IEEE TRANSACTIONS ON SMART GRID 1 Dynamic Economic ...static.tongtianta.site/paper_pdf/47d67e4e-e6ee-11e8-aa5f-00163e08bb86.pdfalgorithm namely random drift particle swarm optimization

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IEEE TRANSACTIONS ON SMART GRID 1

Dynamic Economic Dispatch Problem IntegratedWith Demand Response (DEDDR) Considering

Non-Linear Responsive Load ModelsHamdi Abdi, Ehsan Dehnavi, and Farid Mohammadi

Abstract—Intelligent implementation of demand response pro-grams (DRPs) not only decreases electricity price in electricitymarkets, but also improves network reliability. In this paper, thedynamic economic dispatch (DED) problem has been optimallyintegrated with the incentive-based DRPs. Moreover, mathemati-cal load modeling can be so effective in the load curve estimationwith the lowest error. So, economic models of the linear andnon-linear responsive loads (power, exponential, and logarithmic)have been developed for time-based and incentive-based DRPsand integrated with DED. Also, a procedure to select the mostconservative responsive load model for the load estimation hasbeen presented too. Also, determining the optimal incentive inthe incentive-based DRPs is one of the independent system oper-ator’s challenges. In the proposed combined model, the fuel costis minimized and the optimal incentive is determined simulta-neously. Valve-point loading effect, prohibited operating zones,spinning reserve requirements, and the other non-linear practi-cal constraints make the combined problem into a complicated,non-linear, non-smooth, and non-convex optimization problem,which has been solved with a population-based meta-heuristicalgorithm namely random drift particle swarm optimization algo-rithm. The proposed combined model is applied on a ten unitstest system. Results indicate the practical benefits of the proposedmodel.

Index Terms—Demand response, dynamic economic dispatch,DEDDR, incentive-based demand response programs (DRPs),non-linear responsive loads modelling, optimal incentive, randomdrift particle swarm optimization (RDPSO).

I. INTRODUCTION

DEMAND response (DR) has gotten much attention allover the world due to its impressive benefits such as the

price reduction and reliability improvement. Demand responseprograms (DRPs) are divided into two main categories namelythe incentive-based and time based programs and each cat-egory has some programs too. DRPs are categorized asfollowing [1], [2].Incentive-based DRPs:

• Emergency demand response program (EDRP/DLC)• Direct Load Control (DLC)• Interruptible / Curtail able Service (I/C)

Manuscript received August 31, 2015; revised September 9, 2015 andNovember 7, 2015; accepted December 10, 2015. This work was sup-ported by the Iran Energy Efficiency Organization under Grant 93/270.Paper no. TSG-01043-2015.

The authors are with the Department of Electrical Engineering,Faculty of Engineering, Razi University, Kermanshah 67149-67346,Iran (e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TSG.2015.2508779

• Ancillary Services Market (A/S)• Capacity Market (CAP)• Demand Bidding (DB)

Time-based:• Time of Use (TOU)• Real Time Pricing (RTP)• Critical Peak Pricing (CPP)This paper mainly focuses on those common

incentive-based DRPs, i.e., emergency demand responseprogram (EDRP) and direct load control (DLC) by whichincentives are paid to the customers to reduce or cut theirconsumption during peak hours. These two programs can beimplemented voluntary or compulsory.

DRPs’ modeling based on the price elasticity matrix (PEM)is one of the most common and powerful methods in thisfield [3]–[6]. As will be shown in this paper, sometimes con-sidering just the linear model of the responsive loads maylead to significant mistakes in the load estimation and conse-quently may impose enormous costs. In other words selectingthe most reliable and conservative responsive load model hasa great realistic insight in carrying out some tasks in powersystems such as the load estimation and reserve procurementetc. In this paper the way of selecting the most conservativeresponsive load model is presented too. Linear and non-linearmodels of time based DRPs with a focus on TOU have beendeveloped in [7]. They showed that DRPs can reduce cost andimprove reliability. Their developed models can be applied onthe time based DRPs and cannot be applied on the incentivebased DRPs, i.e., EDRP/DLC. Therefore, in this paper non-linear models of responsive load, i.e., power, exponential, andlogarithmic models have been developed both for the incen-tive based and time based DRPs and their proofs are given inthe Appendix A. These extracted models can be applied onboth incentive based and time based DRPs. Moreover deter-mining the optimal incentive in incentive-based DRPs shouldbe based on a feasible approach. It is because of the fact thatif it is determined inappropriately, it may impose high addi-tional cost at the supply side or it may create new peaks whenDRP ends [8], and decrease the network reliability [9] whichis not based on independent system operator’s (ISO’s) pointof view.

Due to the natures of the DRPs and dynamic economicdispatch (DED) problem which focus on the demand sideand supply side respectively, for a more comprehensive andeffective investigation, integrating these two problems seems

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2 IEEE TRANSACTIONS ON SMART GRID

very useful. By the way, the supply side is connected to thedemand side. Therefore, it prevents high additional cost, newpeaks or valley. In other words, in the combined problem thefuel cost is minimized and the optimal scheduling of units isappointed and, also the optimal incentive is determined.

DRPs prevent undesirable effects which impose financialcost and decrease social welfare. So, investigating the effectsof DRPs on improving the security of the restructuring powersystems is one of the ISO’s goals. In the presented combinedproblem, i.e., DEDDR the effects of the incentive-based DRPsin reducing demand side and supply side costs and improvingthe network reliability are investigated too.

There are few works related to the economic dispatch (ED)problem integrating with the DRPs. In [10] a combinationof the ED problem and demand side management (DSM)with a focus on the abstraction of houses, renewable energyresources (RES), in a localized setting which user preferencesand generation capacity are encoded as constraints of the sys-tem for residential loads in a micro-grid, has been presentedbased on the genetic algorithm (GA). They showed that theirproposed approach can effectively reduce operating costs ina single and multiple facility micro-grids for both suppliersand consumers alike [10]. In [11] a novel model of genera-tion dispatch in regional grids is integrated with demand sideresponse. In their combined model the concept of the dissatis-faction degree for customers and the utility is introduced intothe problem formulation to determine a proper compensationto be paid to customers participating in DR. They investigatedtime of use (TOU) demand response program which is a timebased DRP. In their combined model the optimal price is deter-mined by which the total generation cost at the supply sideand also the cost of electricity consumption at the demand sidedecrease [11].

Nwulu and Xia investigated the game theory based DR inte-grating with the economic and environmental dispatch [12].Their combined model is not so clear and by paying incen-tives to the customers, at the all periods (even at the valleyperiod) of the operation, the demand is decreased whichmay not be always realistic, practical, and economical (themain goal of implementing DRPs is decreasing demand atthe peak and critical hours not at the all hours of opera-tion). Also, this may not be based on the ISO’s point ofview. Also, the valve-point loading effect, prohibited operatingzones (POZs), and spinning reserve requirements (SRRs) havenot been considered in their model. Moreover, like the previ-ous works, just linear model of the responsive loads has beentaken into account, while in this paper non-linear responsiveload models including power, exponential, and logarithmichave been developed and taken into account in the combinedproblem.

DRPs have a high capability of improving the SRRs andconsequently improving the network reliability. In this paperthis concept has been investigated too.

DED problem including some non-linear practical con-straints such as ramp rate limits, POZs, SRRs etc. integratedwith the incentive-based DRPs is a complicated, non-linear,non-convex, and non-smooth optimization problem. Most ofthe traditional optimization methods cannot successfully solve

such a complicated problem. These methods have a high sensi-tivity to the initial estimation. So, they may converge to a localoptimal solution. On the other hand due to their large num-ber of the calculations, they have low speed. Population-basedmeta-heuristic algorithms (PBMHA) have a good performancein solving economic load dispatch problems [13], [14]. Thesemethods have higher capability of solving the non-linear prob-lems than the traditional methods and usually can optimallysolve non-convex and non-smooth cost functions. Particleswarm optimization (PSO) [15], the honey bee colony opti-mization (BCO) [16], cuckoo search algorithm (CSA) [17],and gravitational search algorithm (GSA) [18], are some ofthese optimization algorithms. In this regard, a modified algo-rithm has been introduced in [19] which is based on thePSO algorithm namely random drift particle swarm optimiza-tion (RDPSO) algorithm. Due to the investigations carried outin [19], this method has a better performance than the otheralgorithms in solving ED problem subject to non-smooth andnon-convex objective function and different non-linear con-straints. So, this method, i.e., RDPSO has been used in thispaper to solve DEDDR. It should be noted that its code hasbeen written in MATLAB environment.

The main contributions of this paper are as following.(I) Extracting the non-linear mathematical models (power,exponential, and logarithmic) of the incentive based and timebased DRPs and presenting a procedure to find the most reli-able and conservative responsive load model. (II) Intelligentintegration of DED and EDRP/DLC, i.e., DEDDR consider-ing some practical non-linear constraints such as the SRRs,POZs, etc. by which the optimal incentive in the EDRP/DLC isdetermined and the generation units’ power outputs are sched-uled simultaneously. (III) Investigating the effects of DRPson improving spinning reserve requirements (SRRs) and alsothe effects of not choosing the reliable load model whichhave been taken into account for the first time in this paper.(IV) Presenting the solution method and handling constraintsof the combined problem, i.e., DEDDR by meta-heuristicalgorithms with a focus on RDPSO.

The rest of this paper is organized as follows. InSection II the problem formulation through integrating DEDand EDRP/DLC is presented. The solution method of theDEDDR problem is given in Section III. The characteristicsof the test system are introduced in Section IV. Numericalsimulation and results are presented in Section V. Finally, inSection VI the conclusion is drawn.

II. PROBLEM FORMULATION

The Dynamic Economic Dispatch (DED) is one of theimportant optimization problems used in power systems toobtain the optimal operation schedule of the generation unitsover the entire dispatch period. The cost function of the DEDproblem considering value-point effects is as (1) [14], [20].

Fi(Pi,t) = ai + biPi,t + ciPi,t

2 + ∣∣disin(ei(Pi

min − Pi,t))∣∣ (1)

Where Pi,t is the power output of the ith unit at the tthinterval, Ng is the number of the generating units, ai, bi, ci arethe fuel cost coefficients of the ith unit, Pi

min is the minimum


ABDI et al.: DEDDR CONSIDERING NON-LINEAR RESPONSIVE LOAD MODELS 3

power generation, and di, ei indicate the valve point loadingeffect.

The cost of implementing EDRP/DLC is as (2).

CDR(t) = (d0(t) − d(t))inc(t) (2)

Finally the objective function of the proposed combinedmodel, i.e., DEDDR is minimization of (3).

TOF(Pi,t) =

T∑

t=1

⎧⎨

⎩

⎡

⎣Ng∑

i=1

{ai + biPi,t + ci

(Pi,t)2

+ ∣∣disin(ei(Pmin

i − Pi,t))∣∣}⎤

⎦+ CDR(t)

⎫⎬

⎭(3)

A. Constraints

The proposed model should satisfy the following equalityand inequality constraints.

1) Power Balance Constraint:

Ng∑

i=1

Pi(t) = d(t) + PL(t) t = 1, . . . , T (4)

Where d(t) and PL,t are the load demand and the powerloss of transmission line at the tth time interval.

d(t) is calculated by (15)-(18) and PL(t) is calculated byKron’s loss formula which can be given as (5).

PL(t) =Ng∑

i=1

Ng∑

j=1

Pi,tBl(i, j)Pj,t (5)

Where Bl(i, j) is the power loss coefficient of the transmis-sion network.

2) Incentive Limits:

inc(t)min ≤ inc(t) ≤ inc(t)max (6)

Referring to [21] inc(t)min and inc(t)max are usually consid-ered to be 0.1×ρ0(t) and 10×ρ0(t) respectively.

3) Power Generation Limits:

Pimin ≤ Pi,t ≤ Pi

maxi = 1, 2 . . . Ng (7)

Where Pimin and Pi

max are the lower and upper generationlimits for the ith unit.

4) Prohibited Operation Zones (POZs) Constraint: Theallowed practical operation zones of the ith generator areas (8).

⎧⎪⎨

⎪⎩

Pmin(i,t) ≤ Pi,t ≤ Pl

(i,t),1 or i = 1, . . . , Ng

Pu(i,t),q−1 ≤ Pi,t ≤ Pl

(i,t),q or t = 1, . . . , TPu

(i,t),ni≤ Pi,t ≤ Pmax

(i,t) q = 2, 3, . . . , ni

(8)

Where Pi,ql and Pi,q

u are the lower and upper limits of theqth POZ respectively, ni is the number of POZs for the ith unit.

5) Generator Ramp Rate Limits: The increase and decreaserates of the generator power output are usually calledthe Ramp Up (RU) and Ramp Down (RD) respectively.

So, the operating ranges of the ith unit are as (9).{

PI,t − PI,t−1 ≤ RUi

PI,t−1 − PI,t ≤ RDI(9)

Where RUi and RDI are the Ramp Up and Ramp Downlimits of the ith unit, respectively and are usually expressedin MW/h.

6) Spinning Reserve Requirements (SRRs): SRRs for DEDproblem are expressed by (10)-(12) [22], [23].

D1t =Ng∑

i=1

Pmaxi − (d(t) + PL,t + SRRt

) ≥ 0, t = 1, . . . , T

(10)

D2t =Ng∑

i=1

(min(Pmax

i − Pi,t, RUi))− SRRt ≥ 0, t = 1, . . . , T

(11)

D3t =Ng∑

i=1

(min

(Pmax

i − Pi,t,RUi

6

))− SRR

′t ≥ 0, t = 1, . . . , T

(12)

Where SRRt and SRR′t are the spinning reserve requirements

for 60 and 10 minutes compensation time in the tth hour andare expressed in MW.

B. Economic Models of the Responsive Load

To obtain the optimal consumption at the demand side, theelasticity is defined as the sensitivity of the demand respect tothe price as (13) [3]–[6].

E(t, t′) = ρ0(t′)d0(t)

∂d(t)

∂ρ(t′)

{E(t, t′) ≤ 0 if t = t′

E(t, t′) ≥ 0 if t �= t′

(13)

Where E is the elasticity, d(t) and d0(t) are the cus-tomer’s demands after implementing DRP and before it, duringperiod t, ρ(t′) and ρ0(t′) are the elasticity price and the initialelectricity price during period t

′respectively.

For 24 hours in a day, self and cross elasticity values canbe given as a 24×24 matrix as (14).⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�d(1)

d0(1)�d(2)

d0(2)�d(3)

d0(3). . .

�d(24)

d0(24)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=⎡

⎢⎣

E(1, 1) · · · E(1, 24)...

. . ....

E(24, 1) · · · E(24, 24)

⎤

⎥⎦×

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�ρ(1)

ρ0(1)�ρ(2)

ρ0(2)�ρ(3)

ρ0(3). . .

�ρ(24)

ρ0(24)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

The economic linear model of the responsive load isas (15) [4], [5], [8].

dlin(t) = d0(t) ×{

1 +24∑

t′=1

E(t, t′)

×[ρ(t′)− ρ0

(t′)+ inc

(t′)− pen

(t′)]

ρ0(t′)

}

(15)



The economic power, exponential, and logarithmic modelsof responsive loads are as (16)-(18) respectively. The proofsof their models are given in the Appendix A.

d pow(t) = d0(t) ×24∏

t′=1

(ρ(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)E(t,t′)

(16)

dexp(t) = d0(t)

× exp

(24∑

t′=1

(ρ(t′)− ρ0

(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)

E(t, t′))

(17)

dlog(t) = d0(t) ×24∑

t′=1

((

lnρ(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)

E(t, t′))

(18)

III. SOLUTION METHOD FOR THE DEDDR PROBLEM

In this part a general procedure for solving theDEDDR problem by the population-based meta-heuristic algo-rithms (PBMHAs) is presented. The possible solutions inRDPSO are called particles, in honey bee colony optimiza-tion bees, in cuckoo search algorithm cuckoos, in gravitationalsearch algorithm masses etc.

In all evolutionary algorithms, a set of possible solutionsof optimization problem namely initial population should bechosen to start the optimization procedure. Each member ofpopulation (candidate) has two main parts. One part includesthe population’s parameters (the amount of optimization vari-ables) and the other one includes the objective function. Then,each algorithm by its own specific procedure, changes the pop-ulation’s parameters with consecutive iterations and subject tothe different constraints in a way that the amount of the objec-tive function (the solution with least amount of the objectivefunction in the minimization problems) be close to the optimalsolution. In the proposed combined problem, i.e., DEDDR, thepopulation’s parameters are the same power outputs of genera-tion units which should be determined in a way that minimizethe objective function in (3).

In fact, in DEDDR, every scheduled generating units outputat each hour comprises a candidate. The kth candidate (PGk)

at each hour is defined as (19).

PGk = [Pk,1, Pk,2, . . . , Pk,j, . . . , Pk,Ng

], k = 1, 2 . . . PS (19)

Where PGk is the current position of the kth vector, Ng isthe number of generation units, PS is the population size, j isthe generator number, and Pk,j is the power output of the jthgeneration unit.

The power balance constraint (See (4)) can be met by addinga penalty factor to the (3). By the way the evaluation functionof DEDDR is minimization of (20).

EF(Pi,t) =T∑

t=1

⎧⎨

⎩

Ng∑

i=1

TOF(Pi,t)

+ Kn.abs

⎛

⎝Ng∑

i=1

Pi,t − d(t) − PL,t

⎞

⎠

⎫⎬

⎭(20)

Where Kn is the penalty factor which is a positive realnumber (it is considered to be 1000 in this paper). If the con-straint (4) is nonzero, the amount of the second term in (20)will be nonzero too. In other words a candidate which doesn’tmeet the constraint (4) will have a large evaluation functionand more likely will be discarded. On the other hand a can-didate which meets the constraint (4) will have a relativelysmall evaluation function and consequently will be kept.

To meet the other constraint and also the way of using (20),the general procedure of solving the DEDDR problem ispresented in the form of the nine steps as following.

A. Solving the Combined Problem and Handling Constraints

The proposed combined problem (DEDDR) is solvedby RDPSO in nine steps as following. Also, the simpli-fied flowchart of the DEDDR solution method is shown inFig. 1 based on the following steps.

Step 1: Defining the initial data such as the characteris-tics of generation units, initial load curve, initial electricityprice, PEM, initial incentive, determining the dispatch inter-val (T=24 hours), determining the optimization iteration whichis set to be 100 in this paper.

Step 2: Increasing the amount of incentive by ISO. Inthis paper the step of changing incentive is considered to be0.25 $/MWh (inc(t + 1) = inc(t) + 0.25).

Step 3: Setting the number of the hour to zero (t=0).Step 4: t=t+1.Step 5: Determining the hourly demand in 24 hours

by (15)-(18).Step 6: Determining the initial population of optimization

algorithm (some possible solutions of DEDDR) as following:1- K=1, k is the number of candidate.2- Based on pervious explanation, the kth candidate (the

output of generation units for the kth component of pop-ulation) is generated randomly in the permissible rangesfor each unit described in (7).

3- Calculating the transmission line losses for the kthcandidate based on (5).

4- For the kth candidate the following constraints areevaluated.

Constraint one: the power output of each generating unitsshould not be in POZs (See (8)).

Constraint two: The ramp rate limits are evaluated so thatincrease and decrease rates of each generating units from theprevious hour be in acceptable ranges defined by (9). If theinitial power outputs of generating units are not given, it issupposed thatinitial power outputs of all generating units arein acceptable ranges and there is no need to consider thisconstraint at the first hour.

Constraints three: The power outputs of generating unitsmeet SRRs based on (10)-(12).

5- If all above constraints are met, add one unit to k(k=k+1).

6- If the amount of k is smaller or equal to the selectedpopulation size (PS), return to number 2 in this step.

Step 7: Execution of the main loop of selected optimizationalgorithm.



Fig. 1. Simplified flowchart of DEDDR solution method.

It should be noted that the main difference between the evo-lutionary algorithms is the way of population’s convergence tothe optimal solution. In this paper RDPSO has been used inwhich every population member (candidate) is a particle. Aftergenerating the initial particles based on steps 1-6, every parti-cle moves toward its best personal experience (best) and bestgroup’s experience (gbest) with a specific speed. Best expe-rience means when a particle has the least objective function(in the minimization problem). If during the process of parti-cles’ movement, the current position of a particle is better thanits best personal position, then it will be selected as the bestpersonal experience. If its current position is better than thegroup’s experience, then it will be selected as the best group’sparticle. After this procedure, one iteration of RDPSO algo-rithm finishes. This procedure continues until the number ofiterations finishes and then the best group’s experience willbe selected as the solution of problem. For more informationabout the particles’ movement and speed refer to [19].

To meet the constraints of DEDDR, the main loop ofRDPSO algorithm is executed as following.

1- iteration=12- k=1 (k is the number of particle in population)3- Calculation of transmission line losses for the kth parti-

cle as (5).4- Calculating the cost of implementing EDRP/DLC as (2).5- Calculating the evaluation function for kth particle

by (20).

TABLE ISELF AND CROSS ELASTICITY VALUES

6- Moving of the kth particle toward its best personalexperience and best group’s experience.

7- Evaluation of following constraints for the kth particle.Constraint one: the output power of each generating unit

should not be POZs (See (8)) otherwise modified toward thenear margin of the feasible solution.

Constraint two: Same as Step 6, Number 4, Constraint 2;but if violated, then should be modified toward the near marginof the feasible solution.

Constraint three: The power outputs of generation unitsmeet SRRs described in (10) to (12).

8- If all the above constraints are meet, plus one unit to k(k=k+1).

9- If the amount of k is smaller or equal to the selectedpopulation size (PS), return to 3 in this step.

10- Determining the best personal experience of the kthparticle (bestk) and also best group’s experience (gbest).

11- Plus one unit to the iteration (iteration=iteration+1).12- If the number of iterations has net been finished return

to number 2 in this step. Otherwise save the bestparticle (gbest) as the solution of problem in the tth hour.

Step 8: If the tth hour is not equal to T (t �= T), go tostep 4. Otherwise save the final solution (optimal generationpower outputs over the whole dispatch period, i.e., 24 hours)for the related incentive.

Step 9: If the amount of incentive has not reached to itsmaximum value (See (6)), return to step 2. Otherwise select theincentive related to the best solution as the optimal incentiveand save the related parameters to this incentive as the optimaloutputs and solution of the DEDDR problem.

IV. TEST SYSTEM

To show the correctness, features, and practical benefits ofthe proposed model (DEDDR), it is applied on the ten unitstest system. The test system is taken from [24] with somemodifications. The characteristics of the test system is givenin the Appendix B. Also, the transmission line coefficientsare available in [12]. To consider the SRRs, the SRRs for60 and 10 minutes compensation time (SRRt and SRR

′t) have

been set to 5% and ( 1060 ) × 5% of the load demand as shown

in (10)-(12). Also, the elements of PEM which are determinedexperimentally are as Table I. This PEM is actually a commonone used in some related papers [3]–[5], [7], [8].

It is assumed that the EDRP/DLC is implemented vol-untary and there is no penalty for not participating inEDRP/DLC. Also, the daily load curve is divided into thepeak period (10 A.M.–16 P.M. & 20 P.M.–24 P.M.), the off-peak period (6 A.M.–9 A.M. & 17 P.M.–19 P.M), and thevalley period (0 A.M.–5 A.M.). Participation percentage ofEDRP/DLC is considered to be 20%. It means that 20 percent



TABLE IIDETAILS OF SCENARIOS

of the total load participates in the EDRP/DLC. The electricityprice in the valley, off-peak, and peak periods are 15, 25, and35 ($/MWh) respectively.

Three different groups with different values of PEM havebeen taken into account. Also, non-linear load models havebeen considered for each group. Details of scenarios aresummarized in Table II.

Some factors are defined as following (See (21)-(23)) toshow the effects of implementing DRPs on the load curveimprovement. Smoothness of the load curve is evaluated bythe Load Factor (See (21)). This factor is ideally 100% whichshows that demand is constant and does not change withthe time.

Load − factor% = 100 ×( ∑T

t=1 d(t)

T × dmax(t)

)

(21)

Peak to Valley is another factor which shows the distanceratio between peak to valley as (22).

Peak − to − Valley% = 100 ×(

dmax(t) − dmin(t)

dmax(t)

)(22)

Finally, Peak Compensate is defined as (23) and is thenormalized amount of compensated peak after implement-ing DRPs. In other words, after implementation of DRPs theamount of peak decreases; this factor, i.e., Peak Compensateactually demonstrates the normalized amount of this reduction.

Peak − Compensate% = 100 ×(

d0max(t) − dmax(t)

d0max(t)

)(23)

V. NUMERICAL SIMULATION AND RESULTS

Sixteen scenarios have been defined (Table II) to investi-gate the effects of implementing EDRP/DLC on the overallcost of the generation units. Results are shown in Table III.In all scenarios after implementing EDRP/DLC, the total costreduces. Implanting EDRP/DLC imposes an additional cost(CDR) which is paid as the incentive to the customers. But,the total cost which is sum of the cost of the generating unitsand the total incentive, reduces. It is because of the fact thatcustomers who participate in the EDRP/DLC reduce their con-sumption during the peak hours. From Table III, it can be

TABLE IIIEFFECTS OF IMPLEMENTING DEDDR IN DIFFERENT SCENARIOS

concluded that different load models have different incentivesin each group. As expected, by increasing elasticity, the totalincentive which should be afforded by the ISO is increasedand vice versa. Scenario 13 has the most cost reduction by14155.1282 $ (676747.2978-662592.1696) and scenario 7 hasthe least one by 3155.5637 $ (676747.2978-673591.7341). Onthe other hand the customer’s benefit in each group increaseswith the amount of PEM (scenario 10 has the most totalincentive by 12751.7143 $) and decreases with the gener-ation cost of units (scenario 9 has the least total incentiveby 4319.1868 $). Also, in all scenarios after implementingEDRP/DLC total losses decrease too.

All characteristics of the load curve are improved for sce-narios as shown in Table IV. The load curves before andafter implementing EDRP/DLC for the linear model in theall groups is illustrated in Fig. 2. As is clear from Fig. 2, thedemand at the peak hour decreases and whatever the elasticityis more the reduction at the peak hours is more too.

Also, the optimal generators power output after implement-ing EDRP/DLC (scenario 2) and its equality error are givenin the Appendix C.

To compare DED and DEDDR in meeting three SRRs con-straints described in (10)-(12), the amounts of D1t, D2t, andD3t (See (10)-(12)) for the base case (without implement-ing EDRP/DLC, scenario 1) and the linear model of groupone (after implementing EDRP/DLC, scenario 2) are givenin Fig. 3. As is clear from Fig. 3, all three constraints are



TABLE IVLOAD CURVE’S CHARACTERISTICS IN

DIFFERENT SCENARIOS

Fig. 2. Load curve before and after implementing EDRP/DLC.

Fig. 3. SRRs violation (MW).

bigger than zero. Also, after implementing EDRP/DLC theseamounts get bigger values at peak hours. In other words,after implementing EDRP/DLC the SRRs improves and con-sequently the network reliability is less jeopardized at the peakhours.

The error of the load curve estimation which is the differ-ence between the amount of linear and non-linear responsiveload models (dlin−dnon−lin) is given in Fig. 4. From Fig. 4, themost difference happens for scenario 14 (difference betweenlinear and power models). The negative values mean that theamount of the real demand is more than the supposed demand.

TABLE VTOTAL COST OBTAINED BY DIFFERENT METHODS

Fig. 4. The error of load curve estimation.

In other words, if the incorrect (not conservative and reliable)load model is used for the load estimation, system may notmeet the load demand and consequently the problem may notbe solved. In other words, in this state, the cost of the systemwill be enormous. So, not carefully choosing the load modelmay have significant impacts on the results. In other words themain goal of developing these non-linear models is to help ISOto select the most conservative load model. The procedure forchoosing the most conservative and reliable economic loadmodels is in such a way that the model which has the mostpredicted amount of demand would be selected as the mostreliable and conservative one. Here the most conservative andreliable load model is the power model. Because, it has themost difference with the linear model and consequently hasthe most predicted peak. So, if the power model is selectedfor the loads estimation, actually the most reliable and conser-vative one would be selected. In other words ISO should haschosen the power model for the load estimation instead of thelinear model.

A. Validation

As there is not a similar work to compare results, the totalcost for group one of customers (scenario 1-5) has been cal-culated by different optimization methods [15]–[18]. Table Vvalidates results and show that the used method, i.e., RDPSOhas better results than the other ones. It should be noted thatthis paper does not focus on introducing a new optimiza-tion algorithm, i.e., RDPSO [19], it just has been applied tosolve the combined problem (DEDDR). However, the solu-tion method of the combined problem (DEDDR) has beenpresented in Part IV. Moreover, [11] shows that intelligentimplementation of DRPs can effectively decreases supply sideand demand side costs.



VI. CONCLUSION

Demand response programs (DRPs) play a more importantrole in electricity price reduction and reliability improvement.In this paper dynamic economic dispatch (DED) problem wasintegrated to the EDRP/DLC to minimize the fuel cost anddetermine the optimal incentive simultaneously. In fact thecombined model (DEDDR) is a win-win situation for boththe customers and the generation companies. It is becauseof the fact that intelligent implementation of DRPs not onlydecreases electricity price in electricity markets but alsoincreases the customer’s benefit and network reliability. Byapplying the proposed model on the ten-unit system, someanalyses were carried out to investigate the impacts of someimportant factors such as the amount of elasticity and the loadmodel on the results. Improving SRRs is another importantbenefit of DRPs. Also, different non-linear responsive loadmodels were developed and a procedure for choosing the mostconservative and reliable load model was presented. Moreover,it was shown that not carefully choosing the correct loadmodel (the most conservative and reliable) may have signifi-cant impacts on the results. As the combined model (DEDDR)is a non-linear, complicated, non-convex, and non-smoothoptimization algorithm, it was solved by a meta-heuristic algo-rithm namely RDPSO algorithm and its solution method waspresented too. Results showed the effectiveness and practicalbenefits of the proposed model. In the future work, the time-based DRPs such as the TOU and RTP will be taken intoaccount in DED and a new method to determine the optimalprice in different periods (valley, off-peak, and peak periods)will be presented.

APPENDIX A

PROOF OF THE ECONOMIC NON-LINEAR

MODELS OF RESPONSIVE LOAD

The total revenue for the customers who participate in theDRPs is given in (A1) based on the hourly incentive rate.

INC(�d(t)) = inc(t) × [(�d(t)] (A1)

Where inc (t) and �d(t) are the amount of the incentive perMWh and reduced load.

The total penalty which is paid as the penalty by customersis given in (A2).

PEN(�d(t)) = pen(t) × {IC(t) − [�d(t)]} (A2)

Where pen(t) is the penalty at the tth hour and IC (t) isthe amount of the demand which the customer is responsiveto reduce or shift.

The net-profit of the customer is as (A3).

NP(t) = B(d(t)) − d(t)ρ(t) + INC(�d(t)) − PEN(�d(t))

(A3)

Where B is the profit which customers obtain by consum-ing power.

To maximize the customer benefit, the derivative of (A3)should be zero.

∂NP(t)

∂d(t)= ∂B(d(t))

∂d(t)− ρ(t) + ∂INC

∂d(t)− ∂PEN

∂d(t)= 0 (A4)

∂B(d(t))

∂d(t)= ρ(t) + inc(t) − pen(t) (A5)

Taylor series of B for the power, exponential, and logarith-mic benefit functions are as (A6)-(A8) respectively [25], [26].

B(d pow(t)

) ∼= B(d0(t))

+ ρ0(t)d pow(t)

1 + E(t, t)−1

[(d pow(t)

d0(t)

)E(t,t)−1

− 1

]

(A6)

B(dexp(t)

) ∼= B(d0(t)) + ρ0(t)dexp(t)

×[

1 + E(t, t)−1(

lndexp(t)

d0(t)− 1

)](A7)

B(

dlog(t)) ∼= B(d0(t))

+ ρ0(t)d0(t)E(t, t)e

(dlog(t)−d0(t)

d0(t)E(t,t) −1

)

(A8)

A.1. Power Modeling

For the power model by differentiating, (A6) changes to:

∂B(d pow(t))

∂d pow(t)= ρ0(t)

1 + E(t, t)−1

×[(

d pow(t)

d0(t)

)E(t,t)−1

− 1 + E(t, t)−1(

d pow(t)

d0(t)

)E(t,t)−1]

(A9)

Combining (A5) and (A9) yields:

ρ(t) + inc(t) − pen(t) = ρ0(t)

1 + E(t, t)−1

×[(

d pow(t)

d0(t)

)E(t,t)−1

− 1 + E(t, t)−1(

d pow(t)

d0(t)

)E(t,t)−1]

(A10)

Simplifying:

ρ(t) + inc(t) − pen(t)

ρ0(t)=(

d pow(t)

d0(t)

)E(t,t)−1

−(

1

1 + E(t, t)−1

)(A11)

The second term in (A11) is very small value and can beignored. From (A11) and for the single model of the load:

d pow(t) = d0(t) ×(


ρ0(t)

)E(t,t)

(A12)

For the multi-period model of the load:

d pow(t) = d0(t) ×∏24

t′ = 1t′ �= t

(ρ(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)E(t,t′)

(A13)

Finally the combined model including the single and multi-period models of the load is as (16).



TABLE VITEN-UNIT TEST SYSTEM CHARACTERISTICS

TABLE VIIBEST DISPATCH FOUND BY DEDDR FOR THE LINEAR MODEL IN GROUP ONE (SCENARIO 2, AFTER IMPLEMENTING EDRP/DLC)

A.2. Exponential Modeling

Differentiating from (A7) yields:

∂B(dexp (t))

∂dexp (t)= ρ0(t) ×

[1 + E(t, t)−1

(ln

dexp(t)

d0(t)− 1

)]

+ ρ0(t)dexp(t) ×

[E(t, t)−1 × 1

d0(t)× d0(t)

dexp(t)

]

(A14)

Combining (A14) and (A5) yields:

ρ(t) − ρ0(t) + inc(t) − pen(t)

= ρ0(t) ×[

1 + E(t, t)−1(

lndexp(t)

d0(t)− 1

)]+ ρ0(t)d

exp(t)

×[

E(t, t)−1 × 1

d0(t)× d0(t)

dexp(t)

](A15)

By simplifying (A15):

ρ(t) − ρ0(t) + inc(t) − pen(t)

ρ0(t)× E(t, t) =

(ln

dexp(t)

d0(t)

)

(A16)

For the single-period model of the load:

dexp(t) = d0(t)exp

((ρ(t) − ρ0(t) + inc(t) − pen(t)

ρ0(t)

)× E(t, t)

)

(A17)

And the multi-period model of the load:

dexp(t) = d0(t)

× exp

⎛

⎜⎜⎜⎝

24∑

t′ = 1t′ �= t

(ρ(t′)− ρ0

(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)

E(t, t′)

⎞

⎟⎟⎟⎠

(A18)

Finally the combined model including the single and multi-period models of the load is as (17).

A.3. Logarithmic Model

Differentiating from (A8) yields:

∂B(dlog (t)

)

∂dlog (t)= ρ0(t) × d0(t) × E(t, t)

×(

1

d0(t) × E(t, t)

)× e

(dlog(t)−d0(t)

d0(t)E(t,t)

)

(A19)

Combining (A5) and (A19), for the single-period model:


ρ0(t)= e

(dlog(t)−d0(t)

d0(t)E(t,t)

)

(A20)



Simplifying:

dlog(t) − d0(t)

d0(t) × E(t, t)= ln

(ρ(t) + inc(t) − pen(t)

ρ0(t)

)(A21)

So, the single model of the load is as (A22).

d(t) = d0(t) ×[1 + E(t, t) × ln

(ρ(t) + inc(t) − pen(t)

ρ0(t)

)](A22)

Multi-period:

dlog(t)=d0(t)

⎛

⎜⎜⎜⎝

1 +24∑

t′ = 1t′ �= t

(

lnρ(t′)+ inc

(t′)− pen

(t′)

ρ0(t′)

)

E(t, t′)

⎞

⎟⎟⎟⎠

(A23)

Finally the combined model is as (18).

APPENDIX B

THE CHARACTERISTICS OF THE TEST SYSTEM

See Table VI.

APPENDIX C

GENERATORS’ OPTIMAL SCHEDULING

See Table VII.

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Hamdi Abdi was born in Kermanshah, Iran, in1973. He received the B.Sc. degree from TabrizUniversity, Tabriz, Iran, in 1995, and the M.Sc. andPh.D. degrees from Tarbiat Modares University,Tehran, Iran, in 1999 and 2006, respectively, all inelectrical engineering. He is currently an AssistantProfessor with the Department of ElectricalEngineering, Razi University, Kermanshah.

His research interests include power systemoptimization, operation and planning, transmissionexpansion planning, renewable energies, smart grids,

energy efficiency, demand response, and design of electrical and controlsystems for industrial plant.

Ehsan Dehnavi was born in Kermanshah, Iran, in1991. He received the B.Sc. and M.Sc. degrees fromRazi University, Kermanshah, in 2013 and 2015,respectively, both in electrical engineering.

His research interests include smart grids, demandside management, demand response, power systemsoperation, planning and optimization, energy effi-ciency, distributed generations, and reconfiguration.

Farid Mohammadi was born in Kermanshah, Iran,in 1991. He received the B.Sc. and M.Sc. degreesfrom Razi University, Kermanshah, in 2013 and2015, respectively, both in electrical engineering.

His research interests include smart grids, powersystem operation and restructuring, demand sidemanagement, demand response, energy efficiency,distributed generations, optimization methods, andevolutionary algorithms.

http://www.FERC.gov

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