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    Electrical Power

    System OperationUnit Commitment (part 2)

    Week#6

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    Unit Commitment Solution

    MethodsDynamic programmingchief advantage over enumeration schemes is

    the reduction in the dimensionality of theproblem  in a strict priority order scheme there are only N

    combinations to try for an N unit system

     a strict priority list !ould result in a

    theoretically correct dispatch and commitmentonly if   the no"load costs are ero  unit input"output characteristics are linear

     there are no other limits constraints or restrictions  start"up costs are a $%ed amoun

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    Unit Commitment Solution

    Methods Dynamic programming  the follo!ing assumptions are made in this

    implementation of the D& approach

     a state consists of an array of units  !ith speci$ed units operating and the rest decommitted (o'"line)  a feasible state is one in !hich the committed units can supply

    the reuired load and meets the minimum capacity for each period

     start"up costs are independent of the o'"line or do!n"time

     ie it is a $%ed amount !rt time  no unit shutting"do!n costs  a strict priority order !ill be used !ithin each interval  a speci$ed minimum amount of capacity must be

    operating !ithin each interval

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    Unit Commitment Solution

    Methods *he for!ard D& approach  runs for!ard in time from the initial hour to the $nal hour  the problem could run from the $nal hour back to the initial

    hour  the for!ard approach can handle a unit+s start"up costs that

    are a function of the time it has been o'"line (temperaturedependent)  the for!ard approach can readily account for the system+s history

     initial conditions are easier to speci$ed !hen going for!ard

     the minimum cost function for hour K !ith combination I,-cost(./)0min1&cost Fcost(K,I)=min[Pcost(K,I)]+Scost(K-1,L:K,I)+Fcost(K-1.L)]

    345  F cost(K  I) 0 least total cost to arrive at state (K  I)  Pcost(K  I) 0 production cost for state (K  I)

     Scost(K 7 L, K  I) 0 transition cost from state (K 7 L) to (K  I)

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    Unit Commitment Solution

    Methods *he for!ard D& approach  state (K  I) is the Ith commitment combination in hour K   a strategy is the transition or path from one state at a

    given hour to a state at the ne%t hour    is de$ned as the number of states to search each period  N is de$ned as the number of strategies to be saved at each

    step  these variable allo! control of the computational e'ort  for complete enumeration the ma%imum value of  or N is 2N  7

     for a simple priority"list ordering the upper bound on 8 is n thenumber of units  reducing N means that information is discarded about the

    highest cost schedules at each interval and saving only thelo!est N paths or strategies  there is no assurance that the theoretical optimum !ill be found

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    Unit Commitment SolutionMethods  *he for!ard D& approach

     restricted search paths  9 0 :  8 0 ;

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    Unit Commitment Solution

    Methods  hour load

    patternUnit Pmax Pmin IncrementalHeat Rate

    No-loadCost

    Full-load Ave. Cost

    Min. Time(h

    (M! (M! ("tu # $!h (% # h (% # m!h U& 'on

    ) *+ , )++ ,)/.++ ,/. ,

    , ,+ 0+ 1+++ *.0, ,+./ /

    / /++ 2 *2/+ 0*.2 )1.2

    0+ ,+ ))1++ ,,.++ ,*.++ ) )

    Unit

    InitialCondition

    o33(- #on(4(h

    5tart-u& Costs

    Hot (% Cold(% Cold start(%

    3 8 500 1100 5

    4 -6 0 0.02 0

     Hour 6oad (M!

    1 450

    2 530

    3 600

    4 540

    5 400

    6 280

    7 290

    8 500

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    Unit Commitment SolutionMethods

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    Unit Commitment SolutionMethods  Case 7, ?trict priority"list ordering

     the only states e%amined each hour consist of the listed four,  state ;, unit : state 72, : @ 2 state 7=, : @ 2 @ 7 state 7;, all four

     all possible commitments start from state 72 (initial condition)  minimum unit up and do!n times are ignored  in hour 7,

     possible states that meet load demand (=;A BW), 72 7= 7;5tate Unit 5tatus Ca&acit7

    5 0010 300 MW

    12 0110 550 MW

    14 1110 630 MW

    15 1111 690 MW

    Pcost()8) 9F)(,4F,()+4F/(/++4F)(, :conomic 'is&atch :;.9)2/./04,+.**(,4)*()+4)2.0(/++4,/.*+(,+91*0)./0

    Fcost()8)9Pcost()8)45cost(+8),

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    Unit Commitment SolutionMethods Case 7 in hour 7,  minimum at state 72 (2A>)

     in hour 2,  possible states that meet load demand (;:A BW), 72

    7= 7;

     ? Pcost 5cost Fcost

    15 9861 350 10211

    14 9493 350 9843

    12 9208 0 9208

    cos 1 2 3 1

    cos cos cos

    (2,15) (25) (185) (300) (20)

    1735 2088(25) 18(185) 17046(300) 2380(20) 11301

    (2,15) (2,15) (1, : 2,15)

    350 9208

    11301 min 0 9843 208

    0 10211

    t t t 

     P F F F F 

     F P S L

     DP state transition equation

    = + +

    = + + + + =

    = +

    − − −

    + = + + =

    +

    59

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    Unit Commitment Solution

    Methods  Case 7, D& diagram  total cost, E:=:

     priority order list up"times and do!n"times neglected

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    Unit Commitment Solution

    Methods Case 2, complete enumeration (2;6 F 7A

    possibilities)  fortunately most are not feasible because they do not

    supply suGcient capacity

     in this case the true optimal commitment isfound  the only di'erence in the t!o traHectories occurs in

    hour : it is less e%pensive to turn on the less eGcient peaking unit

    #= for three hours than to start up the more eGcient unit#7 for that same time period

     only minor improvement to the total cost  case 7, E:=:

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    Unit Commitment Solution

    Methods Case 2, D& diagram

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    Unit Commitment Solution

    Methods 4agrange Iela%ation dual variables and dual optimiation  consider the classical constrained optimiation

    problem  primal problem, minimie ! ( " 7J "n) subHect to K( " 7

    J "n) 0 A  the 4agrangian function, L( " 7J "n) 0 ! ( " 7J "n) @ L

    K( " 7J "n)  de$ne a dual function

     then the Mdual problemN is to $nd  the solution involves t!o separate optimiation

    problems in the case of conve% functions this rocedure is

    1 2

    1 2,

    ( ) min ( , , ) x x

    q L x xλ λ =

    *

    0( ) max ( )q q

    λ 

    λ λ ≥

    =

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    Unit Commitment Solution

    Methods 

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    Unit Commitment Solution

    Methods /terative form of 4agrange rela%ation

    methodthe optimiation may contain non"linear or non"

    conve% functionsiterative process based on incremental

    improvements of L is reuired to solve theproblem select a arbitrary starting L

    solve the dual problem such that #(L) becomes larger update L using a gradient adHustment,

    $nd closeness to the solution by comparing the gapbet!een the MprimalN function and the dual function

    primal function, $O 0 min L relative duality gap, in practice the gap

    ( )t t    dq

    λ λ λ α 

    λ 

    = +

    * *

    *

    ( ) j q

    q

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    Unit Commitment Solution

    Methods 4agrange rela%ation for unit commitmentloading constraint

    unit limits

    unit minimum up"time and do!n time

    constraintsthe obHective function

    1

    0

    1....

     N t t t 

    load i i

    i

     P P U 

    t T 

    =

    − =

    ∇ =

    min max

    1.... & 1....

    t t t 

    i i i i iU P P U P  

    i N t T  

    ≤ ≤

    ∇ = =

    ... ,

    1 1

    ( ) ( )T N 

    t t t t  

    i i start up i t i i i

    t i

     F P S U F P U −= =

    + + = ∑∑

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    Unit Commitment Solution

    Methods -ormation of the 4agrange function  in a similar !ay to the economic dispatch problem

     unit commitment reuires that the minimiation of the4agrange function subHect to all the constraints  the cost function and the unit constraints are each separated

    over the set of units  !hat is done !ith one unit does not a'ect the cost of running

    another unit as far as the cost function unit limits and the up"timeand do!n"time constraints are concerned

     the loading constraint is a coupling constraint across all theunits

     the 4agrange rela%ation procedure solves the unit

    commitment by temporarily ignoring the couplingconstrain

    1 1

    ( , , ) ( , )T N 

    t t t t t  

    i i i i i

    t i

     L P U F P U P P U λ λ 

    = =

    = + −

    ∑ ∑

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    Unit Commitment Solution

    Methods *he dual procedure attempts to reach the

    constrainedoptimum by ma%imiing the4agrangian !ith respect to the4agrange multiplier

    done in t!o basic steps ?tep 7, $nd a value for each lt !hich moves #(L)

    to!ards a larger value ?tep 2, assuming that Lt found in ?tep 7 is $%ed $nd

    the minimum of L by adHusting the values of Pt and %t minimiing L

    *

    ,

    ( ) max ( )... ... ( ) min ( , ,)t t t 

    i i P U 

    q q where q L P U  λ 

    λ λ λ λ  = =

    , ,1 1 1 1 1

    ( )T N T T N  

    t t t t t t  

    i t i i t i load i it i t t i

     L F P S U P P U λ λ = = = = =

    = + + − ∑∑ ∑ ∑∑

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    Unit Commitment SolutionMethodsseparation of the units from one anotherP the

    inside term can

    no! be solved independently for each generating

    unit the minimum of the 4agrangian is found bysolving for the minimum for each

    generating unit over all time periodssubHect to the up"time and do!n"time constraints andthis is easily solved as a t!ostate dynamic programming

    problem of one variable

    ,

    1

    ( )T 

    t t t t  

    i i i t i i i

     F P S U P U λ =

    + − ∑

    ,

    1 1

    min ( ) min {[ ( ) ] } N T 

    t t t t t  

    i i i t i i i

    i i

    q F P S U P U  λ λ = =

    = + −∑ ∑min max

    1...

    t t t 

    i t i i t  U P P U P  

    t T 

    ≤ ≤

    ∀ =

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    Unit Commitment SolutionMethodsMinimizing the function with respect

    to Pitat the Uit 0 A state the minimiation is trivial and

    euals ero

    at the Uit 0 7 state the minimiation !rt is,

    there are three cases to be considered for &iopt andthe limits

    min[ ( ) ]

    [ ( ) ] [ ( )]

    0

    t t 

    i i i

    t t 

    t i i i i i

    t t i i

     F P P 

    d F P P d F P  

    dP dP  

    λ 

    λ λ 

    +

    +

    = + =

    min min min

    min max

    max max max

    [ ( ) ] [ ( ) ]

    [ ( ) ] [ ( ) ]

    [ ( ) ] [ ( ) ]

    opt t t t  

    i i i i i i i i

    opt t t opt t opt  

    i i i i i i i i i

    opt t t t  

    i i i i i i i i

     fP P ThenMin F P P F P P 

    ifP P P ThenMin F P P F P P  

    ifP P ThenMin F P P F P P  

    λ λ 

    λ λ 

    λ λ 

    ≤ − = −

    ≤ ≤ − = −

    ≤ − = −

    [ ( ) ] 0t i i i F P P λ − <

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    Unit Commitment Solution

    Methods QdHusting L L must be carefully adHusted to ma%imie (L) various techniues use a mi%ture of heuristic strategies

    and gradient search methods to achieve a rapid

    solution Lt for the unit commitment problem L is a vector of Lt +s to

    be adHusted each hour

     simple techniue

     gradient component, !here

     heuristic component,

    ( )t t    dq

    λ λ λ α 

    λ 

    = +

    1

    ( )t    N 

    t t t 

    load i i

    i

    dq P P U 

    λ 

    λ    =

    = − ∑( )

    0.01 ~ ~

    ( )

    0.01 ~ ~

    dqwhen IsPositive

    dq

    when IsNegatif    d 

    λ α 

    λ 

    λ 

    α  λ 

    =

    =

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    Unit Commitment Solution

    Methods *he relative duality gap used as a measure of the closeness to the solution  for large real"sie po!er"systems unit"commitment

    calculations the duality gap becomes uite small as thedual optimiation proceeds  the larger the commitment problem the smaller the gap

     the convergence is unstable at the end  some units are being s!itched in and out

     the process never comes to a de$nite end  there is no guarantee that !hen the dual solution process

    stops it !ill be at a feasible solution

     the gap euation,  * *

    *

    ( ) J q

    q

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    Unit Commitment Solution

    Methods4agrange rela%ation algorithm using dual

    optimiation

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    Unit Commitment Solution

    Methods 

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    Unit Commitment SolutionMethods /teration 7,(L)0AARO0=AAAA (RO"O)SO0unde$ned

     /teration 2,

     dynamic programming for unit #:

     Hour U) U, U/ P) P, P/ ';(# &)edc &,edc &/edc

    1 0 0 0 0 0 0 0 170 0 0 0

    2 0 0 0 0 0 0 0 520 0 0 0

    3 0 0 0 0 0 0 0 1100 0 0 0

    4 0 0 0 0 0 0 0 330 0 0 0

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    Unit Commitment Solution

    Methods /teration 2,(L)07=>2 RO0=AAAA (RO"O)SO076E

     Hour U) U, U/ P) P, P/ ';(#

    &)ed

    c &,edc &/edc

    1 1.7 0 0 0 0 0 0 170 0 0 0

    2 5.2 0 0 0 0 0 0 520 0 0 0

    3 11 0 1 1 0 400 200 500 0 0 0

    4 3.3 0 0 0 0 0 0 330 0 0 0

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    Unit Commitment Solution

    Methods /teration :,(L)07>:== RO0:6A2= (RO"O)SO0A6;

     Hour U) U, U/ P) P, P/ ';(#

    &)ed

    c

    &,ed

    c

    &/ed

    c1 3.4 0 0 0 0 0 0 170 0 0 0

    2 10.4 0 1 1 0 400 200 -80 0 320 200

    3 16 1 1 1 600 400 200 -100 500 400 200

    4 6.6 0 0 0 0 0 0 330 0 0 0

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    Unit Commitment Solution

    Methods /teration =,(L)0727= RO02>A6 (RO"O)SO0A;A2

     Hour U) U, U/ P) P, P/ ';(# &)edc &,edc &/edc

    1 5.1 0 0 0 0 0 0 170 0 0 0

    2 10.24 0 1 1 0 400 200 -80 0 320 200

    3 15.8 1 1 1 600 400 200 -100 500 400 200

    4 9.9 0 1 1 0 380 200 -250 0 130 200

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    Unit Commitment Solution

    Methods /teration ;,(L)07;:2 RO0:6A2= (RO"O)SO0A>==

     Hour U) U, U/ P) P, P/ ';(# &)edc &,edc &/edc

    1 6.8 0 0 0 0 0 0 170 0 0 0

    2 10.08 0 1 1 0 400 200 -80 0 320 200

    3 15.6 1 1 1 600 400 200 -100 500 400 200

    4 9.4 0 0 1 0 0 200 130 0 0 200

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    Unit Commitment SolutionMethods /teration 6,(L)07==2 RO02A7EA (RO"O)SO0AA:E

     Iemarksthe commitment schedule does not change signi$cantly

    !ith further iterationsh th l ti i t t bl ( ill ti f it 2)

     Hour U) U, U/ P) P, P/ ';(# &)edc &,edc &/edc

    1 8.5 0 0 1 0 0 200 -30 0 0 170

    2 9.92 0 1 1 0 384 200 -64 0 320 200

    3 15.4 1 1 1 600 400 200 -100 500 400 200

    4 10.7 0 1 1 0 400 200 -270 0 130 200