Economic dispatch and unit commitment

In short

Economic dispatch (ED) decides how much each already-running generator should produce so that the total fuel cost of meeting the load is minimum, while unit commitment (UC) decides which units should be running at all over the next hours or day, including when to start and stop them. ED works in seconds to minutes within fixed commitment; UC works over 24 hours to a week and must respect start-up costs, minimum up/down times and reserve requirements. Both are core to load despatch centre operation — Grid-India (NLDC/RLDCs) and the State Load Despatch Centres run merit order dispatch under the Indian Electricity Grid Code, with Security Constrained Economic Dispatch run by Grid-India for inter-State stations. The topic is a reliable source of numerical questions in GATE (EE), university power systems papers, and in interviews at NTPC, POWERGRID, Grid-India, state transcos/gencos, Tata Power, Adani and energy-analytics firms.

Explained

Cost curves and the dispatch problem

• Input-output curve: fuel input (kcal/h, Mkcal/h or ₹/h) plotted against output P (MW). For a thermal unit it is approximately quadratic: F(P) = a + bP + cP².
• a is the no-load / fixed running cost (₹/h), b is roughly the linear fuel term (₹/MWh), c is the curvature (₹/MW²h) that makes the unit progressively less efficient near full load.
• Heat rate = heat input per unit output (kcal/kWh); the heat-rate curve is U-shaped, minimum near 80–100% of rating. Incremental heat rate = dH/dP.
• Incremental fuel cost (IFC or IC) = dF/dP = b + 2cP, in ₹/MWh — the cost of the *next* MWh, and the quantity that is equalised in economic dispatch. It is monotonically increasing for a convex cost curve.
• Variable (energy) cost in Indian practice = station heat rate × fuel price ÷ GCV + secondary fuel oil + auxiliary consumption adjustment; it is what merit order stacking uses. Fixed/capacity charges are not part of dispatch — they are sunk once the unit is committed.
• The ED problem: minimise Σ F_i(P_i) subject to Σ P_i = P_D + P_L and P_i,min ≤ P_i ≤ P_i,max.

Economic dispatch neglecting losses — the equal incremental cost criterion

• Form the Lagrangian: L = Σ F_i(P_i) + λ(P_D − Σ P_i).
• Setting ∂L/∂P_i = 0 gives dF_i/dP_i = λ for every unit: all units in economic operation run at the same incremental fuel cost, called the system λ (₹/MWh). ∂L/∂λ = 0 restores the power balance.
• λ is the system's marginal cost — the cost of supplying one more MW. It is the theoretical basis of marginal pricing in electricity markets.
• Intuition: if unit A's IC is lower than unit B's, shifting 1 MW from B to A saves money; savings stop only when the ICs are equal.
• With generation limits (Kuhn-Tucker conditions): dF_i/dP_i = λ when P_i,min < P_i < P_i,max; dF_i/dP_i ≤ λ when P_i = P_i,max; dF_i/dP_i ≥ λ when P_i = P_i,min. Units pinned at a limit drop out of the equalisation.
• The cost curve must be convex (c > 0) for equal-λ to give the true minimum; non-convex curves (valve points, prohibited zones) need other methods.

Economic dispatch including transmission losses

• Losses make a remote cheap unit effectively costlier. Balance becomes Σ P_i = P_D + P_L, with P_L a function of generation.
• Coordination equation: dF_i/dP_i + λ ∂P_L/∂P_i = λ, i.e. L_i × dF_i/dP_i = λ, where L_i = 1/(1 − ∂P_L/∂P_i) is the penalty factor and ∂P_L/∂P_i the incremental transmission loss (ITL).
• L_i > 1 for a unit that increases losses (electrically far from load); its IC is penalised, so it is backed down. L_i < 1 for a unit that reduces losses.
• Kron's loss formula (B-coefficients): P_L = ΣΣ P_i B_ij P_j, or the full form with B_i0 and B_00 terms. B_ij are in MW⁻¹ and are computed from a base-case load flow; they are valid only near that operating point.
• Loss coordination is an approximation to the optimal power flow (OPF), which handles the full AC network, voltage limits and line flows; Security-Constrained OPF/ED adds contingency (N−1) limits.

Constraints and solution methods for ED

• Equality: power balance. Inequality: P_min (technical minimum) and P_max; ramp-up/ramp-down rate (MW/min); line flow and voltage limits; spinning-reserve contribution; fuel/emission caps.
• Technical minimum matters in India: CERC has allowed thermal ISGS to be scheduled down to 55% of MCR (reduced from 70% by the 2016 IEGC amendment) with compensation for part-load efficiency loss — the exact level and compensation formula are set by CERC regulations and have been revised, so quote the current order.
• Lambda-iteration method: (1) guess λ; (2) compute P_i = (λ − b_i)/(2c_i), clamped to limits; (3) find mismatch ΔP = P_D − ΣP_i; (4) raise λ if ΔP > 0, lower it if ΔP < 0 (bisection or Newton: Δλ = ΔP / Σ(1/2c_i)); (5) repeat until |ΔP| < tolerance.
• Other methods: gradient / Newton's method, linear and quadratic programming, dynamic programming for non-convex curves, and metaheuristics (GA, PSO, differential evolution) for valve-point and multi-fuel cases.
• Variants examined: combined economic and emission dispatch (minimise cost + weighted NOx/CO₂), multi-area dispatch with tie-line limits, dynamic economic dispatch (ramp-coupled across intervals), and hydro-thermal coordination, where a water conversion factor γ links the incremental water rate to fuel cost: dF_th/dP_th = γ (dW/dP_hydro), subject to a fixed volume of water over the scheduling period.

Unit commitment — the problem and its constraints

• UC: choose the on/off status u_i(t) ∈ {0,1} of every unit for every interval t over 24 h–1 week, and the dispatch within it, minimising running cost + start-up cost + shut-down cost.
• Needed because load varies over the day; keeping all units on wastes no-load fuel, while starting a large thermal unit costs lakhs of rupees and takes hours.
• Constraints:
  • Load plus reserve: Σ u_i P_i,max ≥ P_D(t) + R(t); spinning reserve is synchronised headroom, commonly planned to cover at least the largest unit on bar. IEGC requires primary, secondary (AGC) and tertiary reserves; exact MW/percentage figures are fixed by CERC regulation and have been revised.
  • Minimum up time and minimum down time (hours) — thermal metallurgy limits cycling.
  • Start-up cost, which depends on how long the unit was off: hot start (cheap, 1–3 h), warm, cold start (dear, 8–12 h for a large coal unit).
  • Ramp rates, crew/manpower limits on simultaneous starts, must-run units (nuclear, run-of-river hydro, renewables), fuel contracts/take-or-pay, hydro water availability, transmission constraints.
• Start-up cost models: *cooling* — the boiler is allowed to cool and reheated: C = C_c(1 − e^−t/α) + C_f; *banking* — fuel is burnt to keep the boiler warm: C = C_b·t + C_f. Banking is cheaper for short shutdowns, cooling for long ones.
• Combinatorial size: with N units and T intervals the number of commitment schedules is up to (2^N − 1)^T — exhaustive search is impossible for real systems.

Unit commitment solution methods

• Priority list: rank units by full-load average cost FLAC = F(P_max)/P_max (or by variable cost); commit in that order until load + reserve is met, de-commit in reverse. Simple and fast, used as a first cut, but ignores start-up cost and min up/down times, so it is sub-optimal.
• Dynamic programming (forward DP): states = feasible combinations at each hour; recursion F_cost(K, I) = min over L [ P_cost(K, I) + S_cost(K−1, L : K, I) + F_cost(K−1, L) ], where P_cost is the economic-dispatch production cost of combination I in hour K and S_cost the transition (start-up) cost. Strategy saving (keeping only the best X states) controls the explosion.
• Lagrangian relaxation: relax the coupling load and reserve constraints with multipliers; the problem decomposes into one cheap single-unit DP per generator. Multipliers are updated by a sub-gradient method; the duality gap is small for large systems. Scales far better than DP.
• Mixed-Integer Linear Programming (MILP) with branch-and-cut is today's industry standard (CPLEX, Gurobi) — it handles binary status, min up/down, ramping and network constraints directly and gives a proven optimality gap.
• Stochastic/robust UC is used where wind and solar forecast error is large — increasingly relevant to the Indian grid.

Indian practice: merit order, SCED and the regulatory frame

• Merit order dispatch (MOD): SLDCs stack the available generators by variable cost (energy charge) in ₹/kWh and schedule the cheapest first; MOD stacks are published by SLDCs/Grid-India. This is ED applied with piecewise-constant incremental costs.
• Scheduling and despatch follow the Indian Electricity Grid Code (CERC IEGC Regulations, 2023 replacing the 2010 code) in 15-minute time blocks, with the Deviation Settlement Mechanism (DSM Regulations, 2022) pricing departures from schedule against frequency/area-clearing price — this gives the commercial incentive to follow the dispatch.
• Security Constrained Economic Dispatch (SCED): operated by Grid-India (formerly POSOCO) as a pilot from 1 April 2019 on inter-State generating stations, re-optimising regional dispatch every 15 minutes above/below schedule and settling the fuel-cost gain/loss between stations. Its scope and whether it has been made a permanent, wider mechanism are governed by CERC orders that have been extended and amended over time.
• Ancillary services: CERC Ancillary Services Regulations, 2022 cover primary (free governor), Secondary Reserve Ancillary Service (SRAS) delivered through Automatic Generation Control, and TRAS — AGC is the closed-loop cousin of ED, moving units along the merit order every few seconds.
• Must-run renewables: solar and wind are must-run under IEGC, so dispatchable optimisation falls on the thermal and hydro fleet; this raises cycling, ramping and reserve requirements and makes UC harder.
• Statutory base: Electricity Act, 2003 — SLDC (§31–32) and RLDC (§26–28) are the apex despatch bodies, their directions are binding; CERC/SERCs set tariffs, grid code and DSM. A Market Based Economic Dispatch (MBED) reform to pool day-ahead dispatch nationally has been discussed by CERC/Ministry of Power but its status is still evolving — check the current position before quoting it.

Key points

• Economic dispatch – allocating load among *running* units at minimum fuel cost; a continuous optimisation over minutes.
• Unit commitment – deciding *which* units run over the next hours/day; a mixed-integer optimisation with start-up costs.
• Incremental fuel cost – dF/dP = b + 2cP in ₹/MWh; the quantity equalised in dispatch.
• Equal incremental cost criterion – at the optimum (no losses) dF₁/dP₁ = dF₂/dP₂ = … = λ.
• System lambda – the marginal cost of the next MW; the shadow price of the power balance constraint.
• Generator limits – a unit at P_max has IC ≤ λ; a unit at P_min has IC ≥ λ; only free units equalise.
• Penalty factor – L_i = 1/(1 − ∂P_L/∂P_i); the loss-coordinated condition is L_i·dF_i/dP_i = λ.
• B-coefficients – P_L = ΣΣ P_i B_ij P_j, from a base-case load flow; valid only near that operating point.
• Lambda iteration – guess λ, compute P_i, correct λ by Δλ = ΔP/Σ(1/2c_i) until balance.
• Spinning reserve – synchronised headroom on running units, typically at least the largest unit on bar.
• Minimum up/down time – thermal cycling limits that make UC a time-coupled problem.
• Start-up cost – cooling model C_c(1 − e^−t/α) + C_f versus banking model C_b·t + C_f.
• Priority list – commit by full-load average cost F(P_max)/P_max; fast but ignores start-up cost and time constraints.
• Dynamic programming / Lagrangian relaxation / MILP – exact-ish UC methods; MILP is the industry standard.
• Merit order dispatch – Indian practice: stack by variable cost (₹/kWh), fixed charges excluded.
• SCED – Grid-India's 15-minute security-constrained re-dispatch of ISGS, piloted from April 2019.
• Fixed cost is sunk – no-load and capacity charges never enter the ED decision, only the commitment decision.
• OPF – the full network version of ED, with voltage, line-flow and contingency constraints.

Formulas and examples

Example 1 — Variable cost for merit order. A coal unit has station heat rate 2,450 kcal/kWh, coal GCV 3,500 kcal/kg, delivered coal price ₹4,200/tonne (= ₹4.20/kg).

• Specific coal consumption = 2,450 / 3,500 = 0.70 kg/kWh.
• Fuel cost = 0.70 × 4.20 = ₹2.94/kWh = ₹2,940/MWh. This (plus oil and auxiliary adjustment) is the number the SLDC stacks in the merit order; the capacity charge is excluded.

Example 2 — Two-unit economic dispatch, no losses. F₁ = 0.2P₁² + 1800P₁ + 250,000 ₹/h; F₂ = 0.3P₂² + 1900P₂ + 180,000 ₹/h; P_D = 600 MW.

• IC₁ = 0.4P₁ + 1800; IC₂ = 0.6P₂ + 1900.
• Using λ = [P_D + Σb/2c]/Σ(1/2c): Σb/2c = 1800/0.4 + 1900/0.6 = 4,500 + 3,166.67 = 7,666.67; Σ1/2c = 2.5 + 1.6667 = 4.1667.
• λ = (600 + 7,666.67)/4.1667 = ₹1,984/MWh.
• P₁ = (1984 − 1800)/0.4 = 460 MW; P₂ = (1984 − 1900)/0.6 = 140 MW (sum 600 ✓).
• Cost: F₁ = 42,320 + 828,000 + 250,000 = ₹11,20,320/h; F₂ = 5,880 + 266,000 + 180,000 = ₹4,51,880/h; total ₹15,72,200/h.
• Equal sharing 300/300: F₁ = ₹8,08,000, F₂ = ₹7,77,000, total ₹15,85,000/h → economic dispatch saves ₹12,800/h ≈ ₹3.07 lakh/day.

Example 3 — Same system with a generation limit. Now P₁,max = 400 MW.

• Unconstrained P₁ = 460 > 400, so fix P₁ = 400 MW, P₂ = 600 − 400 = 200 MW.
• λ = IC₂ = 0.6(200) + 1900 = ₹2,020/MWh; check IC₁(400) = 0.4(400) + 1800 = 1,960 ≤ λ ✓ (consistent with a unit at its upper limit).
• Cost = (32,000 + 720,000 + 250,000) + (12,000 + 380,000 + 180,000) = 10,02,000 + 5,72,000 = ₹15,74,000/h, i.e. ₹1,800/h more than the unconstrained optimum — the price of the binding limit.

Example 4 — Loss coordination with penalty factors. Same cost curves. Total generation 600 MW meets a load of 580 MW plus 20 MW of losses, with ∂P_L/∂P₁ = 0.05 and ∂P_L/∂P₂ = 0.

• L₁ = 1/(1 − 0.05) = 1.0526; L₂ = 1.
• 1.0526(0.4P₁ + 1800) = 0.6(600 − P₁) + 1900 → 0.4211P₁ + 1894.74 = 2260 − 0.6P₁ → 1.0211P₁ = 365.26.
• P₁ = 357.7 MW, P₂ = 242.3 MW, λ ≈ ₹2,045/MWh. Check: IC₁ = 1,943.1; 1.0526 × 1943.1 = 2,045 ✓.
• Losses shift ~102 MW away from the cheaper but loss-causing unit 1 — the whole point of penalty factors.

Example 5 — B-coefficients. B₁₁ = 1.0×10⁻⁴, B₁₂ = B₂₁ = 0.5×10⁻⁴, B₂₂ = 3.0×10⁻⁴ MW⁻¹, with P₁ = 300 MW, P₂ = 200 MW.

• P_L = 1.0e-4(300²) + 2(0.5e-4)(300×200) + 3.0e-4(200²) = 9 + 6 + 12 = 27 MW.
• ∂P_L/∂P₁ = 2(1.0e-4)(300) + 2(0.5e-4)(200) = 0.06 + 0.02 = 0.08 → L₁ = 1.087.
• ∂P_L/∂P₂ = 2(3.0e-4)(200) + 2(0.5e-4)(300) = 0.12 + 0.03 = 0.15 → L₂ = 1.176.

Example 6 — Priority-list unit commitment over 3 hours.

Priority order A → B → C. Loads: 450, 700, 900 MW; only A is initially on.

• Hour 1 (450 MW): A alone. Cost = 2,00,000 + 1,900(450) = ₹10,55,000.
• Hour 2 (700 MW): A+B (800 MW capacity). A at 500 (cheaper incremental), B at 200 (≥ P_min ✓). Running cost = 11,50,000 + 5,40,000 = 16,90,000; plus B start-up 2,50,000 → ₹19,40,000.
• Hour 3 (900 MW): A+B = 800 < 900, so commit C. A = 500, B = 300, C = 100. Running cost = 11,50,000 + 7,50,000 + 3,80,000 = 22,80,000; plus C start-up 80,000 → ₹23,60,000.
• Three-hour total = ₹53,55,000. Note the reserve check: at hour 3 committed capacity is 950 MW against 900 MW load, so spinning reserve is only 50 MW — far less than the largest unit (500 MW), so a real UC would have to commit extra capacity or buy reserve.

Example 7 — Banking versus cooling. C_c = ₹5,00,000, α = 5 h, C_b = ₹80,000/h, C_f = ₹50,000.

• Off for 2 h: cooling = 5,00,000(1 − e^−0.4) = 5,00,000(0.3297) = ₹1,64,840; banking = 80,000 × 2 = ₹1,60,000 → bank the boiler.
• Off for 6 h: cooling = 5,00,000(1 − e^−1.2) = 5,00,000(0.6988) = ₹3,49,400; banking = ₹4,80,000 → let it cool. (Add C_f = ₹50,000 to both.) Crossover ≈ 2.3 h.

Where it is used

• Load despatch centres — NLDC/RLDCs (Grid-India) and all SLDCs run merit order stacking, SCED, AGC and reserve management; shift engineer and despatcher roles.
• Generation companies — NTPC, NHPC, state gencos and IPPs declare availability and variable cost, plan unit start/stop around the merit order and manage technical-minimum and ramping compensation.
• Power trading and market operations — IEX/PXIL day-ahead and real-time market bidding, portfolio optimisation for discoms, DSM exposure management; analyst and trader roles.
• EMS/SCADA product vendors — Siemens, Hitachi Energy, GE Vernova, Honeywell and Indian integrators build the ED/UC/AGC application suites deployed at Indian LDCs.
• Captive and industrial power — steel, cement, refinery and fertiliser plants dispatch captive boilers, gas turbines and DG sets against grid import cost on the same λ logic.
• Renewable integration studies and consultancies — stochastic UC, reserve adequacy and flexibility studies for high-RE Indian scenarios; also microgrid and battery dispatch controllers.

Interview questions

• Q: State the condition for economic dispatch neglecting losses. A: All running generators must operate at the same incremental fuel cost λ, subject to their MW limits and the power balance.
• Q: What is the physical meaning of λ? A: It is the system marginal cost in ₹/MWh — the extra fuel cost of supplying one more MW — and is the shadow price of the power balance constraint.
• Q: How are generator limits handled in equal-λ dispatch? A: Units that hit a limit are fixed there and removed from the equalisation; a unit at P_max has IC ≤ λ, one at P_min has IC ≥ λ, and the rest still share equal IC.
• Q: What is a penalty factor and when is it greater than one? A: L_i = 1/(1 − ∂P_L/∂P_i); it exceeds one when increasing that generator increases transmission losses, so its incremental cost is penalised and it is backed down.
• Q: Why are fixed costs ignored in economic dispatch but not in unit commitment? A: Once a unit is synchronised, its fixed and no-load costs are sunk and do not change with MW output; the decision to incur them belongs to the commitment stage.
• Q: Difference between economic dispatch and unit commitment? A: ED allocates load among already-running units over minutes; UC is the integer decision of which units to run over hours or days, including start-up costs and min up/down times.
• Q: Why can't you just use a priority list for unit commitment? A: It ignores start-up costs, minimum up/down times, ramp rates and reserve, so it may start an expensive-to-start unit for one hour when keeping a costlier-running unit on would be cheaper overall.
• Q: What is spinning reserve and how much is usually kept? A: Synchronised unloaded capacity able to pick up load quickly; a common planning rule is at least the capacity of the largest unit on bar, with the exact Indian requirement set by IEGC and CERC ancillary services regulations.
• Q: Explain banking versus cooling of a boiler. A: Banking burns fuel to keep the boiler hot, costing C_b per hour; cooling lets it cool and pays an exponentially saturating reheat cost — banking is cheaper for short shutdowns, cooling for long ones.
• Q: What is SCED in the Indian grid? A: Security Constrained Economic Dispatch — Grid-India's 15-minute optimisation that re-dispatches inter-State generating stations around their schedules to reduce overall fuel cost, with the gains settled between the stations; it was introduced as a pilot in April 2019 and its scope has since been extended by CERC orders.
• Q: How do must-run renewables affect dispatch? A: Solar and wind are scheduled first at near-zero variable cost, so the thermal fleet operates at lower and more variable loading, increasing cycling, ramping needs and reserve requirements.
• Q: Why is MILP preferred over dynamic programming for large unit commitment problems? A: DP's state space explodes with the number of units, whereas MILP with branch-and-cut handles binary status, time-coupling and network constraints directly and reports a proven optimality gap.
• Q: What does an optimal power flow add to economic dispatch? A: The full AC network — bus voltages, reactive power, line thermal limits and, in SCOPF, contingency constraints — instead of a single lumped loss term.

Common mistakes

• Equalising total cost or average cost per MWh instead of incremental cost dF/dP.
• Forgetting to check P_min/P_max after a lambda iteration, or continuing to include a limited unit in the equal-λ equations.
• Mixing units: writing c in ₹/MW²h but P in kW, or quoting λ in ₹/kWh while costs are in ₹/h.
• Including fixed/no-load cost or capacity charges in the dispatch comparison — they do not vary with MW.
• Writing the penalty factor as (1 − ∂P_L/∂P_i) instead of its reciprocal, or applying B-coefficients far from the load flow case they were derived for.
• Treating unit commitment as hour-by-hour independent economic dispatch, ignoring start-up cost, minimum up/down time and the reserve requirement that couple the hours.
• Assuming the equal-λ solution is optimal for a non-convex cost curve (valve points, multi-fuel, prohibited operating zones).

Related topics

Power system operation, SCADA and EMS · Electricity markets and regulation in India · Power generation: thermal, hydro, nuclear and renewable · Load flow and fault analysis · Smart grid and renewable integration · Power system stability.