Unit Commitment (UC)

Unit Commitment determines the optimal schedule of generator on/off decisions and power outputs over multiple time periods, typically for day-ahead market clearing or operational planning.

flowchart TB
    A[24-Hour Load Forecast] --> B[Unit Commitment]
    C[Generator Constraints] --> B
    D[Reserve Requirements] --> B
    
    B --> E{Mixed-Integer Optimization}
    
    E --> F[On/Off Schedule u_it]
    E --> G[Power Dispatch p_it]
    E --> H[Reserves r_it]
    E --> I[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 (24+)	Single	Single
Commitment	✗	✓ Binary	✗	✗
Startup Costs	✗	✓	✗	✗
Ramping Limits	✗	✓	✗	✗
Min Up/Down Time	✗	✓	✗	✗
Reserve Requirements	✗	✓	✗	✗
Solve Time	Fastest	Slow	Fast	Medium
→ Detailed Comparison

Mathematical Formulation

Complete UC Model

Sets and Indices

Symbol	Description	Example (KPG193)
	Set of generators
	Set of time periods	(hours)
	Set of buses
	Generator index
	Time period index	(noon)
	Bus index

Decision Variables

Binary Variables

Variable	Domain	Description
		Commitment status: 1 if generator is online at time , 0 otherwise
		Startup indicator: 1 if generator starts up at time , 0 otherwise
		Shutdown indicator: 1 if generator shuts down at time , 0 otherwise

Continuous Variables

Variable	Unit	Description
	p.u.	Power output of generator at time
	p.u.	Upward reserve provided by generator at time
	p.u.	Downward reserve provided by generator at time

Parameters

Generator Technical Parameters

Parameter	Unit	Description
	p.u.	Minimum generation when online
	p.u.	Maximum generation capacity
	p.u.	Power available during startup
	p.u.	Power available during shutdown
	p.u./h	Maximum ramp-up rate
	p.u./h	Maximum ramp-down rate
	h	Minimum up time
	h	Minimum down time

Cost Parameters

Parameter	Unit	Description
	$/h	Generation cost function
	$	Startup cost
	$	Shutdown cost

System Parameters

Parameter	Unit	Description
	p.u.	Demand at bus at time
	p.u.	System upward reserve requirement at time
	p.u.	System downward reserve requirement at time

Formulation Explanation

(1a) Objective Function

Minimize total operational cost:

Three cost components:

Generation cost : Fuel and variable O&M
Startup cost : Turbine warmup, auxiliary systems
Shutdown cost : Turbine cooldown, inspection

For KPG 193:

Coal startup: 2,521-15,127 thousand KRW
LNG startup: 21,545-53,323 thousand KRW
Nuclear: 0 (stays online continuously)

Constraint Explanation

(1b) Minimum Generation Constraint

Generator must produce at least minimum when online:

Physical meaning:

If (online):
If (offline): Constraint becomes

Why minimum generation?

Turbine stability (minimum steam flow)
Emissions control performance
Equipment protection

For KPG 193:

Coal:
LNG:
Nuclear: (baseload)

(1c) Maximum Generation with Startup/Shutdown

Generator limited by capacity and transition states:

Three cases:

Normal operation (, ):

Full capacity if online

During startup ():

Reduced capacity while warming up

Before shutdown ():

Reduced capacity while cooling down

graph LR
    A["(Startup) P ≤ P^SU"] --> B["(Normal) P ≤ P̄"]
    B --> C["(Shutdown) P ≤ P^SD"]
    C --> D["(Offline) P = 0"]

(1d) Ramp-Up Constraint

Limit rate of power increase:

Physical meaning:

Cannot increase output faster than turbine/boiler ramp rate
Protects equipment from thermal stress
Includes reserves (must be able to ramp to cover reserve)

(1e) Ramp-Down Constraint

Limit rate of power decrease:

Physical meaning:

Cannot decrease output faster than safe rate
Turbine blade cooling limits
Boiler pressure management

(1f) Minimum Up Time Constraint

Once started, must stay online:

Interpretation:

If unit started in last hours, it must be online now
Prevents rapid cycling (thermal stress)
Protects equipment lifetime

Timeline example ( hours):

timeline
    title Minimum Up Time Constraint (TU = 4 hours)
    Hour 1 : Offline
    Hour 2 : Startup v=1
    Hour 3 : Must stay on
    Hour 4 : Must stay on  
    Hour 5 : Must stay on
    Hour 6 : Can shutdown

For KPG 193:

Coal: 6 hours
LNG: 4 hours
Nuclear: 8 hours

(1g) Minimum Down Time Constraint

Once stopped, must stay offline:

Interpretation:

If unit shut down in last hours, it must stay offline
Allows equipment to cool properly
Reduces maintenance costs

For KPG 193:

Coal: 12 hours
LNG: 3 hours
Nuclear: 12 hours

(1h) Logical Linking Constraint

Connect commitment, startup, and shutdown:

Four cases:

				Meaning
0	0	0	0	Stays offline
0	1	1	0	Startup
1	1	0	0	Stays online
1	0	0	1	Shutdown

Ensures:

and cannot both be 1
Correct startup/shutdown indicators

(1i) & (1j) Reserve Requirements

System must carry sufficient spinning reserves:

Reserve types:

Upward reserve :

Headroom to increase generation
Covers contingencies (generator/line outages)
Meets forecast errors

Downward reserve :

Ability to decrease generation
Absorbs excess renewable generation
Balances sudden load drops

(1k) Power Balance Constraint

Generation equals demand at each time:

Must hold for every hour in the optimization horizon.

Solution Timeline

gantt
    title 24-Hour UC Solution (Example Day)
    dateFormat HH
    axisFormat %H
    
    section Nuclear (22 units)
    All online 24/7: 00, 24h
    
    section Coal Baseload (30 units)
    Online all day: 00, 24h
    
    section Coal Mid-Merit (8 units)
    Start H05: 05, 19h
    
    section Coal Cycling (8 units)
    Morning peak: 07, 6h
    Evening peak: 17, 6h
    
    section LNG Baseload (10 units)
    Online all day: 00, 24h
    
    section LNG Morning (8 units)
    Morning ramp: 06, 7h
    
    section LNG Evening (10 units)
    Evening peak: 16, 9h

Key observations:

Nuclear runs continuously (baseload)
Coal provides mid-merit with some cycling
LNG follows load (flexible peaking)
Startups timed to avoid min up/down violations

Startup Cost Trajectories

Hot/Warm/Cold Starts

graph TB
    A[Unit Offline] --> B{Time Offline?}
    
    B -->|< 3h| C[Hot Start Cost × 1.0 Fast startup]
    B -->|3-6h| D[Warm Start Cost × 1.1 Medium startup]
    B -->|> 6h| E[Cold Start Cost × 1.2 Slow startup]
    
    C --> F[Online in 1-2 hours]
    D --> G[Online in 2-4 hours]
    E --> H[Online in 4-6 hours]

Startup cost formula:

For KPG193:

Fuel	Delay 1	Delay 2	Delay 3
LNG	3	6	12
Coal	12	24	48
Nuclear	8	16	32

Cold start typically 1.2× hot start cost

Results Interpretation

Commitment Schedule Analysis

Example output:

Hour | Nuclear | Coal | LNG | Total | Demand | Reserve
-----|---------|------|-----|-------|--------|--------
  1  |   22   |  35  | 12  |  69   | 48,000 | 5,200
  6  |   22   |  38  | 18  |  78   | 58,000 | 6,100
 12  |   22   |  40  | 24  |  86   | 72,000 | 7,500
 18  |   22   |  38  | 28  |  88   | 85,000 | 8,800
 24  |   22   |  36  | 15  |  73   | 52,000 | 5,500

Insights:

Nuclear: Always online (baseload)
Coal: Moderate cycling (35-40 units)
LNG: High cycling (12-28 units, follows load)

Extensions

Security-Constrained UC (SCUC) enforce N-1 security constraints in the unit commitment schedule.
Stochastic UC couple commitment decisions across multiple demand and renewable generation scenarios.
Multi-Area UC coordinate unit commitment across multiple regions or control areas.

Next Steps

Understand network constraints: DC-OPF →
Full AC model: AC-OPF →
Compare all models: Solver Comparison →
Run UC: Getting Started →