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[2020.2] PSOC - Unit_Commitment.pptx
1.
Unit Commitment © 2011
Daniel Kirschen and the University of Washington 1
2.
OVERVIEW OF UNIT
COMMITMENT (Vertical System) © 2011 Daniel Kirschen and the University of Washington 2
3.
Economic Dispatch: Problem
Definition • Given load • Given set of units on-line • How much should each unit generate to meet this load at minimum cost? © 2011 Daniel Kirschen and the University of Washington 3 A B C L
4.
Typical summer and
winter loads © 2011 Daniel Kirschen and the University of Washington 4
5.
Unit Commitment • Given
load profile (e.g. values of the load for each hour of a day) • Given set of units available • When should each unit be started, stopped and how much should it generate to meet the load at minimum cost? © 2011 Daniel Kirschen and the University of Washington 5 G G G Load Profile ? ? ?
6.
A Simple Example •
Unit 1: • PMin = 250 MW, PMax = 600 MW • C1 = 510.0 + 7.9 P1 + 0.00172 P1 2 $/h • Unit 2: • PMin = 200 MW, PMax = 400 MW • C2 = 310.0 + 7.85 P2 + 0.00194 P2 2 $/h • Unit 3: • PMin = 150 MW, PMax = 500 MW • C3 = 78.0 + 9.56 P3 + 0.00694 P3 2 $/h • What combination of units 1, 2 and 3 will produce 550 MW at minimum cost? • How much should each unit in that combination generate? © 2011 Daniel Kirschen and the University of Washington 6
7.
Cost of the
various combinations © 2011 Daniel Kirschen and the University of Washington 7
8.
Observations on the
example: • Far too few units committed: Can’t meet the demand • Not enough units committed: Some units operate above optimum • Too many units committed: Some units below optimum • Far too many units committed: Minimum generation exceeds demand • No-load cost affects choice of optimal combination © 2011 Daniel Kirschen and the University of Washington 8
9.
A more ambitious
example • Optimal generation schedule for a load profile • Decompose the profile into a set of period • Assume load is constant over each period • For each time period, which units should be committed to generate at minimum cost during that period? © 2011 Daniel Kirschen and the University of Washington 9 Load Time 12 6 0 18 24 500 1000
10.
Optimal combination for
each hour © 2011 Daniel Kirschen and the University of Washington 10
11.
Matching the combinations
to the load © 2011 Daniel Kirschen and the University of Washington 11 Load Time 12 6 0 18 24 Unit 1 Unit 2 Unit 3
12.
Issues • Must consider
constraints – Unit constraints – System constraints • Some constraints create a link between periods • Start-up costs – Cost incurred when we start a generating unit – Different units have different start-up costs • Curse of dimensionality © 2011 Daniel Kirschen and the University of Washington 12
13.
Unit Constraints • Constraints
that affect each unit individually: –Maximum generating capacity –Minimum stable generation –Minimum “up time” –Minimum “down time” –Ramp rate © 2011 Daniel Kirschen and the University of Washington 13
14.
Notations © 2011 Daniel
Kirschen and the University of Washington 14 u(i,t): Status of unit i at period t x(i,t): Power produced by unit i during period t Unit i is on during period t u(i,t) =1: Unit i is off during period t u(i,t) = 0 :
15.
Minimum up- and
down-time • Minimum up time – Once a unit is running it may not be shut down immediately: • Minimum down time – Once a unit is shut down, it may not be started immediately © 2011 Daniel Kirschen and the University of Washington 15 If u(i,t) =1 and ti up < ti up,min then u(i,t +1) =1 If u(i,t) = 0 and ti down < ti down,min then u(i,t +1) = 0
16.
Ramp rates • Maximum
ramp rates – To avoid damaging the turbine, the electrical output of a unit cannot change by more than a certain amount over a period of time: © 2011 Daniel Kirschen and the University of Washington 16 x i,t +1 ( )- x i,t ( )£ DPi up,max x(i,t)- x(i,t +1) £ DPi down,max Maximum ramp up rate constraint: Maximum ramp down rate constraint:
17.
System Constraints • Constraints
that affect more than one unit – Load/generation balance – Reserve generation capacity – Emission constraints – Network constraints © 2011 Daniel Kirschen and the University of Washington 17
18.
Load/Generation Balance Constraint ©
2011 Daniel Kirschen and the University of Washington 18 u(i,t)x(i,t) i=1 N å = L(t) N : Set of available units
19.
Reserve Capacity Constraint •
Unanticipated loss of a generating unit or an interconnection causes unacceptable frequency drop if not corrected rapidly • Need to increase production from other units to keep frequency drop within acceptable limits • Rapid increase in production only possible if committed units are not all operating at their maximum capacity © 2011 Daniel Kirschen and the University of Washington 19 u(i,t) i=1 N å Pi max ³ L(t)+ R(t) R(t): Reserve requirement at time t
20.
How much reserve? •
Protect the system against “credible outages” • Deterministic criteria: – Capacity of largest unit or interconnection – Percentage of peak load • Probabilistic criteria: – Takes into account the number and size of the committed units as well as their outage rate © 2011 Daniel Kirschen and the University of Washington 20
21.
Types of Reserve •
Spinning reserve – Primary • Quick response for a short time – Secondary • Slower response for a longer time • Tertiary reserve – Replace primary and secondary reserve to protect against another outage – Provided by units that can start quickly (e.g. open cycle gas turbines) – Also called scheduled or off-line reserve © 2011 Daniel Kirschen and the University of Washington 21
22.
Types of Reserve •
Positive reserve – Increase output when generation < load • Negative reserve – Decrease output when generation > load • Other sources of reserve: – Pumped hydro plants – Demand reduction (e.g. voluntary load shedding) • Reserve must be spread around the network – Must be able to deploy reserve even if the network is congested © 2011 Daniel Kirschen and the University of Washington 22
23.
Cost of Reserve •
Reserve has a cost even when it is not called • More units scheduled than required – Units not operated at their maximum efficiency – Extra start up costs • Must build units capable of rapid response • Cost of reserve proportionally larger in small systems • Important driver for the creation of interconnections between systems © 2011 Daniel Kirschen and the University of Washington 23
24.
Environmental constraints • Scheduling
of generating units may be affected by environmental constraints • Constraints on pollutants such SO2, NOx – Various forms: • Limit on each plant at each hour • Limit on plant over a year • Limit on a group of plants over a year • Constraints on hydro generation – Protection of wildlife – Navigation, recreation © 2011 Daniel Kirschen and the University of Washington 24
25.
Network Constraints • Transmission
network may have an effect on the commitment of units – Some units must run to provide voltage support – The output of some units may be limited because their output would exceed the transmission capacity of the network © 2011 Daniel Kirschen and the University of Washington 25 Cheap generators May be “constrained off” More expensive generator May be “constrained on” A B
26.
Start-up Costs • Thermal
units must be “warmed up” before they can be brought on-line • Warming up a unit costs money • Start-up cost depends on time unit has been off © 2011 Daniel Kirschen and the University of Washington 26 SCi (ti OFF ) = ai + bi (1 - e - ti OFF t i ) ti OFF αi αi + βi
27.
Start-up Costs • Need
to “balance” start-up costs and running costs • Example: – Diesel generator: low start-up cost, high running cost – Coal plant: high start-up cost, low running cost • Issues: – How long should a unit run to “recover” its start-up cost? – Start-up one more large unit or a diesel generator to cover the peak? – Shutdown one more unit at night or run several units part- loaded? © 2011 Daniel Kirschen and the University of Washington 27
28.
Summary • Some constraints
link periods together • Minimizing the total cost (start-up + running) must be done over the whole period of study • Generation scheduling or unit commitment is a more general problem than economic dispatch • Economic dispatch is a sub-problem of generation scheduling © 2011 Daniel Kirschen and the University of Washington 28
29.
Flexible Plants • Power
output can be adjusted (within limits) • Examples: – Coal-fired – Oil-fired – Open cycle gas turbines – Combined cycle gas turbines – Hydro plants with storage • Status and power output can be optimized © 2011 Daniel Kirschen and the University of Washington 29 Thermal units
30.
Inflexible Plants • Power
output cannot be adjusted for technical or commercial reasons • Examples: – Nuclear – Run-of-the-river hydro – Renewables (wind, solar,…) – Combined heat and power (CHP, cogeneration) • Output treated as given when optimizing © 2011 Daniel Kirschen and the University of Washington 30
31.
Solving the Unit
Commitment Problem • Decision variables: – Status of each unit at each period: – Output of each unit at each period: • Combination of integer and continuous variables © 2011 Daniel Kirschen and the University of Washington 31 u(i,t) Î 0,1 { } " i,t x(i,t) Î 0, Pi min ;Pi max é ë ù û { } " i,t
32.
Optimization with integer
variables • Continuous variables – Can follow the gradients or use LP – Any value within the feasible set is OK • Discrete variables – There is no gradient – Can only take a finite number of values – Problem is not convex – Must try combinations of discrete values © 2011 Daniel Kirschen and the University of Washington 32
33.
How many combinations
are there? © 2011 Daniel Kirschen and the University of Washington 33 • Examples – 3 units: 8 possible states – N units: 2N possible states 111 110 101 100 011 010 001 000
34.
How many solutions
are there anyway? © 2011 Daniel Kirschen and the University of Washington 34 1 2 3 4 5 6 T= • Optimization over a time horizon divided into intervals • A solution is a path linking one combination at each interval • How many such paths are there?
35.
How many solutions
are there anyway? © 2011 Daniel Kirschen and the University of Washington 35 1 2 3 4 5 6 T= Optimization over a time horizon divided into intervals A solution is a path linking one combination at each interval How many such path are there? Answer: 2N ( ) 2N ( )… 2N ( ) = 2N ( )T
36.
The Curse of
Dimensionality • Example: 5 units, 24 hours • Processing 109 combinations/second, this would take 1.9 1019 years to solve • There are 100’s of units in large power systems... • Many of these combinations do not satisfy the constraints © 2011 Daniel Kirschen and the University of Washington 36 2N ( ) T = 25 ( ) 24 = 6.21035 combinations
37.
How do you
Beat the Curse? Brute force approach won’t work! • Need to be smart • Try only a small subset of all combinations • Can’t guarantee optimality of the solution • Try to get as close as possible within a reasonable amount of time © 2011 Daniel Kirschen and the University of Washington 37
38.
Main Solution Techniques •
Characteristics of a good technique – Solution close to the optimum – Reasonable computing time – Ability to model constraints • Priority list / heuristic approach • Dynamic programming • Lagrangian relaxation • Mixed Integer Programming © 2011 Daniel Kirschen and the University of Washington 38 State of the art
39.
A Simple Unit
Commitment Example © 2011 Daniel Kirschen and the University of Washington 39
40.
Unit Data © 2011
Daniel Kirschen and the University of Washington 40 Unit Pmin (MW) Pmax (MW) Min up (h) Min down (h) No-load cost ($) Marginal cost ($/MWh) Start-up cost ($) Initial status A 150 250 3 3 0 10 1,000 ON B 50 100 2 1 0 12 600 OFF C 10 50 1 1 0 20 100 OFF
41.
Demand Data © 2011
Daniel Kirschen and the University of Washington 41 Hourly Demand 0 50 100 150 200 250 300 350 1 2 3 Hours Load Reserve requirements are not considered
42.
Feasible Unit Combinations
(states) © 2011 Daniel Kirschen and the University of Washington 42 Combinations Pmin Pmax A B C 1 1 1 210 400 1 1 0 200 350 1 0 1 160 300 1 0 0 150 250 0 1 1 60 150 0 1 0 50 100 0 0 1 10 50 0 0 0 0 0 1 2 3 150 300 200
43.
Transitions between feasible
combinations © 2011 Daniel Kirschen and the University of Washington 43 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 2 3 Initial State
44.
Infeasible transitions: Minimum
down time of unit A © 2011 Daniel Kirschen and the University of Washington 44 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 2 3 Initial State TD TU A 3 3 B 1 2 C 1 1
45.
Infeasible transitions: Minimum
up time of unit B © 2011 Daniel Kirschen and the University of Washington 45 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 2 3 Initial State TD TU A 3 3 B 1 2 C 1 1
46.
Feasible transitions © 2011
Daniel Kirschen and the University of Washington 46 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 2 3 Initial State
47.
Operating costs © 2011
Daniel Kirschen and the University of Washington 47 1 1 1 1 1 0 1 0 1 1 0 0 1 4 3 2 5 6 7
48.
Economic dispatch © 2011
Daniel Kirschen and the University of Washington 48 State Load PA PB PC Cost 1 150 150 0 0 1500 2 300 250 0 50 3500 3 300 250 50 0 3100 4 300 240 50 10 3200 5 200 200 0 0 2000 6 200 190 0 10 2100 7 200 150 50 0 2100 Unit Pmin Pmax No-load cost Marginal cost A 150 250 0 10 B 50 100 0 12 C 10 50 0 20
49.
Operating costs © 2011
Daniel Kirschen and the University of Washington 49 1 1 1 1 1 0 1 0 1 1 0 0 1 4 3 2 5 6 7 $1500 $3500 $3100 $3200 $2000 $2100 $2100
50.
Start-up costs © 2011
Daniel Kirschen and the University of Washington 50 1 1 1 1 1 0 1 0 1 1 0 0 1 4 3 2 5 6 7 $1500 $3500 $3100 $3200 $2000 $2100 $2100 Unit Start-up cost A 1000 B 600 C 100 $0 $0 $0 $0 $0 $600 $100 $600 $700
51.
Accumulated costs © 2011
Daniel Kirschen and the University of Washington 51 1 1 1 1 1 0 1 0 1 1 0 0 1 4 3 2 5 6 7 $1500 $3500 $3100 $3200 $2000 $2100 $2100 $1500 $5100 $5200 $5400 $7300 $7200 $7100 $0 $0 $0 $0 $0 $600 $100 $600 $700
52.
Total costs © 2011
Daniel Kirschen and the University of Washington 52 1 1 1 1 1 0 1 0 1 1 0 0 1 4 3 2 5 6 7 $7300 $7200 $7100 Lowest total cost
53.
Optimal solution © 2011
Daniel Kirschen and the University of Washington 53 1 1 1 1 1 0 1 0 1 1 0 0 1 2 5 $7100
54.
Notes • This example
is intended to illustrate the principles of unit commitment • Some constraints have been ignored and others artificially tightened to simplify the problem and make it solvable by hand • Therefore it does not illustrate the true complexity of the problem • The solution method used in this example is based on dynamic programming. This technique is no longer used in industry because it only works for small systems (< 20 units) © 2011 Daniel Kirschen and the University of Washington 54
55.
OVERVIEW OF UNIT
COMMITMENT (Unbundled System) © 2011 Daniel Kirschen and the University of Washington 55
56.
Market-Clearing Price Aggregate Demand Aggregate Supply Price ($) MCP Quantity (MW) Stochastic
Model of Market-Clearing Price: MCP determined by load and availability of generating units prevailing at a particular time. Units are loaded in a predetermined Loading order MCP is marginal cost of the last unit loaded to meet the load.
57.
Electricity Prices
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