# Controllability Analysis of an Aggregate Demand Response System

Authors: Kazuhiro Sato; Shun-Ichi Azuma
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

In order to characterize the controllability of the aggregate demand response, that is, the aggregated consumer behaviors,
the paper introduces the concept of controllability index which expresses the lowest occurrence probability in total electric consumption which can be changed by electric price.
As the number of consumers is larger, it becomes difficult to directly analyze the controllability index. To resolve the difficulty of the large number of consumers, by using the central limit theorem,
the paper approximates the controllability index and gives the solution maximizing the approximated controllability index.


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## Body

## 1. Introduction

Real-time pricing (RTP) is a price based program on demand response [1–7] defined as the changes in electricity usage patterns of consumers in response to changes in electric price or to incentive payments [1]. By controlling electricity usage, consumers can produce the effectiveness in the same way that an electric power plant generates electricity. Thus it is important to implement efficacious RTP. From the viewpoint of control theory, RTP system can be regarded as a feedback system as shown in Figure 1. Here, output and reference signals are total electric consumption and electricity supply in a community, respectively, and an input signal is electric price. RTP researches can be categorized into(i)
the stability analysis of power grids under RTP [6, 8],(ii)
the design of electric price [9–12],(iii)
the controllability analysis of the aggregate consumers.Figure 1
Real-time pricing system.To clarify electric usage patterns of consumers when we can reduce the peak of total electric consumption by RTP, this paper studies (iii) which is a new problem on RTP. To this end, this paper considers that(i)
every consumer probabilistically has a value1 or 0, which means that a consumer uses electricity or not,(ii)
probabilities of electric usages of consumers depend on electric price.As a result, the total electric consumption which expresses the aggregate demand response also probabilistically varies corresponding to electric price.In order to discuss the controllability of the aggregate demand response, we introduce the concept of controllability index which expresses “the lowest occurrence probability in total electric consumption which can be changed by electric price.” When the controllability index is large, we can largely change the aggregate demand response by adjusting electric price.Unfortunately, as the number of consumers is larger, the controllability index becomes complicated, and thus it is difficult to study the probabilities maximizing the controllability index. To resolve the problem, by using the central limit theorem, this paper approximates the controllability index and gives the probabilities maximizing the approximated controllability index.Finally, we note that the recent works of [13, 14] are strongly related to this study. Reference [13] has studied good consumers for performing RTP by focusing on the aggregate demand response through some simulations. Reference [14] has given the design principle of randomized automated demand response machines in the viewpoint of control theory. However, [13, 14] have not given the performance index of the controllability of the aggregate demand response and have not mathematically studied the controllability.The paper is organized as follows. Section2 defines the aggregate demand response system and the controllability index. Moreover, the controllability problem is presented. Section 3 approximates the controllability index by applying the central limit theorem. Section 4 gives an approximate solution of the controllability problem. Section 5 gives a validation of the approximations used in Section 3. The conclusion is presented in Section 6.The notation used throughout the paper is stated in Notation section.


## 2. Controllability of Aggregate Demand Response

### 2.1. Aggregate Demand Response System

We consider the aggregate demand response system composed ofN consumers as shown in Figure 2 and study its controllability to answer how total electric consumption changes by adjusting electric price.Figure 2
Aggregate demand response system.The system corresponds to the group of consumers of the RTP system illustrated in Figure1. For each consumer i∈{1,2,…,N}, the input is electric price u(t)∈[0,∞) and the output is electric consumption xi(t)∈{0,1}, where t∈R. Here, xi(t)=0 means that consumer i does not use electricity at time t. On the other hand, xi(t)=1 means that consumer i uses electricity at time t. Then the total electric consumption is denoted by(1)yt≔x1t+x2t+⋯+xNt.We consider that the outputxi(t) of consumer i∈{1,2,…,N} is a random variable which has the following probability distribution:(2)Pxit=1∣0≤ut<γ=pi,Pxit=0∣0≤ut<γ=1-pi,Pxit=1∣ut≥γ=qi,Pxit=0∣ut≥γ=1-qi,where pi,qi∈[0,1] are constants and γ>0. Here, γ denotes the threshold on the electric price u(t). As a result, the total electric consumption y(t) also becomes a random variable which has a probability distribution.Throughout the paper, we assume(3)q_≤qi<pi≤p¯for any i∈{1,2,…,N}, where q_ and p¯ are constants satisfying 0<q_<1, 0<p¯<1, and q_<p¯. This assumption expresses the consumer buying behavior. Furthermore, we assume that the random variables x1(t),x2(t),…,xN(t) are conditionally independent given u∗∈[0,∞); that is,(4)Px1t=ξ1,x2t=ξ2,…,xNt=ξN∣ut=u∗=∏i=1NPxit=ξi∣ut=u∗for any ξ1,ξ2,…,ξN∈{0,1}. This means that each electric usage pattern of consumers is not influenced from another consumer. The assumption is valid for the following reasons:(i)
For the aggregate demand response system illustrated in Figure2, if consumers can communicate with each other, the independence is not satisfied. However, under the situation in which consumers cannot communicate with each other, the assumption is valid.(ii)
In the real society, the independence may not be satisfied due to the environmental factors such as the temperature and the weather. However, under a constant environment, the assumption is valid.Remark 1.
We consider that the outputxi of each consumer only takes two values in {0,1}. The simple setting can be considered as an approximation of the real society when we regard 1 as the average electric consumption of electric devices.Remark 2.
In this paper, we assume that electric usage of each consumer is modeled by (2). The model (2) can express the behavior of the randomized automated demand response (ADR) machine proposed in [14]. If such ADR machines prevail in households, it is meaningful to consider the model (2).


### 2.2. Controllability Index

In order to study the optimal consumer behavior, this paper studies the controllability of the aggregate demand response system as shown in Figure2. As an index of controllability, we introduce(5)Cp,q≔miny∗∈0,1,…,N⁡maxu∗∈0,∞⁡cp,q,y∗,u∗for(6)cp,q,y∗,u∗≔Pyt=y∗∣ut=u∗,where p≔(p1,p2,…,pN)∈[q_,p¯]N and q≔(q1,q2,…,qN)∈[q_,p¯]N denote the vectors of probabilities.The controllability indexC(p,q) in (5) means the lowest occurrence probability in the total electric consumption. For example, Figure 3 illustrates (6) when N=10 and(7)p1=p2=⋯=p10=0.8,q1=q2=⋯=q10=0.15.In Figure 3, (8)maxu∗∈0,∞⁡cp,q,y∗,u∗=cp,q,y∗,u∗∀y∗∈5,6,…,10if0≤u∗<γ,cp,q,y∗,u∗∀y∗∈0,1,…,4ifu∗≥γ=cp,q,y∗,0ify∗∈5,6,…,10,cp,q,y∗,γify∗∈0,1,…,4and thus (9)Cp,q=min⁡cp,q,5,0,cp,q,4,γ=cp,q,5,0≈0.0264.Figure 3
Illustration of (6) under (7).If controllability index (5) is sufficiently large, we can approximately coincide the total electric consumption y(t) with any value in {0,1,…,N} by adjusting the electric price u(t). Hence if controllability index (5) is sufficiently large, we can decide to perform RTP for a community such as an apartment composed of many consumers. Furthermore, we can use the controllability index (5) as the design index of ADR machines [14, 15] for performing RTP. Therefore we want to consider the following problem.Controllability Problem. Find p,q maximizing the controllability index C(p,q).By solving the above problem, we can obtain the optimal consumer behavior from the viewpoint of the controllability of the aggregate demand response.


### 2.3. Difficulty of the Controllability Problem

As the number of consumersN is larger, the optimization variables pi, qi, i=1,2,…,N for the controllability problem increase. As a result, when N is large, we cannot solve the controllability problem even if we use any numerical methods. In fact, as shown in Appendix A, we obtain(10)Cp,q=miny∗∈0,1,…,N⁡max⁡c~1p,y∗,c~2q,y∗for(11)c~1p,y∗≔∑ξ1,ξ2,…,ξN∈Sy∗∏i=1NPxit=ξi∣0≤ut<γ,(12)c~2q,y∗≔∑ξ1,ξ2,…,ξN∈Sy∗∏i=1NPxit=ξi∣ut≥γ,where (13)Sy∗≔ξ1,ξ2,…,ξN∈0,1N∣∑i=1Nξi=y∗.By substituting (2) into P(xi(t)=ξi∣0≤u(t)<γ) in (11) and P(xi(t)=ξi∣u(t)≥γ) in (12), we obtain(14)c~1p,y∗=∑i1,i2,…,iN∈S~y∗pi1pi2⋯piy∗1-piy∗+1×1-piy∗+2⋯1-piN,(15)c~2q,y∗=∑i1,i2,…,iN∈S~y∗qi1qi2⋯qiy∗1-qiy∗+1×1-qiy∗+2⋯1-qiN,where (16)S~y∗≔i1,i2,…,iN∈1,2,…,NN∣12⋯Ni1i2⋯iN∈Sym⁡N,i1<i2<⋯<iy∗,iy∗+1<iy∗+2<⋯<iN.Hence, the controllability index (5) is calculated by solving a minimax problem for the higher-order polynomial defined by (14) and (15).To resolve the difficulty on the number of consumersN, in the next section, we approximate the controllability index (5).


## 2.1. Aggregate Demand Response System

We consider the aggregate demand response system composed ofN consumers as shown in Figure 2 and study its controllability to answer how total electric consumption changes by adjusting electric price.Figure 2
Aggregate demand response system.The system corresponds to the group of consumers of the RTP system illustrated in Figure1. For each consumer i∈{1,2,…,N}, the input is electric price u(t)∈[0,∞) and the output is electric consumption xi(t)∈{0,1}, where t∈R. Here, xi(t)=0 means that consumer i does not use electricity at time t. On the other hand, xi(t)=1 means that consumer i uses electricity at time t. Then the total electric consumption is denoted by(1)yt≔x1t+x2t+⋯+xNt.We consider that the outputxi(t) of consumer i∈{1,2,…,N} is a random variable which has the following probability distribution:(2)Pxit=1∣0≤ut<γ=pi,Pxit=0∣0≤ut<γ=1-pi,Pxit=1∣ut≥γ=qi,Pxit=0∣ut≥γ=1-qi,where pi,qi∈[0,1] are constants and γ>0. Here, γ denotes the threshold on the electric price u(t). As a result, the total electric consumption y(t) also becomes a random variable which has a probability distribution.Throughout the paper, we assume(3)q_≤qi<pi≤p¯for any i∈{1,2,…,N}, where q_ and p¯ are constants satisfying 0<q_<1, 0<p¯<1, and q_<p¯. This assumption expresses the consumer buying behavior. Furthermore, we assume that the random variables x1(t),x2(t),…,xN(t) are conditionally independent given u∗∈[0,∞); that is,(4)Px1t=ξ1,x2t=ξ2,…,xNt=ξN∣ut=u∗=∏i=1NPxit=ξi∣ut=u∗for any ξ1,ξ2,…,ξN∈{0,1}. This means that each electric usage pattern of consumers is not influenced from another consumer. The assumption is valid for the following reasons:(i)
For the aggregate demand response system illustrated in Figure2, if consumers can communicate with each other, the independence is not satisfied. However, under the situation in which consumers cannot communicate with each other, the assumption is valid.(ii)
In the real society, the independence may not be satisfied due to the environmental factors such as the temperature and the weather. However, under a constant environment, the assumption is valid.Remark 1.
We consider that the outputxi of each consumer only takes two values in {0,1}. The simple setting can be considered as an approximation of the real society when we regard 1 as the average electric consumption of electric devices.Remark 2.
In this paper, we assume that electric usage of each consumer is modeled by (2). The model (2) can express the behavior of the randomized automated demand response (ADR) machine proposed in [14]. If such ADR machines prevail in households, it is meaningful to consider the model (2).


## 2.2. Controllability Index

In order to study the optimal consumer behavior, this paper studies the controllability of the aggregate demand response system as shown in Figure2. As an index of controllability, we introduce(5)Cp,q≔miny∗∈0,1,…,N⁡maxu∗∈0,∞⁡cp,q,y∗,u∗for(6)cp,q,y∗,u∗≔Pyt=y∗∣ut=u∗,where p≔(p1,p2,…,pN)∈[q_,p¯]N and q≔(q1,q2,…,qN)∈[q_,p¯]N denote the vectors of probabilities.The controllability indexC(p,q) in (5) means the lowest occurrence probability in the total electric consumption. For example, Figure 3 illustrates (6) when N=10 and(7)p1=p2=⋯=p10=0.8,q1=q2=⋯=q10=0.15.In Figure 3, (8)maxu∗∈0,∞⁡cp,q,y∗,u∗=cp,q,y∗,u∗∀y∗∈5,6,…,10if0≤u∗<γ,cp,q,y∗,u∗∀y∗∈0,1,…,4ifu∗≥γ=cp,q,y∗,0ify∗∈5,6,…,10,cp,q,y∗,γify∗∈0,1,…,4and thus (9)Cp,q=min⁡cp,q,5,0,cp,q,4,γ=cp,q,5,0≈0.0264.Figure 3
Illustration of (6) under (7).If controllability index (5) is sufficiently large, we can approximately coincide the total electric consumption y(t) with any value in {0,1,…,N} by adjusting the electric price u(t). Hence if controllability index (5) is sufficiently large, we can decide to perform RTP for a community such as an apartment composed of many consumers. Furthermore, we can use the controllability index (5) as the design index of ADR machines [14, 15] for performing RTP. Therefore we want to consider the following problem.Controllability Problem. Find p,q maximizing the controllability index C(p,q).By solving the above problem, we can obtain the optimal consumer behavior from the viewpoint of the controllability of the aggregate demand response.


## 2.3. Difficulty of the Controllability Problem

As the number of consumersN is larger, the optimization variables pi, qi, i=1,2,…,N for the controllability problem increase. As a result, when N is large, we cannot solve the controllability problem even if we use any numerical methods. In fact, as shown in Appendix A, we obtain(10)Cp,q=miny∗∈0,1,…,N⁡max⁡c~1p,y∗,c~2q,y∗for(11)c~1p,y∗≔∑ξ1,ξ2,…,ξN∈Sy∗∏i=1NPxit=ξi∣0≤ut<γ,(12)c~2q,y∗≔∑ξ1,ξ2,…,ξN∈Sy∗∏i=1NPxit=ξi∣ut≥γ,where (13)Sy∗≔ξ1,ξ2,…,ξN∈0,1N∣∑i=1Nξi=y∗.By substituting (2) into P(xi(t)=ξi∣0≤u(t)<γ) in (11) and P(xi(t)=ξi∣u(t)≥γ) in (12), we obtain(14)c~1p,y∗=∑i1,i2,…,iN∈S~y∗pi1pi2⋯piy∗1-piy∗+1×1-piy∗+2⋯1-piN,(15)c~2q,y∗=∑i1,i2,…,iN∈S~y∗qi1qi2⋯qiy∗1-qiy∗+1×1-qiy∗+2⋯1-qiN,where (16)S~y∗≔i1,i2,…,iN∈1,2,…,NN∣12⋯Ni1i2⋯iN∈Sym⁡N,i1<i2<⋯<iy∗,iy∗+1<iy∗+2<⋯<iN.Hence, the controllability index (5) is calculated by solving a minimax problem for the higher-order polynomial defined by (14) and (15).To resolve the difficulty on the number of consumersN, in the next section, we approximate the controllability index (5).


## 3. Approximation of the Controllability Index

This section approximates controllability index (5) by the central limit theorem [16, 17] (see Appendix H).To this end, we define(17)μ1≔Eyt∣0≤ut<γ=∑i=1Npi,(18)σ1≔Vyt∣0≤ut<γ=∑i=1Npi-pi2,(19)μ2≔Eyt∣ut≥γ=∑i=1Nqi,(20)σ2≔Vyt∣ut≥γ=∑i=1Nqi-qi2,where μ1 and σ1 represent the expectation and the standard deviation of the total electric consumption y(t) when 0≤u(t)<γ and μ2 and σ2 represent them of y(t) when u(t)≥γ. The derivation is shown in Appendix B.The following lemma can derived by applying the central limit theorem as shown in AppendixH.Lemma 3.
Fori∈{1,2}, y(t)-μi/σi converges in distribution to the standard Gaussian random variable.The proof of Lemma3 is given in Appendix C.Furthermore, as shown in AppendixD, we obtain(21)c~1p,y∗=Pyt=y∗∣0≤ut<γ,(22)c~2q,y∗=Pyt=y∗∣ut≥γ,where c~1(p,y∗) and c~2(q,y∗) are defined by (11) and (12), respectively.Hence ifN is sufficiently large, by Lemma 3, (21), and (22), we can consider that(23)c~1p,y∗≈f1y∗,p,c~2q,y∗≈f2y∗,qfor(24)f1y∗,p≔12πσ1exp⁡-y∗-μ122σ12,f2y∗,q≔12πσ2exp⁡-y∗-μ222σ22.We also write f1(y∗,p) and f2(y∗,q) as f1(y∗,p1,p2,…,pN) and f2(y∗,q1,q2,…,qN), respectively. Therefore if N is sufficiently large, the controllability index (5) can be approximated as(25)Cp,q≈C~p,qfor(26)C~p,q≔miny∗∈0,1,…,N⁡max⁡f1y∗,p,f2y∗,q.We call (26) the approximated controllability index. In Section 5, we give further validation of the approximations (23) and (25) through simulations.


## 4. Controllability Analysis

This section gives an approximate solution of the controllability problem as mentioned in Section2.2 by using the approximated controllability index (26).In order to studyp and q maximizing the approximated controllability index (26), we first clarify the probabilities p and q such that variances of f1(y∗,p) and f2(y∗,q) are maximized under the constraints that means f1(y∗,p) and f2(y∗,q) are constants. Namely, we first solve(27)max⁡p1,p2,…,pN∑i=1Npi-pi2s.t.∑i=1Npi=αN,where 0≤α≤1.Lemma 4.
Probabilitiesp1,p2,…,pN are the solution of the optimization problem (27) if and only if(28)pi=αi=1,2,…,N.The proof of Lemma4 is given in Appendix E.The following lemma shows that if the mean off1(y∗,p) is equal to that of f2(y∗,q) and if N is sufficiently large, the magnitude relationship of variances between f1(y∗,p) and f2(y∗,q) corresponds to that of values of f1(y∗,p) and f2(y∗,q).Lemma 5.
Let(29)giy≔12πviexp⁡-y-mi22vi2,i=1,2.If m1=m2=kN(0<k<1), v1≥v2, and if(30)N>1β-k2v12v22v12-v22ln⁡v12v22for 0≤β≤1(β≠k), then(31)g1βN≥g2βN.The equality of (31) holds if and only if v1=v2.The proof of Lemma5 is given in Appendix F.By applying Lemmas4 and 5, we can give p and q maximizing the approximated controllability index (26).Theorem 6.
IfN is sufficiently large, p∈(0,1)N and q∈(0,1)N are the solution of(32)maxp,q⁡C~p,qif and only if(33)p1=p2=⋯=pN=34,q1=q2=⋯=qN=14.The proof of Theorem6 is given in Appendix G.If a pair ofp and q is composed of the solution (33), the means of f1(y∗,p) and f2(y∗,q) are μ1=3/4N and μ2=1/4N, respectively. Furthermore, then Lemma 4 implies that the variances of f1(y∗,p) and f2(y∗,q) are maximized under μ1=3/4N and μ2=1/4N and are σ12=σ22=3/16N. As a result, the approximated controllability index C~(p,q) in the case of (33) is larger than the index C~(p^,q^) in other cases with μ1=3/4N and μ2=1/4N. For example, Figure 4 illustrates the cases of (33) and(34)p^1=p^2=⋯=p^500=910,p^501=p^502=⋯=p^1000=610,q^1=q^2=⋯=q^500=110,q^501=q^502=⋯=q^1000=410when the number of consumers N is 1000. As shown in Figure 4, the approximated controllability index C~(p,q) in the case of (33) is larger than the index C~(p^,q^) in the case of (34) because the variances σ12=σ22=3/16N of f1(y∗,p) and f2(y∗,q) in the case of (33) are larger than the variances σ12=σ22=33/200N of f1(y∗,p^) and f2(y∗,q^) in the case of (34). Moreover, Theorem 6 shows that if the means of f1(y∗,p) and f2(y∗,q) are not equal to μ1=3/4N and μ2=1/4N, the approximated controllability index C~(p,q) is smaller than the index in the case of (33). For example, Figure 5 illustrates the cases of (33) and(35)p^1=p^2=⋯=p^1000=45,q^1=q^2=⋯=q^1000=13when the number of consumers N is 1000. As shown in Figure 5, the approximated controllability index C~(p,q) in the case of (33) is larger than the index C~(p^,q^) in the case of (35).Figure 4
Comparison with (33) and (34).Figure 5
Comparison with (33) and (35).From (25) and Theorem 6, we can give an approximate solution of the controllability problem as mentioned in Section 2.2.Approximate Solution of the Controllability Problem. When the number of consumers N is large, an approximate solution of the controllability problem is (33).We note two points on the approximate solution (33):(i)
The approximate solution (33) does not depend on the number of consumers N; that is, we can easily implement this result into the ADR machine for performing RTP.(ii)
The approximate solution (33) does not depend on the threshold γ on the electric price u(t). This is because the controllability index C(p,q) in (5) is invariant for any change of the threshold γ. In fact, as shown in Section 2.3, the controllability index C(p,q) is equal to miny∗∈{0,1,…,N}max(c~1(p,y∗),c~2(q,y∗)) in (10) and the functions c~1(p,y∗) and c~2(q,y∗) are defined by (11) and (12), respectively. It seems that (11) and (12) depend on the threshold γ. However, by substituting (2) into (11) and (12), c~1(p,y∗) and c~2(q,y∗) are given in (14) and (15), which are independent of the threshold γ, respectively. Hence the controllability index C(p,q) is invariant for any change of γ, and thus the approximate solution (33) does not depend on γ.


## 5. Validation of the Approximations (23) and (25)

In Section3, the controllability index (5) has been approximated into the index (26). This section demonstrates a validation of the approximation based on simulations. To this end, first, we examine a validation of the approximation (23). Consider the maximum error of c~1(p,y∗) and f1(y∗,p) on y∗∈{0,1,…,N}:(36)maxy∗∈0,1,…,N⁡c~1p,y∗-f1y∗,pat fixed p1,p2,…,pN. Figures 6 and 7 illustrate the relation of (36) and N in the cases of(37)p1=p2=⋯=pN=56,(38)p1=p2=⋯=pN=34,respectively. On the other hand, Figures 8 and 9 illustrate the relations of (36) and N in the cases of(39)p1=p2=⋯=pN1=45,pN1+1=pN1+2=⋯=pN=23,(40)p1=p2=⋯=pN1=12,pN1+1=pN1+2=⋯=pN=13for N1≔round(N×0.7), where round(·) denotes the round off number of (·). Figures 6, 7, 8, and 9 show that as the number of consumers N is larger, (36) becomes smaller. We have the same conclusion for different p1,p2,…,pN. Hence we can observe that c~1(p,y∗)≈f1(y∗,p) when N is sufficiently large. Similarly, we can also observe that c~2(q,y∗)≈f2(y∗,q) when N is sufficiently large. Hence if the number of consumers N is sufficiently large, we can obtain the relation (23).Figure 6
The relation of (36) and N in the case of (37).Figure 7
The relation of (36) and N in the case of (38).Figure 8
The relation of (36) and N in the case of (39).Figure 9
The relation of (36) and N in the case of (40).From the above discussion, ifN is sufficiently large, (41)Cp,q≈miny∗∈0,1,…,N⁡max⁡f1y∗,p,f2y∗,q=C~p,q;that is, we have (25). As a result, if the number of consumers N is sufficiently large, the approximate solution (33) of the controllability problem is valid.


## 6. Conclusion

We have introduced the controllability index of the aggregate demand response system. By applying the central limit theorem, we have shown that if every consumer uses electricity at probability0.75 when electric price is less than or equal to the threshold and if every consumer uses electricity at probability 0.25 when electric price is greater than the threshold, the controllability index is approximately maximized. The optimal consumer behavior can be implemented to the automated demand response machine proposed in [14] for performing RTP. Currently, we have studied the case with several thresholds on the electric price.



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