Controllability Analysis of an Aggregate Demand Response System

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.


Introduction
Real-time pricing (RTP) is a price based program on demand response [1][2][3][4][5][6][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][10][11][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 1 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. 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.

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 ∈ {1, 2, . . . , }, the input is electric price ( ) ∈ [0, ∞) and the output is electric consumption ( ) ∈ {0, 1}, where ∈ R. Here, ( ) = 0 means that consumer does not use electricity at time . On the other hand, ( ) = 1 means that consumer uses electricity at time . Then the total electric consumption is denoted by We consider that the output ( ) of consumer ∈ {1, 2, . . . , } is a random variable which has the following probability distribution: where , ∈ [0, 1] are constants and > 0. 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 assume for any ∈ {1, 2, . . . , }, where and are constants satisfying 0 < < 1, 0 < < 1, and < . This assumption expresses the consumer buying behavior. Furthermore, we assume that the random variables 1 ( ), 2 ( ), . . . , ( ) are conditionally independent given * ∈ [0, ∞); that is, for any 1 , 2 , . . . , ∈ {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 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. x 2 (t) x N (t) 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).

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 introduce where := ( 1 , 2 , . . . , ) ∈ [ , ] and := ( 1 , 2 , . . ., ) ∈ [ , ] 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 = 10 and In Figure 3, c(p, q, y * , u * ) c(p, q, y * , u * ) Total electric consumption y * C(p, q) If controllability index (5) is sufficiently large, we can approximately coincide the total electric consumption ( ) with any value in {0, 1, . . . , } 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.
By solving the above problem, we can obtain the optimal consumer behavior from the viewpoint of the controllability of the aggregate demand response.

Difficulty of the Controllability Problem. As the number of consumers
is larger, the optimization variables , , = 1, 2, . . . , 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 obtain By substituting (2) into ( ( ) = | 0 ≤ ( ) < ) in (11) and ( ( ) = | ( ) ≥ ) in (12), we obtaiñ wherẽ 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).

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 where 1 and 1 represent the expectation and the standard deviation of the total electric consumption ( ) when 0 ≤ ( ) < and 2 and 2 represent them of ( ) when ( ) ≥ . The derivation is shown in Appendix B.

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 1 ( * , ) and 2 ( * , ) are maximized under the constraints that means 1 ( * , ) and 2 ( * , ) are constants. Namely, we first solve max 1 , 2 ,..., where 0 ≤ ≤ 1.

Mathematical Problems in Engineering
The proof of Lemma 4 is given in Appendix E.
The following lemma shows that if the mean of 1 ( * , ) is equal to that of 2 ( * , ) and if is sufficiently large, the magnitude relationship of variances between 1 ( * , ) and 2 ( * , ) corresponds to that of values of 1 ( * , ) and 2 ( * , ).

Lemma 5. Let
The equality of (31) holds if and only if V 1 = V 2 .
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).
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.

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 of̃1( , * ) and 1 ( * , ) on * ∈ {0, 1, . . . , }: at fixed 1 , 2 , . . . , . Figures 6 and 7 illustrate the relation of (36) and in the cases of respectively. On the other hand, Figures 8 and 9 illustrate the relations of (36) and in the cases of for 1 := round( × 0.7), where round(⋅) 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 1 , 2 , . . . , . Hence we can observe that̃1( , * ) ≈ 1 ( * , ) when is sufficiently large. Similarly, we can also observe that

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 0.75 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.

E. Proof of Lemma 4
Let

F. Proof of Lemma 5
By direct calculation, Thus the relation (30) is satisfied; we have (31).