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Heuristic Optimization for the Discrete VirtualPower Plant Dispatch Problem

Mette K Petersen Lars H Hansen Jan Bendtsen Kristian Edlund and Jakob Stoustrup

AbstractmdashWe consider a virtual power plant which is giventhe task of dispatching a fluctuating power supply to a portfo-lio of flexible consumers The flexible consumers are modeledas discrete batch processes and the associated optimizationproblem is denoted the discrete virtual power plant dispatchproblem (DVPPDP) First the nondeterministic polynomial time(NP)-completeness of the discrete virtual power plant dispatchproblem is proved formally We then proceed to develop tailoredversions of the meta-heuristic algorithms hill climber and greedyrandomized adaptive search procedure (GRASP) The algorithmsare tuned and tested on portfolios of varying sizes We find thatall the tailored algorithms perform satisfactorily in the sense thatthey are able to find sub-optimal but usable solutions to verylarge problems (on the order of 105 units) at computation timeson the scale of just 10 s which is far beyond the capabilitiesof the optimal algorithms we have tested In particular GRASPsorted shows with the most promising performance as it is ableto find solutions that are both agile (sorted) and well balancedand consistently yields the best numerical performance amongthe developed algorithms

Index TermsmdashAlgorithms computation time schedulingsuboptimal control

I INTRODUCTION

GLOBAL EFFORTS to reduce CO2 emissions has driventhe introduction or renewable power generation tech-

nologies into the power system However since solar panelsand wind turbines harvest energy from sun and wind poweravailability becomes changeable and more difficult to predictThe smart grid was born out of the need to maintain the bal-ance between production and consumption in this far morevolatile power system In the smart grid a communicationlink to the consumption side is established such that flexi-ble consumers like electric vehicles heat pumps and heatingventilation and air conditioning (HVAC) systems can be orga-nized and activated to follow power availability and scarcity(see [1] and [2])

A major challenge in developing the smart grid is the sheersize of the optimization problems involved Solving a dispatch

Manuscript received August 8 2013 revised January 23 2014 andApril 26 2014 accepted June 10 2014 Paper no TSG-00619-2013

M K Petersen is with the Department of Electronic Systems Automationand Control Aalborg University Aalborg 9210 Denmark and also withDONG Energy Gentofte 2720 Denmark (e-mail mehpedongenergycom)

L H Hansen and K Edlund are with DONG Energy Gentofte 2720Denmark (e-mail larhadongenergycom krieddongenergycom)

J Bendtsen and J Stoustrup are with the Department of Electronic SystemsAutomation and Control Aalborg University Aalborg 9210 Denmark(e-mail dimonesaaudk jakobesaaudk)

Color versions of one or more of the figures in this paper are availableonline at httpieeexploreieeeorg

Digital Object Identifier 101109TSG20142336261

problem for a traditional power system with tens or hundredsof generators is a challenge which has been researched fordecades (see [3]ndash[5]) Switching to the smart grid howeverwill expand that problem with additional thousands or mil-lions of units This means that the computation time associatedwith determining an optimal solution is very likely to beunacceptable in practice

A review of computation times for optimization in smartgrid literature reveals that computation time is still a very realchallenge Table I summarizes problem size and computationtimes for ten recent scientific publications related to smart gridoptimization Obviously computation times are highly depen-dent on the specific structure of the considered problem andthe software and platform used for calculation NonethelessTable I does give the general impression that computationtimes are still quite a lot longer than what one can expectto be acceptable for a fully deployed smart grid operating inreal time This is the case even though several of the citedreferences investigate heuristic rather than exact optimizationmethods

The fastest results found are given in [8] with computationtimes of just 12 s In [8] however it is shown that the consid-ered method does not scale to larger problems due to memoryoverload

To substantiate the aforementioned assertion this paperintroduces the discrete virtual power plant dispatch problem inwhich batch processes (constant power consumption with fixedduration) must be scheduled to balance a fluctuating powersupply Similar optimization problems have been investigatedin [6]ndash[8] and [16] In these papers batch processes are usedas simple models of electric vehicles dishwashers microwaveovens and more These papers formulate objectives to scheduleunits to balance a fluctuating limited or costly power supply

After formulating the discrete virtual power plant dispatchproblem it is proven formally that the optimization problem isNP-complete [17] Motivated by this knowledge we proceed toinvestigate the performance of two heuristic methods to obtainsolutions which are sub-optimal but fast to compute Thisway a feasible solution will always be available before somepredefined power market gate closure In practice a suboptimalsolution available 2 min before market gate closure is far morevaluable than an optimal one available 2 min after

The methods we will investigate are known in the literatureas Hill Climber and greedy randomized adaptive search proce-dure (GRASP) The main reason for choosing these methodsto solve the discrete virtual power plant dispatch problem isthat the considered cost function can be implemented using

2 IEEE TRANSACTIONS ON SMART GRID

TABLE IOVERVIEW OF COMPUTATION TIMES REPORTED IN RECENT SCIENTIFIC

PUBLICATIONS ON OPTIMIZATION IN SMART GRID APPLICATIONSSIMULATION SAMPLES IS THE NUMBER OF DISCRETE TIME

STEPS THAT THE CONSIDERED SIMULATION HORIZON

HAS BEEN SPLIT INTO

delta evaluation and both Hill Climber and GRASP are ableto exploit that

Delta evaluation means that if the cost associated with oncandidate solution is known and a new solution is obtainedthrough a limited number of permutations of the old solutionthen the cost function for the new solution can be computedonly from the permutations Using delta evaluation can onlyimprove calculation time by a constant factor however thisfactor is usually so large that the actual improvement is farbetter than algorithmic changes

Other methods that could be considered for solving the dis-crete virtual power plant dispatch problem are ant swarm opti-mization [11] genetic algorithms [19] and tabu search [22]However these algorithms are all based on manipulating a setof candidate or tabu solutions They thus have a tendency todrown in logistics as the majority of the available computationtime is spent running through and updating solution sets

The main contributions of the paper is firstly the proof thatthe discrete virtual power plant dispatch problem problem isNP-complete Secondly we also show that even though findingoptimal solutions of the considered problem is indeed verychallenging highly promising results can be obtained in shorttime frames and for very large problems by special adaptationsof the considered heuristic algorithms

The paper is structured as follows Firstly Section IIpresents the discrete virtual power plant dispatch problemincluding flexibility modeling and agility In Section IIIthe computational complexity of the discrete virtual powerplant dispatch problem is explored in several different waysfirstly the NP-completeness is proven formally and then thefeasibility of solving the problem by use of the optimiza-tion package CPLEX [18] is explored In Section IV fourheuristic algorithms are developed namely uniform selec-tion (UHC) weighted selection (WHC) GRASP random andGRASP sorted These algorithms are tuned and compared inSection V Finally Section VI summarizes general conclusionsand suggestions for future work

II OPTIMIZATION PROBLEM

We consider a virtual power plant which is given the taskof satisfying the consumption needs of a portfolio of flexi-ble systems (distributed energy resources) by dispatching afluctuating power supply

A forecast of the fluctuating power supply is denotedPDispatch(k) k = 1 2 K and the flexible consumers aredenoted local units A portfolio of N local units is denotedLUii=12N At sample k we let Pi(k) denote the power tobe dispatched to LUi and any quantity which cannot be dis-patched to the portfolio is denoted S(middot) The objective is tominimize the residual power that is |S(middot)|

The optimization problem can be formulated as

minPi(middot)

Ksum

k=1

wk|S(k)| (1)

st

PDispatch(k) isin R+ k = 1 2 K (2)Nsum

i=1

Pi(k)+ S(k) = PDispatch(k) (3)

and also subject to the dynamics and constraints ofLUii=12N Here K denotes the total simulation horizon

The impatience weights wk isin R have been addedbecause the forecasted power production will rarely fit exactlywith the power needed to satisfy LUii=12N In practicewhen this happens a traditional power plant will have to adjustits power consumption such that the discrepancy betweensupply and demand is compensated As a rule of thumb how-ever the better time the plant operator has to modify theoutput of a traditional power plant the cheaper it can bedone It is therefore desirable to introduce slack as late onthe simulation horizon as possible which can be achieved byintroducing wk k = 1 2 K and requiring that wk1 gt wk2

if k1 lt k2

A Flexibility Modeling

In this paper the flexible units are modeled as batch-processes which are characterized by constant power con-sumption P a run time KRun and a deadline KEnd by whichthe process must be finished Also we let Ts denote the sizeof the time step Each LU can therefore be modeled by

E(k + 1) = E(k)+ TsP(k) (4)

P(k) = Pv(k) (5)

0 le E(k + 1) le E (6)

E(KEnd) = E (7)

0 lek+KRunminus1sum

l=k

v(l)minus KRun (v(k)minus v(k minus 1)) (8)

where P isin R+ k = 1 2 K v(k) isin 0 1 E = PKRunKRun le K and KEnd le K Inequality (8) is the minimumruntime constraint which ensures that if v(k) minus v(k minus 1) isone then v(l) l = k+ 1 k+ 2 KRun minus 1 must also all beone that is once the local units is activated it must completeits consumption immediately

We let LUii=12N denote a portfolio of N flexi-ble consumers Then for each LUii=12N a solutionof problem (1)ndash(3) consists of a set of start timesKStart = (KStart1 KStart2 KStartN) one for eachLU An illustration of the discrete virtual power plantdispatch problem for batch processes is given in Fig 1

3

Fig 1 Solving the discrete virtual power plant dispatch problem essentiallycorresponds to packing rectangles under an arbitrary profile

Fig 2 As time progresses an un-agile solution can become sub-optimalwhen earlier projections of PDispatch turn out to be erroneous

Fig 3 As time progresses an agile solution is more likely to remain optimaleven when earlier projections of PDispatch were erroneous

B Agility

As mentioned earlier PDispatch is a forecast of the powerproduction of some renewable production technology Thismeans that PDispatch will not correspond exactly to the powerwhich will actually be produced over the considered timehorizon To alleviate this issue we want to find a solutionof problem (1)ndash(3) which is as agile as possible Finding anagile solution means prioritizing the most urgent tasks namelythe ones which are closest to their deadline In this way wewill have created more maneuverability if updated forecastsof PDispatch give very different power availability than origi-nally projected We also call this maximizing the agility of theportfolio (see Figs 2 and 3)

The concept of agility is illustrated in Figs 2 and 3 fora portfolio consisting of three units The top illustrations inFigs 2 and 3 both depict optimal solutions of this instance ofthe discrete virtual power plant dispatch problem and if thepredicted progress of PDispatch is correct then it is obviously

unimportant to distinguish between the two If however asignificant portion of the available power is delayed as in thelower illustrations then the top solution in Fig 3 remains opti-mal but the top solution in Fig 2 does not This happensbecause the top solution in Fig 2 has left more maneuverabil-ity for the optimization in later time steps Thus in a senseintroducing agility to the problem solving is an attempt tomaximize the flexibility of the remaining solution space

To find an agile solution of problem (1)ndash(3) the cost functionis extended with a term which adds a penalty to dispatchingeach LU that is

f (P(middot)) = minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1

wik|Pi(k)|)

(9)

st


i=1


The agility weights wik should then be chosen such that LUi

is dispatched before LUj if KEndi minusKRuni lt KEndj minusKRunjTo explain how agility weights are chosen letLUii=12N denote a set of local units sorted accord-ing to deadline minus run time Intuitively agility weightswill penalize increasing the energy term of each LU andthis penalty is proportional to the unit index This meansreplacing the cost function (9) with

minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1

i|Ei(k)|) (12)

However since Ei(k) =sumkl=1 TsPi(l) (12) can be written as

minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1

i(K + 1minus k)Ts|Pi(k)|)

and the agility weights are therefore given by

wik = i(K + 1minus k)Ts (13)

C Portfolio Generation for Simulation

Throughout this paper we consider optimization problemson randomly generated portfolios Each portfolio is charac-terized by the numbers N and K such that Portfolio(N K)

denotes a randomly generated portfolio of N local units withKRun isin 2 3 4 5 P isin 1 2 3 4 and KEnd isin 1 2 KAlso we set Ts = 1 in all simulations and all calculationshave been performed on a standard laptop

Fig 4 depicts a solution of problem (9)ndash(11) for aPortfolio(105 100) computed by use of the algorithm GRASPRandom which is introduced in Section IV-B It can be seenthat the majority of slack is introduced toward the end of thesimulation horizon as the total consumption (blue) is no longerable to follow PDispatch (black) Agilitysortedness is illustratedby green red cyan and magenta lines illustrating the accumu-lated consumption of the first second third and fourth quarterof units respectively when the portfolio is sorted accordingto deadline minus runtime


Fig 4 Solution of problem instance of the discrete virtual power plantdispatch problem with 105 local units computed by use of the algorithmGRASP random introduced later in this paper

III COMPUTATIONAL COMPLEXITY

In this section we will investigate the computational com-plexity of the discrete virtual power plant dispatch problemWe first do this by proving that the problem is NP-CompleteNext we attempt to solve the optimization problem viaCPLEX All calculations are performed on a standard laptop

A Proof of NP-Completeness

Polynomial-time reductions provide a formal means ofshowing that one problem is at least as hard to solve as anotherto within a polynomial-time factor That is if L1 leP L2 thenL1 is not more than a polynomial factor harder than L2 [17]

Definition 1 (Subset-Sum Problem) Let there be given afinite set S isin N and a target t isin N Is there a subset Sprime isin Swhose elements sum to t

Lemma 1 The subset-sum problem is NP-completeProof The proof of NP-completeness of the subset-sum

problem is done by formulating the 3 minus CNF minus SAT (three-conjuncture-normal-form-satisfiability) language Next it isproved that satisfiability of boolean formulas in 3minusCNFminusSATis NP-complete Finally the subset-sum problem is formulatedas boolean formulas in 3 minus CNF minus SAT thus proving thatL3minusCNFminusSAT leP LSubsetsum For full proof see [17]

In the following we show that a simplified version of thediscrete virtual power plant dispatch problem is equivalent tothe subset-sum problem We shall refer to this reduced problemas DVPPDP for brevity

Theorem 1 The discrete virtual power plant dispatch prob-lem is NP-complete

Proof In this proof it is shown that a subset of the classof instances of the discrete virtual power plant dispatch prob-lem is equivalent to the subset-sum problem By polynomialreduction this proves that the discrete virtual power plant dis-patch problem is NP-complete since the subset-sum problemis NP-complete

Firstly we simplify the discrete virtual power plant dispatchproblem by setting agility weights to 0 This reduces problem(9)ndash(11) to

f (Pi(middot)) = minPi(middot)

Ksum

k=1

wi|S(k)| (14)

st

PDispatch(k) isin R+ k = 1 2 K (15)

Fig 5 For the considered instance of the discrete virtual power plant dispatchproblem a solution of problem (14)ndash(16) such that f (Pi(middot)) = 0 can exist ifand only if there also exists a subset of KRuni i = 1 2 N which sumsto K

Nsum

i=1


Flexible units are still modeled by (4)ndash(8)Let K isin N+ and K isin N+ be given and assume without loss

of generality that K lt K Next define portfolio LUii=12N such that Pi = 1 i = 1 2 N KEndi = K i = 1 2 Nand

sumNi=1 KRuni = K+ K Also define PDispatch(k) = 2 k =

1 2 K and PDispatch(k) = 1 k = K + 1K + 2 K(see Fig 5)

To prove NP-completeness we formulate the following deci-sion problem given the discrete virtual power plant dispatchproblem instance constructed above does there exist a solutionof problem (14)ndash(16) for which f (Pi(middot)) = 0

First observe thatsumK

k=1 PDispatch(k) = K + K andsumNi=1 KRuniP = sumN

i=1 KRuni = K + K as well This meansthat exactly two local units must be on at any sample untilsample K and that exactly one LU must be on at any sampleafter sample K in order for a solution with zero slack to existHowever such a solution can exist if and only if there alsoexists a subset of KRuni i = 1 2 N which sums to K

This however corresponds exactly to the subset-sum prob-lem for the set S = KRuni i = 1 2 N and t = K sinceK and K are arbitrarily chosen positive integers Thus ifthere exists a polynomial time algorithm for solving the con-sidered instance of the discrete virtual power plant dispatchproblem then this algorithm could also solve the subset-sumproblem in polynomial time It now follows from Lemma 1that the discrete virtual power plant dispatch problem isNP-complete

B CPLEX Performance

A commonly used option for solving integer problems is touse the software package CPLEX [18] If CPLEX is capableof solving the discrete virtual power plant dispatch problem tooptimality within a reasonable time frame then there will beno reason to apply meta-heuristic methods to the problem Inthis section we will therefore investigate how CPLEX handlesthe discrete virtual power plant dispatch problem

We have tested CPLEX performance on tenPortfolio(25 100) and ten Portfolio(50 100) problems

5

Algorithm 1 Hill Climber (LUii=12N

PDispatch(k)k=1 2K n)

1 Generate initial solution KStart0 by randomly choosing feasiblestart samples for each local unit

2 Repeat3 Select KStart

prime in νn(KStarti) by use of Uniform Selection orWeighted Selection

4 If f (KStartprime) lt f (KStarti) then

5 KStarti+1 = KStartprime

6 Until time-limit is reached

In many cases the computations are terminated with an errormessage stating that the computer has run out of memoryand therefore no optimal solution is found We observed thatfor Portfolio(25 100) five of ten problems were solved withan average computation time of 8 min and for Portfolio(50100) zero of ten problems were successfully solved Since allcalculations finish due to lack of memory for 50 units and100 samples there is little hope that this option will scale tolarger problem instances

IV META-HEURISTIC ALGORITHMS

Since we have illustrated that finding optimal solutionsof the discrete virtual power plant dispatch problem isindeed very challenging we will now investigate the perfor-mance of the meta-heuristic methods Hill Climber [19] andGRASP [20]

A Hill Climber

We first define a neighborhood associated with the discretevirtual power plant dispatch problem which is based on theidea of an n-move Next we develop two variations of thealgorithm denoted UHC and WHC

1) Neighborhood and n-Move Let an instance of the dis-crete virtual power plant dispatch problem be given that isa set of parameters Pi KEndi and KRuni i = 1 2 Nand a dispatch sequence PDispatch(k) isin R k = 1 2 KA solution of the discrete virtual power plant dispatch prob-lem is then given by a set of feasible start times KStart =(KStart1 KStartN) The solution space is given by

S = (KStart1 KStartN) isin NN

|KStarti lt KEndi minus KRuni i = 1 2 Nand we have that

|S| = Ni=1KEndi minus KRuni

le KN

In the discrete virtual power plant dispatch problem a neigh-borhood map νnS rarr SM defines for each solution KStart aneighborhood set Sn(KStart) isin SM consisting of the set ofsolutions that can be obtained from KStart by moving the starttime of any n local units to feasible locations This is calledan n-move In other words a neighborhood map is a map ofthe form

νn(KStart) = KStartprime isin S

| KStartprime is obtained from KStart by an n-move

Algorithm 2 Uniform Selection (LUii=12N n)1 for j = 1 to n do2 Select LUi from LUii=12Nminusj+1 with probability 1

Nminusj+1

3 Select start sample k for LUi with probability 1KStarti

4 Set LUii=12Nminusj = LUii=12Nminusj+1LUk5 end for

Algorithm 3 Weighted Selection (LUii=12N

PDispatch(k)k=12K n)1 for j = 1 to n do

2 Select LUi from LUii=12Nminusj+1 with probability 1Nminusj+1

3 for k = 1 to KEndi minus KRuni do4 vk larr Cost(i k) given that j minus 1 local units have already

been assigned new start samples5 end for

6 Construct vnegative containing all negative numbers in v

7 if Length(vnegative) ge 1 then

8 Select start sample k for LUi with probabilityvnegative(k)sum

vnegative9 else

10 Select start sample k for LUi with probability1

v(k)sum 1v

11 end if

12 Set LUii=12Nminusj = LUii=12Nminusj+1LUk

13 end for

2) Uniform Selection Hill Climber and Weighted SelectionHill Climber Pseudo code for the Hill Climber method isgiven in Algorithm 1 The Hill Climber method first generatesa random initial solution for the considered problem Next asolution in the neighborhood of the current solution is foundIf the cost of the neighboring solution is less than the cost ofthe current solution then the neighbor solution will take itsplace as current solution This procedure is continued until thetime limit is reached

Two tailored versions of the Hill Climber method denotedUHC and WHC will be investigated

In UHC the initial solution KStartprime is found by choosing n

local units from the portfolio with uniform probability and thenselecting feasible start samples for each LU also with uni-form probability Pseudo-code for uniform selection is givenin Algorithm 2 Allowing n to be larger than one means that itis possible to accept a solution where a unit is moved to a lessfavorable start sample as long as other units are simultaneouslymoved to beneficial positions This could help the algorithmescape a local optimum which would not be possible if onlyone LU can be moved at a time

In an alternative implementation denoted WHC the nlocal units are again chosen uniformly from the portfoliobut the start time of each LU is now chosen with probabilityproportional to the benefitcost of moving the start time of theLU to each feasible time slot Pseudo-code for weighted selec-tion is given in Algorithm 3 If improving start times existwe choose an improving time with probability proportionalto the obtained benefit On the other hand if no improving


Algorithm 4 GRASP Random (LUii=12N

PDispatch(k)k=12K m l n)1 repeat

2 for j = 1 to N do

3 for k = 1 to m do4 Unit List(k) larr Select LUi from LUii=12Nminusj+1

with probability 1Nminusj+1

5 for h = 1 to KStartUnit List(k) minus KRunUnit List(k) do6 vkh larr Cost(Unit List(k) h) given that j minus 1 local

units have already been assigned start samples7 end for8 end for

9 CandidateListkh larr The l smallest entries in vkh

10 Select LUk and start sample h from CandidateList withprobability 1

l

11 Set start time of LUk to h and set LUii=12Nminusj =LUii=12Nminusj+1LUk

12 end for

13 Hill-Climber(LUii=12N PDispatch(k)k=12K n)

14 until time-limit is reached

start times exist then we choose a deteriorating time withprobability inversely proportional to the cost

UHC and WHC are tuned and compared in Section V

B GRASP

A definite weakness of the developed Hill Climber methodsis that the initial solution is generated by choosing local unitsand start time with uniform probability This means that thesolutions generated for initial use have absolutely no similarityto PDispatch This problem is mended in GRASP

The idea in GRASP is to construct an initial solution oneelement at a time by use of a greedy algorithm The choiceof next element to be added is determined by constructinga candidate list of most beneficial choices The probabilisticelement in GRASP is then introduced by randomly choosingone of the candidates in the candidate list but not necessarilythe top candidate After an initial solution has been generatedHill Climber is called to achieve a further improvement of thesolution

Again two versions of the algorithm have been investigatednamely GRASP random and GRASP sorted GRASP randomfalls closest to the generic description of the GRASP algo-rithm but as we will see later it has some challenges related tothe discrete virtual power plant dispatch problem To addressthis GRASP sorted is developed as well

Pseudo-code for GRASP random is given in Algorithm 4The idea is that m local units are chosen randomly from theportfolio and placed in the unit list Next the cost of startingeach LU in the unit list at each feasible sample is computedand saved in v The candidate list is then generated by choosingthe l smallest elements from v Finally an element from thecandidate list is chosen with uniform probability

When exploring GRASP random it was discovered that forall problem sizes GRASP random generates initial solutionswhich overshoot PDispatch in the beginning of the horizon and

Fig 6 GRASP random builds an initial solution one unit at a time and aboveit is depicted how the initial solution looks when 14 24 34 and 44 ofthe units in the portfolio have been added Since slack is cheaper toward theend of the horizon GRASP random will first start units early in the horizonWhen 14 of the portfolio has been added a decent fit with PDispatch hasbeen obtained for sample 0ndash18 However there are still units remaining inthe portfolio which have a deadline of 18 or less and now GRASP randomcan only add these in such a way that the accumulated power consumptionbefore sample 18 overshoots PDispatch even further This means that as unitsare added more and more positive slack builds up at the beginning of thesimulation horizon as can be seen from the progression of the figures

fall lower than PDispatch toward the end of the horizon Thisbehavior can be explained as follows

Since slack is cheaper toward the end of the horizon GRASPrandom will first start units early in the horizon as it builds aninitial solution At some point a decent match with PDispatchis obtained for say the first ten samples However if a LUthen remains in LUii=12Nminusj which has deadline 10 orless then it can only be added such that it makes the accu-mulated power consumption overshoot PDispatch somewhere inthe first ten samples When this has happened a number oftimes (see Fig 6) the result is a consumption profile whichovershoots PDispatch in the beginning of the horizon and fallslower than PDispatch toward the end of the horizon

To alleviate this problem the algorithm GRASP sorted wasdeveloped The algorithm is identical to GRASP random exceptthat local units are not initially chosen at random but in sortedorder starting with the n local units with the earliest dead-lines The candidate list is again generated based on the unitlist and an element from the candidate list is chosen with uni-form probability Pseudo-code for GRASP sorted is given inAlgorithm 5

V RESULTS

Before the algorithms can be compared they must be prop-erly tuned When applying a meta-heuristic there will alwaysbe a trade-off between time and performance so computationtime is fixed not a parameter All calculations are performedon a standard laptop The algorithms have been implementedin C [21]

A Tuning

To tune the algorithms a representative training test setof R problem instances is generated and each algorithm is

7

Algorithm 5 GRASP Sorted (LUii=12N


2 Sort LUii=12N according to KEndi minus KRuni

3 for k = 1 to m do4 Unit List(k)larr LUk5 end for

6 for j = 1 to N do

7 for h = 1 to KStartUnit List(k) minus KRunUnit List(k) do8 vkh larr Cost(Unit List(k) h) given that jminus1 local units

have already been assigned start samples9 end for



l

12 Set start time of LUk to h and set UnitList = UnitListLUkcupLUj+1

13 end for

14 Hill-Climber(LUii=12N PDispatch(k)k=12K n)15 until Time limit is reached

TABLE IIPERCENTAGE GAP AND PERCENTAGE DEVIATION FOR ALL

DEVELOPED METHODS

run T times on each training data set In order to be able tocompare performance on problem instances with very differ-ent absolute values we compute the percentage gap and thepercentage deviation Since it is not feasible to determine theoptimal solution of problems of the considered size the per-centage gap and the percentage deviation is computed relativeto the minimum value found over all calculations on eachparticular problem instance

In order to be able to compare performanceon problem instances with very different abso-lute values we compute the percentage gapE = (1S)

sumTj=1(((zj minus zlowast) middot 100)zlowast) and the percentage devi-

ation σ =radic

(1(T minus 1))sumT

j=1

((((zj minus zlowast) middot 100)zlowast)minus E

)2where zlowast is the optimal solution of the problem instance andj indexes the set of T candidate solutions Since zlowast is notknown we will substitute the minimum value found over allcalculations on the particular problem instance

The tuning test is performed on randomly generatedPortfolio(N 100) N = 103 104 105 (see Section II-C) Thetuning set consists of nine instances of the discrete virtualpower plant dispatch problem (sets of portfolio and PDispatch)

Fig 7 Performance of uniform selection hill climber for aPortfolio(105 100) problem

Fig 8 Performance of weighted selection hill climber for aPortfolio(105 100) problem

Fig 9 Performance of GRASP random for a Portfolio(105 100) problem

with three portfolios of 103 104 and 105 local units eachComputation time is fixed at 10 s for all runs of the algo-rithms and T = 10 Results of parameter tuning test are givenin Tables III and IV where A = 103 local units B = 104 localunits and C = 105 local units

B Results

To finally test the algorithms a new Portfolio(N 100) N isin103 104 105 is generated with three portfolios of 103 104and 105 local units each Computation time is still fixed at 10s for all runs of the algorithms and T = 10 The performanceof all four algorithms is given in Table II

It is found that for all problem sizes the Hill Climber meth-ods and GRASP random have very similar performance withno clear winner GRASP sorted on the other hand outper-forms all the other methods by at least an order of magnitudeboth in terms of percentage gap and percentage deviation andfor all problem sizes


Fig 10 Performance of GRASP sorted for a Portfolio(105 100) problem

TABLE IIIDEVELOPED HILL CLIMBER METHODS HAVE ONE PARAMETER NAMELY

THE VALUE OF n IN nminus move TO PROPERLY JUDGE THE PERFORMANCE

OF THE ALGORITHM A SUITABLE VALUE OF THIS PARAMETER MUST

BE FOUND THIS TABLE STATES THE BEST PARAMETER

VALUE FOUND FOR BOTH HILL CLIMBER METHODS

Figs 7ndash10 depict solutions found for aPortfolio(100000 100) problem using each algorithmand the parameters given in Tables III and IV A visualinspection confirms the general conclusions that UHC WHCand GRASP random have very similar performance as thecurves can hardly be distinguished However GRASP sortedis clearly superior It can be seen that particularly at thebeginning of the simulations GRASP sorted has less slackthan the other methods and GRASP sorted has furthermorefound a far more sorted solution which can be seen by thevery steep slopes of the quarter lines in Fig 10

As demonstrated in Section III there exists no efficient strat-egy for determining optimal solutions of the virtual powerplant dispatch problemfor large problem instances One optionwould be to generate artificial structured problem instanceswhere the optimal solution is known However inherent struc-ture in a problem could likely favor one of the methodsin particular GRASP sorted and would thus lead to unfaircomparisons

Our best available optimal solutions are therefore the fivePortfolio(25 100) problem instances which were successfullyfound by CPLEX in Section III-B In Table V GRASP sortedhas been retuned for Portfolio(25 100) problems and the aver-age performance over ten solutions is given for these problemsIt is found that the deviation from optimality is 6 to 13which is fairly good considering the short computation time

VI CONCLUSION

In the vision for the future smart grid not just hundredsbut thousands or even millions of flexible consumers must becoordinated to operate in a sensible interconnected manner Amajor issue in implementing the smart grid however is thatthis must happen in real time In this paper we have there-fore investigated computational speed of large-scale dispatchproblems

TABLE IVDEVELOPED GRASP METHODS HAVE THE FIVE PARAMETERS LISTED

UNDER ldquoPARAMETERrdquo THIS TABLE STATES THE BEST PARAMETER

VALUE COMBINATION FOUND FOR BOTH GRASP METHODS

TABLE VOPTIMAL SOLUTIONS VALUES FOUND IN CPLEX AND SOLUTION

VALUES FOUND BY GRASP SORTED FOR FIVE

PORTFOLIO(25 100) PROBLEMS INSTANCES

Our investigations have concentrated on the discrete vir-tual power plant dispatch problem Firstly the computationalcomplexity was investigated by proving that the problem isNP-complete and investigating the options of solving thediscrete virtual power plant dispatch problem by use ofCPLEX [18] Being NP-complete the discrete virtual powerplant dispatch problem is therefore at least as complex (hard)to solve as the well-known unit commitment problem [23]

We therefore developed heuristic methods for solvingthe problem and specifically looked at Hill Climber andGRASP Four algorithms were developed denoted UHCWHC GRASP random and GRASP sorted After tuning andtesting by far the best results where obtained by GRASPsorted This method can determine solutions which are bothagile (sorted) and well balanced even for problems of 100000units 100 samples and with a computation time of just 10 s

REFERENCES

[1] A Mohsenian-Rad V W S Wong J Jatskevich R Schober andA Leon-Garcia ldquoAutonomous demand-side management based ongame-theoretic energy consumption scheduling for the future smartgridrdquo IEEE Trans Smart Grid vol 1 no 3 pp 320ndash331 Dec 2010

[2] D S Callaway and I A Hiskens ldquoAchieving controllability of electricloadsrdquo Proc IEEE vol 99 no 1 pp 184ndash199 Jan 2011

[3] K Edlund J D Bendtsen and J B Joslashrgensen ldquoHierarchical model-based predictive control of a power plant portfoliordquo Control Eng Practvol 19 no 10 pp 1126ndash1136 Oct 2011

[4] K Aoki T Satoh M Itoh T Ichimori and K Masegi ldquoUnit commit-ment in a large-scale power system including fuel constrained thermaland pumped-storage hydrordquo IEEE Trans Power Syst vol 2 no 4pp 1077ndash1084 Nov 1987

[5] A K Ayuob and A D Patton ldquoOptimal thermal generating unit com-mitmentrdquo IEEE Trans Power App Syst vol 90 no 4 pp 1752ndash1756Aug 1971

PETERSEN et al HEURISTIC OPTIMIZATION FOR THE DISCRETE VIRTUAL POWER PLANT DISPATCH PROBLEM 9

[6] J Huang V Gupta and Y-F Huang ldquoScheduling algorithms for PHEVcharging in shared parking lotsrdquo in Proc 2012 Amer Control ConfMontreal QC Canada 2012 pp 276ndash281

[7] S Chen P Sinha and N B Shroff ldquoScheduling heterogeneous delaytolerant tasks in smart grid with renewable energyrdquo in Proc 51st IEEEConf Decis Control Maui HI USA 2012 pp 1130ndash1135

[8] K C Sou J Weimer H Sandberg and K H Johansson ldquoSchedulingsmart home appliances using mixed integer linear programmingrdquo inProc 50th IEEE Conf Decis Control Eur Control Conf Orlando FLUSA 2011 pp 5144ndash5149

[9] D T Phan J Xiong and S Ghosh ldquoA distributed scheme for fair EVcharging under transmission constraintsrdquo in Proc Amer Control ConfMontreal QC Canada 2012 pp 1053ndash1058

[10] Z A Vale H Morais H Khodr B Canizes and J Soares ldquoTechnicaland economic resources management in smart grids using heuristic opti-mization methodsrdquo in Proc IEEE Power Energy Soc General MeetMinneapolis MN USA 2010 pp 1ndash7

[11] P Faria J Soares Z Vale H Morais and T Sousa ldquoModified par-ticle swarm optimization applied to integrated demand response andDG resources schedulingrdquo IEEE Trans Smart Grid vol 4 no 1pp 606ndash616 Mar 2013

[12] Z Chen L Wu and Y Fu ldquoReal-time price-based demand responsemanagement for residential appliances via stochastic optimizationand robust optimizationrdquo IEEE Trans Smart Grid vol 3 no 4pp 1822ndash1831 Dec 2012

[13] T Logenthiran D Srinivasan and T Z Shun ldquoDemand side manage-ment in smart grid using heuristic optimizationrdquo IEEE Trans SmartGrid vol 3 no 3 pp 1244ndash1252 Sep 2012

[14] S Salinas M Li and P Li ldquoMulti-objective optimal energy consump-tion scheduling in smart gridsrdquo IEEE Trans Smart Grid vol 4 no 1pp 341ndash348 Mar 2013

[15] P Yi X Dong A Iwayemi C Zhou and S Li ldquoReal-time opportunisticscheduling for residential demand responserdquo IEEE Trans Smart Gridvol 4 no 1 pp 227ndash234 Mar 2013

[16] A Subramanianz et al ldquoReal-time scheduling of deferrable electricloadsrdquo in Proc Amer Control Conf 2012 pp 3643ndash3650

[17] T H Cormen C E Leiserson R L Rivest and C Stein Introductionto Algorithms 2nd ed Cambridge MA USA MIT Press 2001

[18] (2014 Jun 18) [Online] Available httpwww-01ibmcomsoftwareintegrationoptimizationcplex-optimization-studio

[19] E K Burke and G Kendall Search Methodologies IntroductoryTutorials in Optimization and Decision Support Techniques New YorkNY USA Springer 2005

[20] T A Feo and M G C Resende ldquoGreedy randomized adaptive searchproceduresrdquo J Global Optim vol 6 no 2 pp 109ndash133 Mar 1995

[21] (2014 Jun 18) [Online] Available httpwwwmsdnmicrosoftcom[22] S I Gass and C M Harris Encyclopedia of Operations Research and

Management Science Springer 1996[23] G Xiaohong Z Qiaozhu and A Papalexopoulo ldquoOptimization based

methods for unit commitment Lagrangian relaxation versus generalmixed integer programmingrdquo in Proc IEEE Power Eng Soc GenerMeet Jul 2003 pp 1095ndash1100

Mette K Petersen received the MSc degree in applied mathematics fromthe Technical University of Denmark Kongens Lyngby Denmark in 2008and is currently pursuing the industrial PhD degree in collaboration withthe Danish utility company DONG Energy Gentofte Denmark and AalborgUniversity Aalborg Denmark

Her research interests include mathematical modeling optimization andnumerical methods

Lars H Hansen received the MSc degree in electrical engineering in 1994and the PhD degree in mathematical modeling in 1997 from the TechnicalUniversity of Denmark Kongens Lyngby Denmark

From 1997 to 2001 he was a Post-Doctoral Researcher at the Wind EnergyDepartment Risoe National Laboratory Roskilde Denmark Since 2001 hehas been with industrial process control and since 2008 he has been withsmart grid research projects and wind energy projects at DONG EnergyGentofte Denmark His research interests include modeling and advancedcontrol and numerical methods for wind power and smart grid

Jan Bendtsen was born in Denmark in 1972 He received the MSc andthe PhD degrees from the Department of Control Engineering AalborgUniversity Aalborg Denmark in 1996 and 1999 respectively

Since 2003 he has been an Associate Professor with the Department ofElectronic Systems Aalborg University In 2005 he was a Visiting Researcherwith Australian National University Canberra ACT Australia Since 2006he has been involved in the management of several national and internationalresearch projects and organizing international conferences In 2012 he was aVisiting Researcher with the University of California San Diego CA USAHis research interests include adaptive control of nonlinear systems closedloop system identification control of energy systems and linear parametervarying systems

Dr Bendtsen was the co-recipient of the Best Technical Paper Awardat American Institute of Aeronautics and Astronautics (AIAA) GuidanceNavigation and Control Conference 2009

Kristian Edlund received the MSc Electrical Engineering (EE) degree andthe PhD degree in control engineering from Aalborg University AalborgDenmark in 2006 and 2010 respectively

He is currently with the Development of Power Hub a commercial smart-grid and flexibility product from the Danish utility company DONG EnergyGentofte Denmark His research interests include the fields of mathematicaloptimization modeling control of energy systems and commodity markettrading in a smart grid context

Jakob Stoustrup received the MSc degree in electrical engineering in 1987 and the PhD degree in applied mathematics in 1991 from the Technical University of Denmark Kongens Lyngby Denmark

From 1991 to 1996 he held several positions in the Department ofMathematics Technical University of Denmark From 1997 to 2013 he was aProfessor of Automation and Control Aalborg University Aalborg Denmarkwhere he has been the Head of Research for the Department of ElectronicSystems since 2006 Since 2014 he has been a Chief Scientist at PacificNorthwest National Laboratory Richland WA USA leading the control ofcomplex systems initiative for the US Department of Energy His researchinterests include robust control theory and the theory of fault tolerant controlsystems


TABLE IOVERVIEW OF COMPUTATION TIMES REPORTED IN RECENT SCIENTIFIC

PUBLICATIONS ON OPTIMIZATION IN SMART GRID APPLICATIONSSIMULATION SAMPLES IS THE NUMBER OF DISCRETE TIME

STEPS THAT THE CONSIDERED SIMULATION HORIZON

HAS BEEN SPLIT INTO

delta evaluation and both Hill Climber and GRASP are ableto exploit that

Delta evaluation means that if the cost associated with oncandidate solution is known and a new solution is obtainedthrough a limited number of permutations of the old solutionthen the cost function for the new solution can be computedonly from the permutations Using delta evaluation can onlyimprove calculation time by a constant factor however thisfactor is usually so large that the actual improvement is farbetter than algorithmic changes

Other methods that could be considered for solving the dis-crete virtual power plant dispatch problem are ant swarm opti-mization [11] genetic algorithms [19] and tabu search [22]However these algorithms are all based on manipulating a setof candidate or tabu solutions They thus have a tendency todrown in logistics as the majority of the available computationtime is spent running through and updating solution sets

The main contributions of the paper is firstly the proof thatthe discrete virtual power plant dispatch problem problem isNP-complete Secondly we also show that even though findingoptimal solutions of the considered problem is indeed verychallenging highly promising results can be obtained in shorttime frames and for very large problems by special adaptationsof the considered heuristic algorithms

The paper is structured as follows Firstly Section IIpresents the discrete virtual power plant dispatch problemincluding flexibility modeling and agility In Section IIIthe computational complexity of the discrete virtual powerplant dispatch problem is explored in several different waysfirstly the NP-completeness is proven formally and then thefeasibility of solving the problem by use of the optimiza-tion package CPLEX [18] is explored In Section IV fourheuristic algorithms are developed namely uniform selec-tion (UHC) weighted selection (WHC) GRASP random andGRASP sorted These algorithms are tuned and compared inSection V Finally Section VI summarizes general conclusionsand suggestions for future work

II OPTIMIZATION PROBLEM

We consider a virtual power plant which is given the taskof satisfying the consumption needs of a portfolio of flexi-ble systems (distributed energy resources) by dispatching afluctuating power supply

A forecast of the fluctuating power supply is denotedPDispatch(k) k = 1 2 K and the flexible consumers aredenoted local units A portfolio of N local units is denotedLUii=12N At sample k we let Pi(k) denote the power tobe dispatched to LUi and any quantity which cannot be dis-patched to the portfolio is denoted S(middot) The objective is tominimize the residual power that is |S(middot)|

The optimization problem can be formulated as

minPi(middot)

Ksum

k=1

wk|S(k)| (1)

st


i=1


and also subject to the dynamics and constraints ofLUii=12N Here K denotes the total simulation horizon

The impatience weights wk isin R have been addedbecause the forecasted power production will rarely fit exactlywith the power needed to satisfy LUii=12N In practicewhen this happens a traditional power plant will have to adjustits power consumption such that the discrepancy betweensupply and demand is compensated As a rule of thumb how-ever the better time the plant operator has to modify theoutput of a traditional power plant the cheaper it can bedone It is therefore desirable to introduce slack as late onthe simulation horizon as possible which can be achieved byintroducing wk k = 1 2 K and requiring that wk1 gt wk2

if k1 lt k2

A Flexibility Modeling

In this paper the flexible units are modeled as batch-processes which are characterized by constant power con-sumption P a run time KRun and a deadline KEnd by whichthe process must be finished Also we let Ts denote the sizeof the time step Each LU can therefore be modeled by

E(k + 1) = E(k)+ TsP(k) (4)

P(k) = Pv(k) (5)

0 le E(k + 1) le E (6)

E(KEnd) = E (7)

0 lek+KRunminus1sum

l=k

v(l)minus KRun (v(k)minus v(k minus 1)) (8)

where P isin R+ k = 1 2 K v(k) isin 0 1 E = PKRunKRun le K and KEnd le K Inequality (8) is the minimumruntime constraint which ensures that if v(k) minus v(k minus 1) isone then v(l) l = k+ 1 k+ 2 KRun minus 1 must also all beone that is once the local units is activated it must completeits consumption immediately

We let LUii=12N denote a portfolio of N flexi-ble consumers Then for each LUii=12N a solutionof problem (1)ndash(3) consists of a set of start timesKStart = (KStart1 KStart2 KStartN) one for eachLU An illustration of the discrete virtual power plantdispatch problem for batch processes is given in Fig 1

3




B Agility






Ksum

k=1

(wk|S(k)| +

Nsum

i=1

wik|Pi(k)|)

(9)

st


i=1




minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1

i|Ei(k)|) (12)


minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1






















Ksum

k=1

wi|S(k)| (14)

st



Nsum

i=1











B CPLEX Performance



5












A Hill Climber






le KN





Nminusj+1











vnegative9 else


v(k)sum 1v

11 end if


13 end for









2 for j = 1 to N do







l


12 end for





B GRASP










V RESULTS


A Tuning


7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES




































3




B Agility






Ksum

k=1

(wk|S(k)| +

Nsum

i=1

wik|Pi(k)|)

(9)

st


i=1




minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1

i|Ei(k)|) (12)


minPi(middot)

Ksum

k=1

(wk|S(k)| +

Nsum

i=1






















Ksum

k=1

wi|S(k)| (14)

st



Nsum

i=1











B CPLEX Performance



5












A Hill Climber






le KN





Nminusj+1











vnegative9 else


v(k)sum 1v

11 end if


13 end for









2 for j = 1 to N do







l


12 end for





B GRASP










V RESULTS


A Tuning


7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES


















































Ksum

k=1

wi|S(k)| (14)

st



Nsum

i=1











B CPLEX Performance



5












A Hill Climber






le KN





Nminusj+1











vnegative9 else


v(k)sum 1v

11 end if


13 end for









2 for j = 1 to N do







l


12 end for





B GRASP










V RESULTS


A Tuning


7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES




































5












A Hill Climber






le KN





Nminusj+1











vnegative9 else


v(k)sum 1v

11 end if


13 end for









2 for j = 1 to N do







l


12 end for





B GRASP










V RESULTS


A Tuning


7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES







































2 for j = 1 to N do







l


12 end for





B GRASP










V RESULTS


A Tuning


7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES




































7





6 for j = 1 to N do





l


13 end for



DEVELOPED METHODS




ation σ =radic

(1(T minus 1))sumT

j=1








B Results













VI CONCLUSION










REFERENCES














































VI CONCLUSION










REFERENCES


































































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Heuristic Optimization for the Discrete Virtual Power ...kom.aau.dk/~jakob/selPubl/papers2014/tsg_2014.pdf · Heuristic Optimization for the Discrete Virtual ... link to the consumption