Verification of Real-Time Pricing Systems Based on Probabilistic Boolean Networks

In this paper, verification of real-time pricing systems of electricity is considered using a probabilistic Boolean network (PBN). In real-time pricing systems, electricity conservation is achieved by manipulating the electricity price at each time. A PBN is widely used as a model of complex systems, and is appropriate as a model of realtime pricing systems. Using the PBN-based model, real-time pricing systems can be quantitatively analyzed. In this paper, we propose a verification method of real-time pricing systems using the PBN-based model and the probabilistic model checker PRISM. First, the PBN-based model is derived. Next, the reachability problem, which is one of the typical verification problems, is formulated, and a solution method is derived. Finally, the effectiveness of the proposed method is presented by a numerical example.


Introduction
In recent years, there has been growing interest in energy and the environment.For problems on energy and the environment such as energy saving, several approaches have been studied (see, e.g., [1] [2]).In this paper, we focus on real-time pricing systems of electricity.A real-time pricing system of electricity is a system that charges different electricity prices for different hours of the day and for different days, and is effective for reducing the peak and flattening the load curve (see, e.g., [3]- [6]).In general, a real-time pricing system consists of one controller deciding the price at each time and multiple electric consumers such as commercial facilities and homes.If electricity conservation is needed, then the price is set to a high value.Since the economic load be-comes high, consumers conserve electricity.Thus, electricity conservation is achieved.
In the existing methods, the price at each time is given by a simple function with respect to power consumptions and voltage deviations and so on (see, e.g., [6]).In order to realize more precisely pricing, it is necessary to use a mathematical model of consumers.
On the other hand, in order to deal with complex systems such as power systems and gene regulatory networks, it is one of the appropriate methods to approximate a complex system by a discrete abstract model (see, e.g., [7]).In addition, human decision making is also complex, and is modeled by a discrete model (see, e.g., [8]).Thus, in analysis and control of complex systems and those with human decision making, a discrete model plays an important role.Several discrete models have been proposed so far (see, e.g., [9]).In this paper, we focus on a Boolean network (BN) [10].In a BN, the state is given by a binary value (0 or 1), and the dynamics are expressed by a set of Boolean functions.Since Boolean functions are used, it is easy to understand the interaction between states.In addition, the behavior of complex systems is frequently stochastic by the effects of noise.From this viewpoint, a probabilistic BN (PBN) has been proposed in [11].In a PBN, a Boolean function is randomly decided at each time among the candidates of Boolean functions.
Under the above backgrounds, the authors have proposed in [12] the PBN-based model of real-time pricing systems.In this model, decision making of electric consumers is modeled by a PBN.That is, decisions of a consumer are modeled by Boolean functions, and one of decisions is selected probabilistically.Selection probabilities are controlled by the price at each time.In [12], an approximate algorithm for solving the optimal control problem has been proposed.However, analysis and verification using the PBN-based model have not been considered.
In this paper, we propose a verification method of real-time pricing systems using the PBN-based model and the probabilistic model checker PRISM [13].Using PRISM, we can verify whether this system satisfies the specification described by probabilistic computation tree logic (PCTL) [14] or not.The reachability problem is considered as one of the typical verification problems, and a numerical example is presented.The proposed method provides us a basic of model-based design of real-time pricing systems.
In Section 2, the outline of real-time pricing systems studied in this paper is explained.In Section 3, the PBN-based model is explained.In Section 4, the verification problem is formulated.In Section 5, a solution method using PRISM is proposed.In Section 6, a numerical example is presented.In Section 7, we conclude this paper.

Real-Time Pricing Systems
In this section, we explain the outline of real-time pricing systems studied in this paper.
Figure 1 shows an illustration of real-time pricing systems studied in this paper.This system consists of one controller and multiple electric consumers such as commercial facilities and homes.For an electric consumer, we suppose that each consumer can monitor the status of electricity conservation of other consumers.In other words, the status of some consumer affects that of other consumers.For example, in commercial facilities, we suppose that the status of rival commercial facilities can be checked by lighting, Blog, Twitter, and so on.Depending on power consumption, i.e., the status of electricity conservation, the controller determines the price at each time.If electricity conservation is needed, then the price is set to a high value.Since the economic load becomes high, consumers conserve electricity.Thus, electricity conservation is achieved.
The price does not depend on each consumer, and is uniquely determined.
In this paper, decision making of electric consumers is modeled by a probabilistic Boolean network (PBN).Here, we suppose that each electric consumer has candidates of a decision in electricity conservation, and one of candidates is selected probabilistically depending on the electricity price at the current time.In such a case, it is appropriate to adopt the PBN-based model.In this paper, the property of real-time pricing systems can be verified using the PBN-based model.

Modeling Using Probabilistic Boolean Networks
In this section, first, we explain the outline of PBNs.Next, each consumer in real-time pricing systems is modeled by a PBN.

Probabilistic Boolean Networks
First, we explain a (deterministic) Boolean network (BN).A BN is defined by where is the state, and 0,1, 2, k =  is the discrete time.The set is a given index set, and the function given Boolean function consisting of logical operators such as AND ( ∧ ), OR ( ∨ ), and NOT ( ¬ ).If Next, we explain a probabilistic Boolean network (PBN) (see [11] for further details).
In a PBN, the candidates of ( ) i f are given, and for each i x , selecting one Boolean function is probabilistically independent at each time.Let Then, the following relation must be satisfied.Probabilistic distributions are derived from experimental results.Fi- .
We show a simple example.
Example 1.Consider the PBN in which Boolean functions and probabilities are given by where ( ) , and hold, and we see that the relation ( 2) is satisfied.Next, consider the state trajectory.Then, for ( ) [ ] , we can obtain In this example, the cardinality of the finite state set { } 0 0 0 , 0 0 1 , 0 1 0 , 1 1 0

Model of Consumers
Consider modeling the set of consumers as a PBN.The number of consumers is given by n.We assume that the state of consumer is binary, and is denoted by i x .The state implies 0 consumer conserves electricity, 1 consumer normally uses electricity.
The binary value of i x is determined by power consumption of consumer i.In addition, we assume that the probability ( ) i l c is time-varying and is changed by the price at each time.That is, the probability is given by where ( ) is the price (the control input).We assume that the set  is a finite set, and for any u ∈  , two conditions (2) and ( ) ( ) must be derived depending on real situations and experimental results.In this paper, as one of examples, we consider the following situation, which will mimic a real situation.
 denote the set of consumers, which affect to con- sumer i.We assume that there exists one leader in the local area.The state of a leader is given by 1 x .Then, for consumer i, we consider the following model i Σ : ( ) , , The Boolean functions ( )

Problem Formulation
In this section, the verification problem described by probabilistic computation tree logic (PCTL) is formulated for the PBN-based model of consumers (see Appendix A for details on PCTL).
Here, the reachability problem is formulated as one of the typical problems.For the  given by ( 3), the output We remark that the output is not the measured signal.First, the reachability problem is given.
Problem 1. Suppose that for the system i Σ ,  given by ( 3), the initial state ( ) 0 0 x x = , the control time N, and the target output f y are given.Then, find a maximum probability p satisfying ( ) ( ) Let max P denote the maximum probability obtained by solving this problem.In this problem, we find a maximum probability that ( ) f y k y = holds within time N.In the conventional reachability problem, only terminal time is focused, and it is checked whether ( ) f y N y = holds or not.In this paper, we focus on not only terminal time N but also other times 0,1, , 1 N −  .Since the system has the control input, we find a maximum probability satisfying the condition.In the case where peak demand is focused on, ( ) f y k y = may be replaced with ( ) ≤ , where γ is a given constant.Furthermore, by solving Problem 1 at each time, a kind of model predictive control (MPC) can be realized (see Section 5.3 for further details).

Solution Method Using PRISM
In this section, we consider a solution method for Problem 1 using the probabilistic model checker PRISM [13].

Preparation: Transformation of Boolean Functions
As a preparation, the following lemma [15] is introduced.Lemma 1.Consider two binary variables 1 2 , δ δ .Then the following relations hold. i) . By using this lemma, a Boolean function can be transformed into a polynomial with binary variables.

Description in PRISM
To solve Problem 1 and the verification problem described by PCTL formulas, the probabilistic model checker PRISM is used.PRISM supports a discrete-time Markov chain (DT-MC), a continuous-time Markov chain (CT-MC), and a Markov decision process (MDP).PRISM consists of three "Model", "Properties", "Simulator".In the "Model" part, a given probabilistic system is described using the PRISM language.
In the "Properties" part, the property specification language incorporates temporal logic such as PCTL, and we can verify whether a given PCTL formula holds or not.In the "Simulator", the state trajectories can be simulated.
Using PRISM, consider modeling the system i Σ , 1, 2, , i n =  given by (3).To ex- plain the PRISM-based method, consider the following model of three consumers: ( ) ( ) ( ) ( ) In addition,  is given by { } dynamics for 1 x (consumer1) are modeled.In line 3, it is described that 1 x takes a binary value, and the initial value of 1 x is given by ( ) holds, then the value of 1 x at the next time is given by 1 with the probability 0.1, 0 with the probability 0.075, 1 x (i.e., the state is not changed) with the probability 0.6, x x (corresponding to ( ) ( ) x k x k ∧ 1 ) with the probability 0.15, and 1 x with the probability 0.15.Similarly, in line 5, the case of ( ) 4 u k = is described.In line 6, the case of ( ) 5 u k = is described.In lines 8-13, the dynamics for 2 x (consumer 2) are modeled.In lines 14-19, the dynamics for 3 x (consumer 3) are modeled.In this sys- tem, a discrete probabilistic distribution is given for each i x .Hence, in PRISM, the dynamics for each i x must be modeled separately.In lines 20-31, the property of the control input is described as a nondeterministic variable.We note here that the initial value of the control input must be given (see line 21).Finally, to associate with each module, [RTP] is described in lines 4-6, 10-12, 16-18, 22-30.
From the above example, we see that the system i Σ ,  given by (3) can be described by PRISM.Finally, we present a procedure for deriving the PRISM source code as follows.In the following procedure, without loss of generality, the input set  is given by Derivation Procedure of PRISM Source Code: Step 1: Transform each Boolean function into a polynomial with binary variables by using Lemma 1.Let ( ) ˆi l f denote the obtained polynomial.
Step 2: Describe that a given system is an MDP.
Step 3: Compute the probability ( ) i l c for each element of  .Let ( ) , i l p c denote the probability for p ∈  .
Step 4: Describe module RTP i, 1, 2, , i n =  as follows.module RTP i; : : Step 5: Describe the control input u as follows.
module input u: [RTP] ( ) ( ) ( ) The above procedure is the improved version of the procedure proposed in [16].

Verification and Application to MPC
Several properties described by PCTL formulas can be verified by using the obtained model on PRISM.We use the "Properties" part in PRISM.From the above results, we see that the verification problem can be easily implemented by using PRISM.The control input sequence ( ) ( ) ( ) tained simultaneously, but in PRISM 4.0.3, the obtained control input sequence cannot be displayed except for the case of N = ∞ .In of N = ∞ , the discrete-time Markov chain can be obtained as the closed-loop system of a given system.The control input sequence can be obtained by exploratory analysis using the simulator in PRISM.
Otherwise, this sequence can be obtained by solving the control problem such as the optimal control problem.In both cases, the verification result will be useful.
On the other hand, the problem of finding max P and a control input sequence can be regarded as a kind of the control problem.Noting that the initial value of the control input must be given, a kind of MPC can be realized by the following procedure.
[Procedure of MPC] Step 1: Set 0 t = , and determine the current state ( ) t x t x = according to power consumption.
Step 2: Find the current control input ( ) Step 3: Apply only the control input at t, i.e., ( ) , to the plant.
Step 4: Set : according to power consumption, and go to Step 2.

Numerical Example
We present a numerical example.For i Σ , 1, 2, , i n =  given by (3), parameters are given as follows: In Problem 1, the control time N, the output, and the target output are given by 10, N = ( ) ( ), 1, 2, , , In this example, we consider the following cases: • Case 1: The initial state is given by ( ) = (all consumers normally use electricity).
Case 1-1: The initial input is given by ( ) Case 1-2: The initial input is given by ( ) • Case 2: The initial state is given by ( ) ( ) Case 2-1: The initial input is given by ( ) Case 2-2: The initial input is given by ( ) • Case 3: The initial state is given by ( ) ( ) Case 3-1: The initial input is given by ( ) Case 3-2: The initial input is given by ( ) Next, we present the computation result.Table 1 shows max P for each case.By checking max P , we can verify the status of electricity conservation.If max P is large, then there is a trend that consumers conserve electricity.From Table 1, we see the following facts: 1) It is desirable that the initial input (price) is given by ( ) 2) Even if one consumer, who is not the leader, conserves electricity, then a contribution to electricity conservation is small.
3) If the leader conserves electricity, then a contribution to electricity conservation is large.
Thus, using the PBN-based model, we can analyze real-time pricing systems in a quantitative way.

Conclusions
In this paper, using a probabilistic Boolean network (PBN), we discussed verification of There are several open problems.It is significant to develop the identification method of Boolean functions and parameters ( ) ( ) , i i l l a b in (3).Once Boolean functions and parameters can be obtained, the proposed method enables us quantitative analysis.
Furthermore, for large-scale systems, there is a possibility that PRISM does not work.
In such a case, we may use the assume-guarantee verification technique [17], which is one of the compositional verification techniques.Details are one of the future efforts.It is also significant to consider extending a PBN to a probabilistic system with multi-valued logic functions (see e.g., [18]- [21] for further details about such probabilistic systems).Since the PBN-based model expresses human decision making in the purchasing behavior, the proposed method is related to analysis of the consumer behavior in economics.It is important to clarify the relation between the proposed method and the existing method in economics.The proposed method is the first step toward mathematical analysis of the consumer behavior.

Figure 1 .
Figure 1.Illustration of real-time pricing systems.

Consider solving Problem 1 (
the reachability problem).Then, we use P max prepared in PRISM.Suppose[] , the number of times that ( ) f y k y = holds is greater than or equal to 1, i.e., this code expresses the reachability problem itself.

(
In these cases, time evolution of the state does not depend on the past state.The Boolean function( )

Table 1 .
Computation result.pricing systems of electricity.The PBN-based model and PRISM enable us an easy and convenient verification.As one of the verification problems, the reachability problem was considered.In addition, application to model predictive control was also discussed.The proposed method provides us verification/control methods for real-time pricing systems.