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.
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.
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 value or , 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. Section 2 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 of consumers as shown in Figure 2 and study its controllability to answer how total electric consumption changes by adjusting electric price.
The system corresponds to the group of consumers of the RTP system illustrated in Figure 1. For each consumer , the input is electric price and the output is electric consumption , where . Here, means that consumer does not use electricity at time . On the other hand, means that consumer uses electricity at time . Then the total electric consumption is denoted by
We consider that the output of consumer is a random variable which has the following probability distribution:where are constants and . Here, denotes the threshold on the electric price . As a result, the total electric consumption also becomes a random variable which has a probability distribution.
Throughout the paper, we assumefor any , where and are constants satisfying , , and . This assumption expresses the consumer buying behavior. Furthermore, we assume that the random variables are conditionally independent given ; that is,for any . 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 Figure 2, 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 output of each consumer only takes two values in . The simple setting can be considered as an approximation of the real society when we regard 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 Figure 2. As an index of controllability, we introduceforwhere and denote the vectors of probabilities.
The controllability index in (5) means the lowest occurrence probability in the total electric consumption. For example, Figure 3 illustrates (6) when andIn Figure 3, and thus
If controllability index (5) is sufficiently large, we can approximately coincide the total electric consumption with any value in by adjusting the electric price . 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 maximizing the controllability index .
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 consumers is larger, the optimization variables , , for the controllability problem increase. As a result, when is large, we cannot solve the controllability problem even if we use any numerical methods. In fact, as shown in Appendix A, we obtainforwhere By substituting (2) into in (11) and in (12), we obtainwhere 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 consumers , 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 definewhere and represent the expectation and the standard deviation of the total electric consumption when and and represent them of when . The derivation is shown in Appendix B.
The following lemma can derived by applying the central limit theorem as shown in Appendix H.
Lemma 3. For , converges in distribution to the standard Gaussian random variable.
The proof of Lemma 3 is given in Appendix C.
Furthermore, as shown in Appendix D, we obtainwhere and are defined by (11) and (12), respectively.
Hence if is sufficiently large, by Lemma 3, (21), and (22), we can consider thatforWe also write and as and , respectively. Therefore if is sufficiently large, the controllability index (5) can be approximated asforWe 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 Section 2.2 by using the approximated controllability index (26).
In order to study and maximizing the approximated controllability index (26), we first clarify the probabilities and such that variances of and are maximized under the constraints that means and are constants. Namely, we first solvewhere .
Lemma 4. Probabilities are the solution of the optimization problem (27) if and only if
The proof of Lemma 4 is given in Appendix E.
The following lemma shows that if the mean of is equal to that of and if is sufficiently large, the magnitude relationship of variances between and corresponds to that of values of and .
Lemma 5. LetIf , , and iffor , thenThe equality of (31) holds if and only if .
The proof of Lemma 5 is given in Appendix F.
By applying Lemmas 4 and 5, we can give and maximizing the approximated controllability index (26).
Theorem 6. If is sufficiently large, and are the solution ofif and only if
The proof of Theorem 6 is given in Appendix G.
If a pair of and is composed of the solution (33), the means of and are and , respectively. Furthermore, then Lemma 4 implies that the variances of and are maximized under and and are . As a result, the approximated controllability index in the case of (33) is larger than the index in other cases with and . For example, Figure 4 illustrates the cases of (33) andwhen the number of consumers is . As shown in Figure 4, the approximated controllability index in the case of (33) is larger than the index in the case of (34) because the variances of and in the case of (33) are larger than the variances of and in the case of (34). Moreover, Theorem 6 shows that if the means of and are not equal to and , the approximated controllability index is smaller than the index in the case of (33). For example, Figure 5 illustrates the cases of (33) andwhen the number of consumers is . As shown in Figure 5, the approximated controllability index in the case of (33) is larger than the index in the case of (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 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 ; 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 on the electric price . This is because the controllability index in (5) is invariant for any change of the threshold . In fact, as shown in Section 2.3, the controllability index is equal to in (10) and the functions and are defined by (11) and (12), respectively. It seems that (11) and (12) depend on the threshold . However, by substituting (2) into (11) and (12), and are given in (14) and (15), which are independent of the threshold , respectively. Hence the controllability index is invariant for any change of , and thus the approximate solution (33) does not depend on .
5. Validation of the Approximations (23) and (25)
In Section 3, 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 and on :at fixed . Figures 6 and 7 illustrate the relation of (36) and in the cases ofrespectively. On the other hand, Figures 8 and 9 illustrate the relations of (36) and in the cases offor , where denotes the round off number of . Figures 6, 7, 8, and 9 show that as the number of consumers is larger, (36) becomes smaller. We have the same conclusion for different . Hence we can observe that when is sufficiently large. Similarly, we can also observe that when is sufficiently large. Hence if the number of consumers is sufficiently large, we can obtain the relation (23).
From the above discussion, if is sufficiently large, that is, we have (25). As a result, if the number of consumers 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 probability when electric price is less than or equal to the threshold and if every consumer uses electricity at probability 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.
Appendices
A. Proof of (10)
The function in (5) can be expressed byThe equality (A.1) obeys (1) and (A.2) follows from the fact that each element of generates mutually exclusive events, and (A.3) obeys (4).
On the other hand, (2) implies that there exists such that and there exists such that Therefore (A.3) implies thatSimilarly,
Equations (A.6) and (A.7) imply thatHence (5) and (A.8) yield (10).
B. Proof of (17), (18), (19), and (20)
First, we prove (17). Equation (2) yields Hence by the definition (1) of ,
Next, we prove (18). The claim that there exists such that is equivalent to . Hence there exists such that Here, the last equality follows from (2). Therefore (4) implies that there exists such that
Similarly, we can prove (19) and (20).
C. Proof of Lemma 3
Let . Then (2) implies that Hence we obtain Therefore ifthenso that the Lyapunov condition is satisfied (see Appendix H). On the other hand, since we have assumed (3) and since is defined by (18), (C.3) is satisfied. Hence, since the Lyapunov condition implies the Lindeberg condition, by Proposition H.1 in Appendix H, converges in distribution to the standard Gaussian random variable.
Similarly, by the central limit theorem, converges in distribution to the standard Gaussian random variable.
D. Proof of (21) and (22)
For all , we have Here, the last equality is derived from (4). Since is any value in and is defined by (11), we obtain (21).
Similarly, we have (22).
E. Proof of Lemma 4
LetThen a necessary condition [18] for to be a solution of optimization problem (27) isEquation (E.2) implies . This and (E.3) yield . Thus we get .
Putting ,where denotes the identity matrix of size . Since and satisfy (E.2) and (E.3), the relation (E.4) is a sufficient condition [18] for (28) to be a solution of optimization problem (27).
F. Proof of Lemma 5
By direct calculation, Thus the relation (30) is satisfied; we have (31).
G. Proof of Theorem 6
We show that if is sufficiently large, for any , ,with equality if and only if (33) holds.
If is sufficiently large, for any satisfying ,In fact, if is sufficiently large, Lemmas 4 and 5 imply for any satisfying , , Furthermore, by a direct calculation, Hence if , Thus if is sufficiently large, for any satisfying , (G.2) holds. Similarly, if is sufficiently large, for any satisfying ,On the other hand, when , satisfy or , (3) yieldsTherefore if is sufficiently large, for any , satisfying or , (G.2), (G.6), and (G.7) yield Hence then we have (G.1).
Next, we show that if is sufficiently large, for any satisfying and , (G.1) also holds. By Lemmas 4 and 5, if is sufficiently large, for any satisfying ,Moreover,Thus if , Therefore (G.9) and (G.11) imply that if is sufficiently large, for any satisfying ,Similarly, we can show that if is sufficiently large, for any satisfying ,On the other hand, when satisfy and , (3) yieldsHence if is sufficiently large, for any satisfying and , (G.12), (G.13), and (G.14) yield Therefore then we have (G.1).
Finally, we show that the equality condition of (G.1) is to hold (33). Lemmas 4 and 5 imply that if and only if . Similarly, Lemmas 4 and 5 imply that if and only if . Therefore and are the solution of (32) if and only if (33) holds.
H. Central Limit Theorem
For the convenience of readers, the appendix summarizes the central limit theorem. We refer [16].
A sequence of random variables is said to converge in distribution to a random variable if for any at which is continuous, where and are the distribution functions of random variables and , respectively. The following proposition is known as the central limit theorem [16].
Proposition H.1. Let be a sequence of independent random variables with and , and let Ifis satisfied for any , converges in distribution to the standard Gaussian random variable, where denotes the indicator function.
Note that Proposition H.1 guarantees that if condition (H.3) called the Lindeberg condition is satisfied, standardized sums of independent random variables converge in distribution to the standard Gaussian random variable without the assumption that the random variables are identically distributed.
The following condition called the Lyapunov condition is a sufficient condition for the Lindeberg condition (H.3) to hold [16]. There exists such thatIn fact, since yields , for any , Hence if the Lyapunov condition (H.5) is satisfied, the Lindeberg condition (H.3) is also satisfied.
Notation
| : | The set of all real numbers |
| : | The symmetric group of elements |
| : | The probability of assuming |
| : | The expectation of assuming |
| : | The variance of assuming . |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by The Japan Science and Technology Agency (JST) Strategic Basic Research Programs (CREST) “Creation of Fundamental Theory and Technology to Establish a Cooperative Distributed Energy Management System and Integration of Technologies across Broad Disciplines toward Social Application.”