Critical peak pricing with load control demand response program in unit commitment problem

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Published in IET Generation, Transmission & DistributionReceived on 18th December 2012Revised on 26th January 2013Accepted on 13th February 2013doi: 10.1049/iet-gtd.2012.0739

ISSN 1751-8687

Critical peak pricing with load control demandresponse program in unit commitment problemJamshid Aghaei, Mohammad-Iman Alizadeh

Department of Electrical and Electronics Engineering, Shiraz University of Technology, 71555-313 Shiraz, Iran

E-mail: [email protected]

Abstract: In this study, critical peak pricing with load control (CPPLC), recently announced by Federal Energy and RegulatoryCommission, is investigated in a cost-emission-based unit commitment (UC) problem. In order to be easily implementable withavailable real market solvers, the non-linear, non-convex problem formulation is converted to multi-objective mixed integer linearprogramming (MMILP). The MMILP problem is then solved through a new modified ε-constraint multi-objective optimisationmethod. Moreover, UC is applied not only to schedule the status of the generating units but also to determine both price deviationsand load profile provided by CPPLC program. Finally, the conventional 10-unit test system is employed to indicate theapplicability of the proposed method through several case studies.

Nomenclature

T index for time.i index for conventional unit.n index for segment in linearised fuel cost

curve.m index for segment in linearised energy

consumption curve.k index for segment in linearised emission

cost curve.A(.), B(.), C(.) energy consumption coefficients in an

hour.a(.), b(.), c(.) fuel cost coefficients of a unit.α(.), β(.), γ(.) emission coefficients of a unit.alin, blin linear demand against price coefficients.D0(.) initial forecasted load demand in an hour.E(.) minimum limit on the emission of the unit.Elast(.) price elasticity of demand.G(.) minimum limit on the energy consumption

in an hour.M, L coefficients of the load deviation’s lower

limit boundary.Mmax maximum of load deviation coefficient

‘M’.MU(.), MD(.) minimum up/down time of a unit.Nseg number of linearisation segments of fuel

cost, energy consumption and emissioncost functions.

slp(.) slope of segment ‘n’ in linearised fuel costcurve of a unit in an hour.

slpg(.) slope of segment ‘m’ in linearised energyconsumption curve in an hour.

slpe(.) slope of segment ‘k’ in linearised emissioncost curve of a unit in an hour.

IET Gener. Transm. Distrib., 2013, Vol. 7, Iss. 7, pp. 681–690doi: 10.1049/iet-gtd.2012.0739

UT(.), TD(.) number of hours a unit has been on/off atthe beginning of the scheduling period.

Ngen number of conventional thermal units.Pmin(.),Pmax(.)

minimum/maximum generating capacityof a unit.

Pr0(.) initial electricity price in an hour.RU(.), RD(.) ramp up/down limit of a unit.SU(.),SD(.) start-up and shutdown cost of a unit.CSC(i),) cold start-up cost of a generation unit.HSC(i) hot start-up cost of a generation unit.CST(i) cold start-up time of a generation unit.Spin(.) spinning reserve in an hour.Pr(.) electricity price in an hour.D(.) final calculated demand in an hour.Pg(.) generation of a unit in an hour.o(.) generation of segment n in linearised fuel

cost curve.g(.) cost of segment m in linearised energy cost

function.u(.) binary unit status indicator.v(.) binary energy cost indicator.y(.) start-up indicator.z(.) shutdown indicator.Costconsum(.) energy consumption cost in an hour.CostGen(.) fuel cost of a unit in an hour.

1 Introduction

These days, a general tendency towards deploying availablepower resources to enhance the efficiency of power systemoperation is exceedingly ascending. Among many differentelectric power resources, renewable energy sources (RESs)and demand-side management (DSM) programs can becounted as two major categories. RESs have their ownadvantages such as low operating costs and no

681& The Institution of Engineering and Technology 2013

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environmental contaminants. These resources, however, havesome restrictions in being applied as dispatchable resourcesbecause of their intermittent behaviours, high capital costsand low availabilities (i.e. wind power generation). Thus, inspite of many advantages RESs may have, employing themas high reliable resources such as conventional fossil fuelsseems to be under some limitations. Besides, DSMprograms are more flexible in performance and more easilymatchable with power scheduling done by independentsystem operators (ISOs). Hence, demand response program(DRP) has been investigated in many recent papers as oneof the major branches of DSM programs.Federal Energy and Regulatory Commission (FERC), in its
recent survey, declared the latest modifications in DRP’sclassifications and definitions [1]. In [1], two new ones add13 versatile conventional DRPs. These two newly addedprograms are critical peak pricing with load control(CPPLC) and system peak response transmission tariff(SPRTT).According to the recent definition announced by FERC,

CPPLC program is a demand-side management thatcombines direct load control with a pre-specified high pricefor use during designated critical peak periods, triggered bysystem contingencies or high wholesale market prices. Tobe more precise, direct load control programs have beenone of the most common DRPs offered since 1968. Directload control programs are most often offered to residentialor small commercial customers to control appliances suchas air conditioning, water heating and pool pumps. Theseprograms help sponsors balance load by remotelycontrolling the appliances during peak periods in a price.Besides, the wholesale market prices might vary in case oflarge deviations in demand profiles. It is applicable toassume when electricity prices are high, system operator(SO) orders a reduction to down-stream load sponsors toreduce demand and consequently load sponsors mayremotely switch-off the curtailable appliances of their clients.To the best of our knowledge, CPPLC as one of the four

dynamic pricing programs (i.e. real-time pricing, time ofuse, critical peak pricing (CPP) and CPPLC) has beenconsidered in few works [2, 3]. In [2], Herter et al.statistically and analytically scrutinised the effects ofintegrating CPP programs in California during specificperiods. They found that significant and clear loadreduction occurred during critical periods for participantswith and without air-conditioning control, respectively, −41and −13%. Moreover, they observed that, large reductionsoccurred during extreme temperatures, suggesting that theprimary source of load reduction are cooling and heatingfacilities. They conclude that the use of CPP to countersystem peak spikes on hot summer days might be veryefficient. In the same direction, Newsham and Bowker [3],researched on some pilot CPP projects and suggested thatthe effective load reductions, in few peak days, can beachieved by educating households along with providingthem with excellent utility support services, without theexpense of new technologies.

2 Literature review

DRPs have been investigated in many recent papers. One ofthe first pioneer papers about demand response and priceelasticity of demand is [4]. In [4], price elasticity of demandin a pool-based electricity market has been taken intoconsideration when scheduling generation is done with


responsive loads and different amount of incentives are paidto curtailable loads. Prior to [4], spot pricing concept wasstated by Schweppe et al. [5]. It is demonstrated, thatcustomers can control their load based on spot electricmarket prices. The concept of demand responsiveness hasbeen being tracked with FERC staff surveys issued everyother year since 2006 [6–8]. In [6], for the first time, DRPswere classified into two major categories namely, time-basedprograms and incentive-based programs. Modifications weredone, in [7], by announcing more detailed sub-branches ofthe incentive-based programs (i.e. voluntary-based programs,mandatory-based programs and market clearing programs),which enhanced the number of different programs up to 12.In the recent issue of the surveys [8], 15 versatile programsare announced without any clustering because of combinedprograms.Based on what announced by FERC [6–8] and the concept

of price elasticity of demand, many leading papers have beenpublished [9–11]. In [9], Aalami et al. proposed an innovativemethod by which customers can participate in interruptible/curtailable programs targeting benefit maximisation. Aalamiet al. [10], proposed a procedure in order to aid systemregulator selecting and prioritising DRPs by means oftechnique for order preference by similarity to ideal solutionmethod. An analytical hierarchy process is used as apowerful method to select the most effective DRP.Q-learning method based on weighting method isimplemented in [11] to lead a comprehensive model ofDR. In [11], all possible demand against price functionscombined to realise DRPs.Unit commitment (UC) problem is also investigated in

some outstanding published papers in the presence ofDRPs. In [12], interruptible loads are investigated in UCproblem. In [13], a new demand response approachdistinguished from demand elasticity platform is presented.In this paper, the amount of curtailed load is restricted withsome realisation constraints such as minimum up/downtime, up/down ramp rates and maximum daily curtailmentsetc. Security constraint UC is considered as the baseproblem for implementing DRPs in [14]. Parvania and hiscoworkers suggested that DRP may pile up discrete retailcustomer responses and submit a bid-quantity offer to theISO. In this area, it is worth noting that, mixed integerlinear programming (MILP) for UC is scrutinised in [15] asone of the leading paper on linearising UC.DRPs are investigated in UC platform in recent papers

[16, 17]. Conejo et al. proposed a real-time pricing programconsidering price uncertainty in [16]. In this paper, robustoptimisation technique is employed to add price uncertaintyto the problem. In the same field, Roscoe and Aultinvestigated real-time DRP in high penetrated renewablegeneration system [17]. Authors recommend that instead ofmandatory load curtailment in case of low renewablegeneration a meaningful balance between real-time pricingand renewable generation can bestow flexibility uponcustomers. In [18], optimal incentive payment to thecurtailable load during incentive-based programs, ispresented in an economic and environment-driven UC.Authors proposed an MMIP for solving demand responseunit commitment problem.To the best of our knowledge, the contributions of this

paper with respect to the previous researches in the area arelisted as follows:

1. No similar CPPLC model in MILP has been proposed inthe technical literature.


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2. CPPLC DRP is investigated in a multi-objective UCformulation problem, which can be employed by ISOs orany other SOs.3. Dynamic elasticity of demand [11], is implemented as abasic platform to define price deviations in multi-objectiveMILP (MMILP).4. The lexicographic optimisation and hybrid augmented-weighted ɛ-constraint method is proposed to solve themulti-objective optimisation problem. The lexicographicoptimisation is used to determine the range of objectivefunctions more effectively compared with the conventionalɛ-constraint method. Moreover, the augmented ɛ-constraintmethod generates only efficient Pareto-optimal solutionsand avoids inefficient ones. In order to select the ‘best’compromised solution among the Pareto-optimal solutionsof multi-objective optimisation problem, a fuzzy decision-making tool is adopted.
The rest of the paper is organised as follows. Section 3indicates the latest modifications in DRP classifications anddefinitions. Section 4 describes the proposed model. Section5 describes modified ɛ-constraint method. Section 6provides and discusses results of a case study. The lastSection 7 is dedicated to the conclusions.

3 Glance at DRP

The recent definition of DR used in [8] is: changes in electricuse by demand-side resources from their normal consumptionpatterns in response to changes in the price of electricity, or toincentive payments designed to induce lower electricity useat times of high wholesale market prices or when systemreliability is jeopardised.This definition substitutes ‘demand-side resources’ for

the phrase ‘end-use customers’ used in previous surveys, toconform to the definition in use by North American ElectricReliability Corporation’s (NERC’s) demand response datatask force in its development of a demand responseavailability data system to collect DRP information. Fifteenrecent DRPs are as follows [8]:

† Direct load control.† Interruptible load.† CPPLC.† Load as capacity resources.† Spinning reserve.† Non-spinning reserve.† Emergency demand response.† Regulation service.† Demand bidding and buyback.† Time-of-use pricing.† CPP.† Real-time pricing.† Peak time rebate.† SPRTT.† Other programs.

4 Problem formulation

To better illustrate the underlying ideas of the proposedmulti-objective framework of UC problem with DRPs,deterministic formulation is adopted in this paper. Also, weconsider DC load flow formulations as did in many famousmarkets in the worldwide without network constraints, asthis model is simpler to describe and analyse. However, the


proposed model can be easily extended to the stochasticformulation with network loss and line flow constraintsbased on the AC power flow formulations.

4.1 Price elasticity of demand

Price elasticity of demand can be defined as the demandsensitivity with respect to the price [19]

Elast(t) = Pr0(t)DD(t)

D0(t)D Pr(t)(1)

According to (1) demand changes can be extracted as follows

DD(t) = D0(t)D Pr(t)Elast(t)

Pr0(t)(2)

Not like [19], instead of fixed elasticity, dynamic elasticity isemployed in the current paper. Meanwhile, among fourdifferent demands against price functions (i.e. linear,exponential, quadratic and logarithmic) linear function isused [11]

D(t) = alin + blin Pr(t) (3)

where according to (1) the linear dynamic elasticity can becounted as

Elast(t) = blinPr0(t)

alin + blin Pr0(t)(4)

consecutively, (2), (4) lead to (5)

DD(t) = blinD0(t)

alin + blin Pr0(t)D Pr(t) (5)

Equation (5) is used, in this paper, as demand changes duringrunning CPPLC program.

4.2 Cost of energy consumption

Cost of electrical energy consumption during a period (e.g.1 h) is extracted from multiplying forecasted demand duringthat period in the real-time price of the same period asindicated below

Costconsum(t) = D(t) Pr(t) (6)

where in this paper demand is bisected to initial and elasticpart. The elastic part is the same as one presented in (5)

D(t) = D0(t)+ DD(t) (7)

Real-time pricing is defined as CPP in this paper to beimplementable in CPPLC programs. The peak prices areobtainable by adding initial prices with price deviationscalculated during solving the optimisation problem as follows

Pr(t) = Pr0(t)+ D Pr(t) (8)

where by substituting (5) in (7) and consecutively (8) and (7)in (6), cost of electric consumption as a quadratic function ofprice deviation is achievable after some simplifications as


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below
Costconsum(t) = A(t)D Pr(t)2 + B(t)D Pr(t)+ C(t) (9)

where A(t), B(t), and C(t) are, respectively

A(t) = blinD0(t)


B(t) = D0(t)+blinD0(t) Pr0(t)


C(t) = D0(t)Pr0(t) (12)

4.3 Objective functions

This two objective function problem formulation, the firstobjective function is dedicated to conventional UCproblem along with modifications and the second objectivefunction considers environmental issues in generationscheduling.

4.3.1 Objective one: The objective of UC is to determineloads provided by CPPLC DRP and schedule commitmentstatus of conventional generating units such that totalgeneration cost, and total deviation in electric energyconsumption cost, as in (9), resulting from changing pricesin critical periods are minimised. The quadratic fuel costfunction can be formulated as

CostGen(i, t) = a(i)+ b(i)Pg(i, t)+ c(i)Pg(i, t)2 (13)

where Pg(i, t) in (MW) is the amount of real power generatedof unit ‘i’ in period ‘t’. As can be observed from (8) and (13),both generation cost and energy consumption cost functionsare non-linear quadratic functions. Linearisation isemployed, as stated in Appendix, to accommodate thesecost functions in an MMILP as below

CostGen(i, t) =∑Nseg

n=1

slp(n, i)o(n, i, t) (14)

Pg(i, t) = Pmin(i)u(i, t)+∑Nseg

n=1

o(n, i, t) (15)

Costconsum(t) =∑Nseg

m=1

slpg(m)g(m, t) (16)

D Pr(t) = G(t)v(t)+∑Nseg

n=1

g(m, t) (17)

Thus, the first objective function is as comes next(see (18))

where SU(i) and SD(i) represent the start-up and shutdowncosts, respectively. SU(i) can be identified by the


following criterion

SU(i)= HSC(i), if MD(i, t)≤TC(i) ≤MD(i, t)+CST(i)CSC(i), if TC(i).MD(i, t)+CST(i)

{(19)

where CSC(i), HSC(i), CST(i) represent cold start-upcosts, hot start-up costs and the cold start-up time of ageneration unit, respectively.

4.3.2 Objective two: In order to schedule generating unitsenvironmental-friendly, emission effects should be takeninto account. Therefore a polynomial emission function isconsidered as contrary objective function with the first oneas below

Em(i, t) = a(i)+ b(i)Pg(i, t)+ g(i)Pg(i, t)2 (20)

This objective function is linearised as declared inAppendix

Em(i, t) =∑Nseg

k=1

slpe(k, i)o(k, i, t) (21)

Pg(i, t) = E(i)u(i, t)+∑Nseg

n=1

o(n, i, t) (22)

4.4 First objective constraints

The first objective function is subject to the followingconstraints. The first and the most important constraint ispower balance that states generated power must satisfy loaddemand.

∑Ngen

i=1

Pg(i, t) = D0 + DD(t) (23)

Elastic part of demand has to be considered with somelimitations. The first one is elastic demand’s range. Thelower side of the range is

DD(t) ≥ −L.D0(t).M (24)

where L and M are specified as

M = Pr0(t)−min [ Pr0(t)]

max [ Pr0(t)](25)

In which M guarantees that load reduction would occur justwhen prices are higher than a specific value. L maintainsthe amount of maximum load reduction remains constant.Moreover, the (min) and (max) amounts are defined overthe entire period. Thus, to determine the value of L,suppose the amount of maximum allowed load reduction is

Min F1 =∑Tt=1

∑Ngen

i=1

CostGen(i, t)+ SU(i)y(i, t)+ SD(i)z(i, t)( )+ Costcustom(t)

{ }(18)


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Rmax then L will be as follows
L = Rmax

Mmax(26)

where Mmax is countable through (24) as

Mmax =max [ Pr0(t)]−min [ Pr0(t)]

max [ Pr0(t)](27)

It is worth noting that L and M are constant parameters.Meanwhile, the upper limit of elastic demand range is zero,which implies that no reduction will occur in off peak andvalley hours.Next constraint would be the price deviation, which has to

be positive in order to increase price during peak hours.Regular constraints regarding UC are outlined in thefollowing. Once the unit is committed/shutdown, it has tobe ‘on/off’ for a minimum number of hours indicated in thefollowing equations

∑UT(i)t=1

(1− u(i, t)) = 0 ∀i [ Ngen

y(i, t)+∑max [T ,t+MU(i)−1]

m=t+1

z(i, m) ≤ 1

∀i [ Ngen

∀t = UT(i)+ 1, . . . , T

(28)

where y(i, t) and z(i, t) are binary variables specifying thestart-up and shutdown status flags, respectively, and UT(i) is

UT(i) = max {0, min [T , MU(i)− TU(i, 0)u(i, 0)]} (29)

Accordingly, the shutdown time constraint can be consideredas given below

∑DT(i)t=1

u(i, t) = 0 ∀i [ Ngen

z(i, t)+∑max [T ,t+MD(i)−1]

m=t+1

y(i, m) ≤ 1

∀i [ Ngen

∀t = UT(i)+ 1, . . . , T

(30)

consecutively, DT(i) is

DT(i) = max{0, min[T , MD(i)− TD(i, 0)(1

− u(i, 0))]} (31)

The joint equation between start-up and shut-down indicatorsare

y(i, t + 1)− z(i, t + 1) = u(i, t + 1)− u(i, t)

y(i, t)+ z(i, t) ≤ 1(32)


Ramping up and down constraints are as indicated below

Pg(i, t + 1)− Pg(i, t) ≤ RU(i) ∀i [ Ngen (33)

Pg(i, t)− Pg(i, t + 1) ≤ RD(i) ∀i [ Ngen (34)

In order to provide a reliable generation scheduling,spinning reserve is required. This parameter is usuallypre-specified that is either equal to the largest unit or agiven percentage of total loads. It should be noted that tosatisfy spinning reserve constraint, the total amount ofmaximum capacity of all committed generating unitssubtracted the load demand must be greater than theamount of spinning reserve.

∑Ngen

i=1

Pmax(i, t)u(i, t) ≥ Spin(t)+ D0(t)+ DD(t) (35)

where Spin(t) represents the amount of overall reservecapacity in an hour.

5 Multi-objective optimisation

In multi-objective programming there is more than oneobjective function and there is no single optimal solutionthat simultaneously optimises all the objective functions. Awell-organised technique to solve multi-objective problemsowning one main objective function among all objectivefunctions is the ɛ-constraint method, which is used to solvethe proposed problem. The modified augmented ɛ-constraintmethod is concisely discussed in the following section.However, more details about this solution method can befound in [20].

5.1 Modified ε-constraint method

In multi-objective optimisation problem (also called,multi-performance, multi-criterion or vector optimisation)the engineer’s goal is to maximise or minimise severalobjective functions simultaneously. The purpose of multi-objective problem in the mathematical programmingframework is to optimise different objective functions, M isthe number, subject to a set of system constraints. For instance

Maximise f (x) = [ f1(x), …, fm(x)]T

Subject to. x∈ X

where x is an n-dimensional vector of decision variables, X isthe decision space and f(x) is a vector of M real-valuedfunction. Objective functions in a multi-objective problemare assumed to be in conflict and incommensurable. Anefficient solution is defined in [21]. The definition says asolution xe∈ X is said to be efficient solution if for any x∈X satisfying fk(x) > fk(xe), fi(x) > fi(xe) for at least one otherindex j≠ k.Among many multi-objective mathematical programming

methods, ɛ-constraint method shows many advantages. Inorder to properly apply the ɛ-constraint method, the rangesof at least M− 1 objective functions are needed that will beused as the additional objective function constraints. Themost common approach is to calculate these ranges fromthe payoff table. A detailed description of calculatingpayoff table and range of objective functions can be foundin [22]. However, two important issues related to


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ɛ-constraint method need to be scrutinised. First, the range ofthe objective function over the efficient set is not optimised.Second, there is no guarantee for Pareto-optimal solution,generated by ɛ-constraint method, to be efficient or notdominated.Lexicographic optimisation can be counted as a suitable
remedy for the first problem. To overcome the secondproblem augmented ɛ-constraint technique is implementedin the current paper.The importance of objective functions in generating

the Pareto solutions has not been seen in augmentedɛ-constraint method. Thus, in order to include the relativeimportance of the objective functions augmented-weightedɛ-constraint approach is proposed. A combination oflexicographic optimisation and hybrid augmented-weightedɛ-constraint technique is performed in the current paper.The following formulation explains augmented-weightedɛ-constraint method, briefly

Min/Max f1(x)+dir1r1w1

( )∑Mi=2

wiski

ri(36)

subject to.

eki = f mini (diri + 1)/2− f max

i (diri − 1)/2+ diririkqi (37)

si [ R+, i = 2, 3, . . . , M k = 0, 1, . . . , qi

where diri is the direction of ith objective function, P is thetotal number of objective functions. When diri is −1 itmeans that the ith objective function needs to beminimised and +1 means that the related objectivefunction have to be maximised. The efficient solutions ofthe problem can be obtained by defining ei as an iterativevariation parameter. si is surplus variable added to theconstraint formula. r1si/ri is added to the second term ofthe objective function to avoid scaling problem. wi is afactor of a decision maker to weight the objectivefunctions. ri, which is the range of the objective function,is countable through payoff table. In the conventionalaugmented ɛ-constraint method, the importance degree ofthe objective functions could not be allocated to theoptimisation problem in generating Pareto solutions. Thus,the augmented-weighted ɛ-constraint method presents thewi as the weight factor for ith objective function. It isworth noting that the proposed method utilises theweighting factor concept, although it is completelydifferent from the weighting method, which is used as anoptimisation approach. To avoid tautology in writing, werefer readers to the references [20, 23] for detailedfunctionality of both lexicographic optimisation andaugmented ɛ-constraint technique.

5.2 Fuzzy decision making

The algorithm described above generates the non-dominatedset of solutions known as the Pareto-optimal solutions. ISOmay have imprecise fuzzy goal for each objective function.The defined membership function mr

n indicates the degreeof optimality for nth objective function in the rthPareto-optimal solution. The whole membership function ofthe rth Pareto-optimal solution (μr) is calculated based on


its individual membership function mrn as below

mrn =

1 Frn ≤ Min(Fn)

Max(Fn)− Frn

Max(Fn)−Min(Fn)Min(Fn) ≤ Fr

n ≤ Max(Fn)

0 Frn ≥ Max(Fn)

⎧⎪⎪⎨⎪⎪⎩

(38)

where Frn and mr

n represent the value of the nth objectivefunction in the rth Pareto-optimal solution and itsmembership function, respectively. wn, in (39), is theweight value of the nth objective function in themulti-objective mathematical problem and M is the numberof Pareto-optimal solutions. Fuzzy decision-making thencan be achieved with the following formula

mr =∑p

n=1 wn.mrn∑M

r=1

∑pn=1 wn.m

rn

(39)

6 Numerical case studies and discussion

In order to investigate the applicability of the proposed method,a conventional 10-unit test system has been used with ascheduling time of 1 day (24 h). Unit characteristic of thementioned test system along with forecasted load profile areillustrated in Table 1 and Fig. 1, respectively, extracted from[18]. Moreover, emission coefficients related to the testsystem are given in Table 2. Linear elasticity of demandagainst price coefficients alin, blin are 1400 and −8.6, whichare extracted from [11] with some modifications. MMILP ismodelled applying CPLEX 11.2.0 solver in GAMSoptimisation software. The rest of this section is dedicated tothree case studies, which are base case, MMILP withoutconsidering DRP and MMILP considering DRP.

6.1 Base case study

This case is performed to obtain initial electricity prices.Initial prices are obtained after solving UC problem without

Table 1 Unit characteristic of conventional 10-unit test system

Units a(i) b(i) c(i) Pmax(i) Pmin(i)

1 1000 16.19 0.00048 455 1502 970 17.26 0.00031 455 1503 700 16.6 0.002 130 204 680 16.5 0.00211 130 205 450 19.7 0.00398 162 256 370 22.26 0.00712 80 207 480 27.74 0.00079 85 258 660 25.92 0.00413 55 109 665 27.27 0.00222 55 1010 670 27.79 0.00173 55 10

Units CSC(i) HSC(i) CST(i) MU(i) MD(i)1 9000 4500 5 8 82 10 000 5000 5 8 83 1100 550 4 5 54 1120 560 4 5 55 1800 900 4 6 66 340 170 2 3 37 520 260 2 3 38 60 30 0 1 19 60 30 0 1 110 60 30 0 1 1


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considering energy consumption and emission costs Fig. 2.MILP-based unit output power for the test system isillustrated in Table 3. It should be noted that the value ofthe total generation cost is 562 965 $ which is less thanwhat presented in [18] (i.e.565 283 $).Emission measures are also considered in the current paper.

Total emission for this problem is 13 994.162 tonnes. Therelated generation status is omitted because of the lack ofspace. The main idea of the proposed method comes frominitial prices which show a rational behaviour as clearlyexplained in the following. The valley hour prices areapproximately same and can be utilised as the base peakpricing. As what stated in Section 4.4, subtracting initialprices from the minimum prices and divide the result on themaximum price in an hour ‘M’ will lead to a proper scalingfactor to determine the quite enough load reduction range inthat hour [i.e. LD0(t)M ]. The base UC problem solved inGAMS optimisation software with 6049 single equations,

Table 2 Emission coefficients of generators

Units α(i), tonne/h β(i), tonne/MWh γ(i), tonne/MW2h

1 1000 16.19 0.000482 970 17.26 0.000313 700 16.6 0.0024 680 16.5 0.002115 450 19.7 0.003986 370 22.26 0.007127 480 27.74 0.000798 660 25.92 0.004139 665 27.27 0.0022210 670 27.79 0.00173

Fig. 2 Initial market clearing price

Fig. 1 Forecasted demand


5771 single variables and 2.880 discrete variables. TheCPU time required to solve the UC problem with a VAIO Eseries laptop computer powered by core i3 processor and3 GB of RAM was less than half a second.

6.2 Case study 1

In the current case study, a complete multi-objective UCproblem considering both fuel and emission costs isinvestigated. The MMILP-based problem is solved bymodified ɛ-constraint optimisation method in GAMSsoftware. Consequently, fuzzy decision-making is appliedon Pareto-optimal solutions and the results are piled up inTable 4. As can be inferred from the Table 4 slight changesoccurred in generation status in compared with Table 3 andas a result generation cost increased from 562 965 $ in thebase case to 563 151 $ in the current case. To be precise,changes in output of power generation are related to hour17 and changes in generation status occurred in hours 22and 23, where units 3 and 4, respectively, still generatingpower up to their maximum limits whereas unit 5 shutdown2 h earlier. Meanwhile, emission in the current case is25 086.23 tonnes, which is meaningfully less than whatstated in [18] (i.e. 26 298 tonnes). This case study isimplemented to show the proposed method efficiency in thenext section. It should be noted that the CPU time requiredto solve this case study is about 24.6 s with 11 091 singleequations, 10 813 single variables and 5280 discretevariables.

6.3 Case study 2

In the second and the last case study, fuel cost, energyconsumption and emission cost functions are considered asMMILP problem. The output of generation scheduling andthe electricity prices are tabulated in Table 5 and Fig. 3,respectively. As can be observed from Table 5, light grey

Table 3 Unit output for the 10-unit generation test system

h Output power, MW

1 2 3 4 5 6 7 8 9 10

1 455 245 0 0 0 0 0 0 0 02 455 295 0 0 0 0 0 0 0 03 455 370 0 0 25 0 0 0 0 04 455 455 0 0 40 0 0 0 0 05 455 390 0 130 25 0 0 0 0 06 455 455 0 130 60 0 0 0 0 07 455 410 130 130 25 0 0 0 0 08 455 455 130 130 30 0 0 0 0 09 455 455 130 130 85 20 25 0 0 010 455 455 130 130 162 33 25 10 0 011 455 455 130 130 162 73 25 10 10 012 455 455 130 130 162 80 25 43 10 1013 455 455 130 130 162 33 25 10 0 014 455 455 130 130 85 20 25 0 0 015 455 455 130 130 30 0 0 0 0 016 455 310 130 130 25 0 0 0 0 017 455 260 130 130 25 0 0 0 0 018 455 360 130 130 25 0 0 0 0 019 455 455 130 130 30 0 0 0 0 020 455 455 130 130 162 48 0 10 10 021 455 455 130 130 110 20 0 0 0 022 455 455 0 130 40 20 0 0 0 023 455 420 0 0 25 0 0 0 0 024 455 345 0 0 0 0 0 0 0 0


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highlighted cells associated with the first two columns specifythe meaningful changes in real power generation but nochanges in generation status. For the next two generators (i.e. generators 3 and 4) the same trend is observable. Theamount of real power generation, however, has very slightchanges for generators 3 and 4 in compared with the sameamounts in the previous case study. The significant effectsof the proposed DRP can be tracked in generators 5 and 6with 18 h on in the previous case study to 16 h on in thecurrent case for generator 5 and consecutively, 9 h on in theprevious case to just 3 h in the current one for the generator 6.

Table 4 Unit output for the first case study

h Output power, MW

1 2 3 4 5 6 7 8 9 10

1 455 245 0 0 0 0 0 0 0 02 455 295 0 0 0 0 0 0 0 03 455 370 0 0 25 0 0 0 0 04 455 455 0 0 40 0 0 0 0 05 455 390 0 130 25 0 0 0 0 06 455 455 0 130 60 0 0 0 0 07 455 410 130 130 25 0 0 0 0 08 455 455 130 130 30 0 0 0 0 09 455 455 130 130 85 20 25 0 0 010 455 455 130 130 162 33 25 10 0 011 455 455 130 130 162 73 25 10 10 012 455 455 130 130 162 80 25 43 10 1013 455 455 130 130 162 33 25 10 0 014 455 455 130 130 85 20 25 0 0 015 455 455 130 130 30 0 0 0 0 016 455 310 130 130 25 0 0 0 0 017 443 272 130 130 25 0 0 0 0 018 455 360 130 130 25 0 0 0 0 019 455 455 130 130 30 0 0 0 0 020 455 455 130 130 162 48 0 10 10 021 455 455 130 130 110 20 0 0 0 022 426 394 130 130 0 20 0 0 0 023 455 315 0 130 0 0 0 0 0 024 455 345 0 0 0 0 0 0 0 0


The counterpart status for peak generators, in the previouscase study are highlighted with light orange to declare thechanges occurred in generator status during this case. Asit can be seen in Table 5, peak hours are targeted and thevery expensive generators are rather off or generate as littleas possible during these hours. Off peak operation forgenerators not only enhances the reliability of the wholepower system, but also releases more generation capacity incase of emergency and lower released contaminants to theair. For instance, unit ‘9’ has not been committed at allduring scheduling period.Accordingly, total generation cost from 563 151 $ in the

first case declined to 541 712 $ in the current case whichmeans 21 439 $ reduction in total generation cost. Samereduction trend is observable for the emission cost functionfrom 25 086.23 tonnes for the first case study to 24 370.56tonnes for the second case.Demand profile presented in Fig. 4 clearly shows that

regarding in what range is the peak demand, loadcurtailment occurred with a rational trend. Moreover,demand curtailment during noon hours is higher than lateevening hours. It should be noted that for this case study

Fig. 3 Market clearing price variations

Table 5 Output power of the second case study

h Output power, MW

1 2 3 4 5 6 7 8 9 10

1 367 333 0 0 0 0 0 0 0 02 417 333 0 0 0 0 0 0 0 03 431 394 0 0 25 0 0 0 0 04 455 455 0 0 38.7 0 0 0 0 05 424.5 424.5 0 126 25 0 0 0 0 06 455 455 0 130 38.7 0 0 0 0 07 440.5 424.5 130 130 25 0 0 0 0 08 455 455 130 130 30 0 0 0 0 09 455 445 130 130 93.5 0 0 10 0 010 426.2 424.5 130 130 148.3 0 25 0 0 011 430.45 424.5 130 130 148.3 0 25 0 0 012 434.7 424.5 130 130 148.3 0 25 46 0 013 436.2 424.5 130 130 148.3 0 25 0 0 014 455 431.3 130 130 107.2 0 0 10 0 015 455 455 130 130 30 0 0 0 0 016 447 333 130 130 0 0 0 10 0 017 455 285 130 130 0 0 0 0 0 018 436 394 130 130 0 0 0 10 0 019 455 435 130 130 0 20 0 10 0 020 455 455 130 130 0 50 31 0 0 14.521 455 455 130 130 0 26 29 0 0 022 455 455 0 130 35 0 25 0 0 023 450.5 424.5 0 0 25 0 0 0 0 024 381 394 0 0 25 0 0 0 0 0


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20% of initial load is allowed as maximum load reduction. Asa result, the amount of L is achieved 0.595. The CPU timerequired for the present case study is about 25 s with 11140 single equations, 10 886 single variables and 5280discrete variables for each set of sub-optimisation problem,which means that the proposed method imposed almost nocomputational burden.

7 Conclusion

This paper presents a new CPPLC mathematical model in anMILP problem. Moreover, the efficiency of the proposedmethod is investigated in a UC problem. Besides, theproposed MIP model can be implemented easily by ISOsand any other SOs. To consider environmental issues in thecurrent paper, an emission-based UC problem in a MMILPproblem is proposed. Considering versatile case studiesverify that tracking electricity price deviation in thepresented form can facilitate DRPs realisation. Standard10-unit generation test system is employed to investigatethe applicability of the model. The elapsed time for solvingthe MMILP problems verifies that the method would notimpose any additional computational burden.Finally, the research work is underway in order to

incorporate uncertainty sources (wind power and priceuncertainty) in the stochastic framework of UC problem inthe presence of DRPs.

8 References

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2 Herter, K., McAuliffe, P., Rosenfeld, A.: ‘An exploratory analysis ofCalifornia residential customer response to critical peak pricing ofelectricity’, Energy, 2007, 32, (1), pp. 25–34

3 Newsham, G.R., Bowker, G.B.: ‘The effect of utility time-varyingpricing and load control strategies on residential summer peakelectricity use: a review’, Energy Policy, 2010, 38, (7), pp. 3289–3296

4 Kirschen, D.S., Strbac, G., Cumperayot, P., Mendes, D.: ‘Factoring theelasticity of demand in electricity prices’, IEEE Trans. Power Syst.,2000, 15, (2), pp. 612–617

5 Schweppe, F.C., Caramanis, M.C., Tabors, R.D., Bohn, R.E.: ‘Spotpricing of electricity’ (Kluwer, Boston, MA, 1989)

6 FERC: Staff Report, ‘Assessment of demand response and advancedmetering’. [Online]. Available at http://www.FERC.gov. August 2006



9 Aalami, H.A., Moghaddam, P.M., Yousefi, G.R.: ‘Demand responsemodeling considering interruptible/curtailable loads and capacitymarket programs’, Appl. Energy, 2010, 87, (1), pp. 243–250

Fig. 4 Demand variations by implementing DRP


10 Aalami, H.A., Parsa Moghaddam, M., Yousefi, G.R.: ‘Modeling andprioritizing demand response programs in power markets’, Electr.Power Syst. Res., 2010, 80, (4), pp. 426–435

11 Yousefi, S., Moghaddam, P.M., Johari Majd, V.: ‘Optimal real timepricing in an agent-based retail market using a comprehensive demandresponse model’, Energy, 2011, 36, (9), pp. 5716–5727

12 Aminifar, F., Fotuhi-Firuzabad, M., Shahidehpour, M.: ‘Unitcommitment with probabilistic spinning reserve and interruptible loadconsideration’, IEEE Trans. Power Syst., 2009, 24, (1), pp. 388–397

13 Khodaei, A., Shahidehpour, M., Bahramirad, S.: ‘SCUC with hourlydemand response considering intertemporal load characteristics’, IEEETrans. Smart Grid, 2011, 2, (3), pp. 564–571

14 Parvania, M., Fotuhi-Firuzabad, M.: ‘Demand response scheduling bystochastic SCUC’, IEEE Trans. Smart Grid, 2010, 1, (1), pp. 89–98

15 Carrión, M., Arroyo, J.M.: ‘A computationally efficient mixed-integerlinear formulation for the thermal unit commitment problem’, IEEETrans. Power Syst., 2006, 21, (6), pp. 1371–1378

16 Conejo, A.J., Morales, J.M., Baringo, L.: ‘Real-time demand responsemodel’, IEEE Trans. Smart Grid, 2010, 1, (3), pp. 236–242

17 Roscoe, A.J., Ault, G.: ‘Supporting high penetrations of renewablegeneration via implementation of real time electricity pricing anddemand response’, IET Renew. Power Gener., 2010, 4, (4), pp. 369–382

18 Abdollahi, A.A., Moghaddam, P.M., Rashidinejad, M., Sheikh-el-eslami, M.K.: ‘Investigation of economic and environmental-drivendemand response measures incorporating UC’, IEEE Trans. SmartGrid, 2012, 3, (1), pp. 12–25

19 Kirschen, D.S., Strabac, G.: ‘Fundamentals of power system economics’(Wiley, Hoboken, NJ, USA, 2004)

20 Mavrotas, G.: ‘Effective implementation of the ε-constraint methodin multiobjective mathematical programming problems modifiedaugmented’, Appl. Math. Comput., 2009, 213, (2), pp. 455–465

21 Afkousi-Paqaleh, M., Rashidinejad, M., Pourakbari-Kasmaei, M.: ‘Animplementation of harmony search algorithm to unit commitmentproblem’, Electr. Eng., 2010, 92, pp. 215–225

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9 Appendix

To reduce computational burden in solving optimisationproblems, linear formulation are preferred. Any continuousfunction including one variable can be linearised withacceptable accuracy, which directly depends on the numberof linear segments. Fig. 5 illustratively shows the linearapproximation of a non-convex function [24]. However, inthe case of non-convex functions, non-linearity needs to beintroduced to the model when solving a minimising(maximising) problem. To formulate the non-linearity, oneapproach is adding binary variables and new inequalitiesto the model where the result will be a MIP. Supposenon-linear function f (x) specified by the points (el f (el)), l∈{0, …, L} let: ul = bl− bl−1, gl = f (el) − f (el−1), ∀ ∈ {0, …,L}. For any b0≤ x ≤ bl, linear approximation model for anon-linear function can be

x = b0 +∑Ll=1

Ml and f (b0)+∑Ll=1

glulyl (40)

0≤Ml≤ ui∀i∈ {1, …, L} and Mi+1 = · · · = ML = 0whenever Mi < ui∀i∈ {1, …, L − 1}. To accommodate theinequalities corresponding to the (40) in a MIP formulation,we can introduce the binary variable ai, i∈ {1, …, L− 1},


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Fig. 5 Non-convex function and its piecewise linearapproximation


along with the following constraints

u1a1 ≤ M1 ≤ u1uiai ≤ Mi ≤ uiai ∀i [ 2, . . . , L− 1{ }0 ≤ ML ≤ uLaL−1

⎫⎬⎭ (41)

where the abbreviated form of the (41) is

uiai ≤ Mi ≤ uiai−1 ∀i [ {2, . . . , L}, a0 = 1

ai [ {0, 1} ∀i [ {2, ..., L}, Mi ≥ 0 ∀i [ {2, ..., L}

(42)

The inequality forces all subsequent ai must be zero if each aiis zero.If ai = ai–1 = 1, both sides of Mi reduce to ui≤Mi≤ uiwhich is Mi = ui. Similarly, if ai = ai−1 = 0, then the two sidesof inequality reduce to Mi = 0. The last case would be ai = 0but ai−1 = 1. In this case, the two-sided inequality will be0≤Mi≤ ui. Generally, the linear counterpart of a non-linearterm can be presented by a combination of (40)–(42).


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Critical peak pricing with load control demand response program in unit commitment problem