, Lagrangian relaxation, and decomposition methods are used in practice. There is a trade-off between exactness and computation time, particularly in real-time applications.

6. Reserves and Security-Constrained Operation

Economic operation must preserve reliability. Operating reserves are additional available capability that can respond to uncertainty or contingencies. Categories include regulating reserve, spinning reserve, non-spinning reserve, and replacement reserve. The required amount depends on forecast error, generator outages, renewable variability, and reliability standards.

Security-constrained economic dispatch (SCED) imposes the normal operating constraints and limits flows or voltages under specified contingencies. In an N−1 criterion, the system should remain within defined operating limits after the loss of any one selected major component, subject to accepted emergency controls.

Security has an economic cost. A dispatch that is cheapest in the intact network may overload a line after a contingency. A more expensive pre-contingency dispatch can reduce redispatch, load shedding, or equipment damage risk. The correct optimization therefore includes the value of reliability and the regulatory security criteria applicable to the system.

7. Electricity Markets and Locational Marginal Pricing

In a centrally coordinated market, generators submit offers and loads may submit bids. The market-clearing optimization selects schedules subject to network and reliability constraints. The locational marginal price (LMP) at a bus is the marginal cost of serving one additional MW of demand at that location, including energy, congestion, and marginal-loss components.

Conceptually,

The exact decomposition depends on the market formulation and settlement rules. When the network is uncongested and losses are neglected, prices may be uniform. Congestion separates prices because increasing injection at one location may require redispatch or may worsen a binding line constraint.

A dispatch optimization and a market settlement are related but not identical. Market rules determine treatment of start-up costs, uplift payments, nonconvex offers, scarcity pricing, financial transmission rights, and out-of-market actions. Engineers must understand both the physical model and the institutional rules before interpreting a price as a complete measure of system cost.

8. Environmental and Multi-Objective Optimization

Economic operation increasingly incorporates emissions. If E_i(P_i) denotes emissions rate, a carbon price τ can be included as τE_i(P_i). Alternatively, emissions may be constrained:

A weighted multi-objective formulation may be written as

where C is operating cost, E is emissions, R is a reliability or risk metric, and the nonnegative weights represent decision priorities. Weighted sums require careful normalization because the objectives have different units. Pareto analysis is often more informative: a solution is Pareto-optimal when no objective can be improved without worsening at least one other objective.

Renewable integration changes the optimization structure. Forecast uncertainty can be addressed with stochastic optimization, scenario-based scheduling, robust optimization, or reserve co-optimization. Storage introduces intertemporal constraints:

where E_t is stored energy, P_c,t and P_d,t are charging and discharging power, η_c and η_d are efficiencies, and ∆t is the interval duration. Bounds on state of charge, charging power, and discharging power are also required. Simultaneous charging and discharging is normally prohibited or penalized.

9. Solution Methods and Computational Practice

Classical dispatch with smooth convex costs can be solved by lambda iteration, gradient methods, or quadratic programming. Lambda iteration adjusts λ until calculated generation equals demand plus losses. Newton-based methods are effective for nonlinear equations but require suitable initialization and derivative information.

Mixed-integer programming is appropriate for unit commitment with binary decisions. AC-OPF commonly uses nonlinear programming, sequential quadratic programming, interior-point methods, or semidefinite and convex relaxations. Heuristic and metaheuristic methods—including genetic algorithms, particle swarm optimization, and simulated annealing—can be useful for difficult nonconvex formulations, but they do not automatically guarantee global optimality. Their results require feasibility checks, sensitivity analysis, and comparison with benchmark methods.

Artificial intelligence can assist load forecasting, renewable forecasting, anomaly detection, contingency screening, and approximate dispatch. However, data-driven models should not replace physical feasibility checks. Explainability, training-data bias, cybersecurity, model drift, and safe fallback operation are important governance concerns. A black-box recommendation that violates voltage or thermal limits is not an acceptable operating solution.

10. Illustrative Case Study: Congested Renewable-Rich Network

Background. Consider an illustrative three-zone system. Zone A contains low-marginal-cost wind generation, Zone B contains a gas generator, and Zone C contains the largest load. The corridor from A to C has limited thermal capacity. Gas generation is more expensive but dispatchable.

Problem. A high-wind forecast suggests that wind should supply most demand. An unconstrained economic dispatch selects wind because its short-run marginal cost is lowest. However, the A–C corridor would exceed its security limit.

Analysis. The system operator solves a security-constrained dispatch. Wind output is curtailed to the maximum transferable level. Gas generation in Zone B increases, and reserve is procured for wind forecast error. The marginal cost in C rises because the next MW of demand must be supplied by local or less efficient generation rather than unconstrained wind. The resulting LMPs differ by zone because of congestion.

Approach and solution. The operator uses a DC-SCED for rapid scheduling, followed by AC power-flow validation. If the AC validation identifies unacceptable voltage or reactive-power conditions, generator reactive limits, voltage-support resources, or a revised dispatch are incorporated. Storage near Zone C may reduce congestion by charging during low demand and discharging during the constrained period.

Outcome. The dispatch is more expensive than the unconstrained schedule but satisfies line limits, preserves reserve, and reduces the risk of emergency curtailment. The case demonstrates that the cheapest generation fleet-wide is not necessarily the cheapest feasible supply at every location.

Learning points. Economic optimization must include network physics, uncertainty, and reliability. Renewable curtailment can be economically rational when transmission expansion is unavailable or more costly. A market price is meaningful only when its network and settlement assumptions are understood.

11. Limitations, Governance, and Best Practice

Optimization results are only as reliable as their data and assumptions. Important sources of error include inaccurate demand forecasts, incorrect generator cost curves, unavailable transmission capacity, unmodeled outages, inconsistent loss factors, and inappropriate reserve requirements. Model validation should include historical back-testing, independent power-flow verification, sensitivity analysis, and explicit treatment of infeasibility.

Operational decisions should document objective functions, constraint sets, data timestamps, solver status, and manual overrides. Cybersecurity controls are essential because manipulation of bids, telemetry, or optimization inputs can produce physical and financial harm. Market participants must comply with applicable grid codes, reliability standards, environmental rules, and non-discrimination requirements.

Best practice includes:

• separating fixed, start-up, no-load, and marginal costs;
• checking units and sign conventions;
• validating every dispatch with a power-flow study;
• modeling reserves and ramping explicitly;
• testing sensitivity to fuel prices, outages, and forecasts;
• using AC validation after simplified DC optimization;
• retaining auditable optimization and settlement records; and
• combining automated optimization with qualified engineering judgment.

Chapter Summary

Economic operation minimizes the cost of meeting demand while respecting generator and system constraints. For smooth convex generator costs and no losses, unconstrained units operate at equal incremental cost. Transmission losses modify this rule through incremental loss coefficients or penalty factors. Optimal power flow extends dispatch by representing voltages, angles, line flows, and network constraints. Unit commitment adds binary decisions, start-up costs, minimum up and down times, and ramp limits.

Security-constrained operation incorporates contingencies and reserves. In electricity markets, locational marginal prices reflect the marginal cost of serving demand at a specific network location, including energy, congestion, and losses. Environmental constraints, renewable uncertainty, storage, and demand response require multi-period and multi-objective formulations. Classical mathematical programming remains central, while AI is useful for forecasting and decision support only when combined with physical validation and governance. Sound practice requires accurate data, transparent assumptions, feasibility checks, reliability analysis, and awareness of regulatory and ethical obligations.

Key Terms

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

Incremental cost: Additional cost of producing one more unit of energy.

Lambda: Lagrange multiplier representing marginal system operating cost.

Unit commitment: Selection of generator on/off schedules over time.

No-load cost: Cost incurred by an online unit independent of its output.

Start-up cost: Cost of bringing a generator from an off state to operation.

Ramp rate: Maximum permitted change in generator output per time interval.

Transmission loss: Electrical power dissipated in network elements.

Penalty factor: Factor adjusting incremental cost for marginal transmission losses.

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

AC-OPF: OPF using nonlinear alternating-current power-flow equations.

DC-OPF: Linearized OPF based mainly on active-power and angle relationships.

Reserve: Available capability held to respond to uncertainty or contingencies.

Security-constrained dispatch: Dispatch satisfying normal and selected contingency constraints.

Locational marginal price: Marginal cost of serving demand at a particular bus.

Congestion: Condition in which a network constraint limits economically preferred transfers.

Pareto optimum: Solution where improving one objective necessarily worsens another.

Stochastic optimization: Optimization explicitly using probability distributions or scenarios.

Robust optimization: Optimization designed to remain feasible over specified uncertainty sets.

Economic curtailment: Reduction of otherwise available generation because of system constraints or costs.

Review Questions

Short Answer Questions

1. What is the difference between economic dispatch and unit commitment?
2. Define incremental cost and explain its role in dispatch.
3. Why can locational marginal prices differ between buses?
4. State two assumptions commonly used in DC optimal power flow.
5. What is the purpose of operating reserves?

Long Answer Questions

1. Derive the equal-incremental-cost criterion using the Lagrangian method and explain the effect of generator limits.
2. Explain how transmission losses modify economic dispatch and discuss the limitations of the B-coefficient loss model.
3. Formulate the principal objective and constraints of a thermal unit-commitment problem.
4. Compare AC-OPF and DC-OPF in terms of physical representation, computational burden, and applications.
5. Discuss how renewable uncertainty, storage, and emissions constraints change conventional economic operation.

Analytical / Application Questions

1. Two unconstrained generators have incremental costs IC₁ = 10 + 0.01P₁ and IC₂ = 8 + 0.02P₂, with demand 300 MW and no losses. Determine the economic dispatch and common incremental cost.
2. A low-cost wind plant is separated from a load center by a binding transmission corridor. Explain how a security-constrained dispatch changes generation, prices, reserves, and possible curtailment.
3. Design an optimization study for a system containing thermal generators, wind, and a battery. Specify the objective, principal constraints, uncertainty treatment, validation steps, and reliability checks.

Multiple Choice Questions with Answers

Question 1

For unconstrained generators with convex cost curves and no transmission losses, the economic-dispatch condition is:

A. Equal total cost for all generators

B. Equal incremental cost for all unconstrained generators

C. Equal power output for all generators

D. Maximum output from every generator

Correct Answer: B. Equal incremental cost for all unconstrained generators

Question 2

For the cost function C(P)=a+bP+cP², the incremental cost is:

A. a+bP

B. b+2cP

C. a+cP²

D. 2a+bP

Correct Answer: B. b+2cP

Question 3

Which feature distinguishes unit commitment from ordinary economic dispatch?

A. It includes generator voltage angles only

B. It uses binary on/off decisions and time-coupled constraints

C. It ignores generator limits

D. It assumes demand is zero

Correct Answer: B. It uses binary on/off decisions and time-coupled constraints

Question 4

In lossless economic dispatch, the Lagrange multiplier for the power-balance constraint represents:

A. Generator maximum output

B. Marginal system operating cost

C. Transmission-line resistance

D. Minimum up time

Correct Answer: B. Marginal system operating cost

Question 5

A binding transmission constraint can cause:

A. Identical prices at every bus in all conditions

B. Different locational marginal prices

C. Zero marginal cost for all generators

D. Removal of all reserve requirements

Correct Answer: B. Different locational marginal prices

Question 6

Which model normally represents voltage magnitudes and reactive power explicitly?

A. AC-OPF

B. Priority-list dispatch

C. Simple merit-order dispatch

D. Unconstrained lambda iteration

Correct Answer: A. AC-OPF

Question 7

If a generator reaches its maximum output during economic dispatch, it should generally be:

A. Fixed at its maximum while remaining units are re-dispatched

B. Removed from the system permanently

C. Assigned negative output

D. Allowed to violate its limit

Correct Answer: A. Fixed at its maximum while remaining units are re-dispatched

Question 8

The principal purpose of spinning reserve is to:

A. Increase fuel consumption intentionally

B. Respond rapidly to contingencies or deviations

C. Replace all transmission lines

D. Eliminate the need for forecasting

Correct Answer: B. Respond rapidly to contingencies or deviations

Question 9

Which constraint is specifically associated with battery intertemporal operation?

A. State-of-charge balance

B. Transformer turns ratio only

C. Generator minimum excitation only

D. Bus numbering constraint

Correct Answer: A. State-of-charge balance

Question 10

A dispatch that minimizes fuel cost but violates a line thermal limit is:

A. A feasible optimal dispatch

B. An infeasible unconstrained result for the actual network

C. Always the market-clearing solution

D. A guaranteed secure schedule

Correct Answer: B. An infeasible unconstrained result for the actual network