^{1}

^{2}

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 real-time 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.

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., [

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., [

Under the above backgrounds, the authors have proposed in [

In this paper, we propose a verification method of real-time pricing systems using the PBN-based model and the probabilistic model checker PRISM [

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.

Notation: For the n-dimensional vector

In this section, we explain the outline of real-time pricing systems studied in this paper.

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.

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

First, we explain a (deterministic) Boolean network (BN). A BN is defined by

where

Next, we explain a probabilistic Boolean network (PBN) (see [

denote the candidates of

Then, the following relation

must be satisfied. Probabilistic distributions are derived from experimental results. Finally,

We show a simple example.

Example 1. Consider the PBN in which Boolean functions and probabilities are given by

where

In this example, the cardinality of the finite state set

Consider modeling the set of consumers as a PBN. The number of consumers is given by n. We assume that the state of consumer

The binary value of

where

Let

The Boolean functions

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 system

Problem 1. Suppose that for the system

by manipulating a control input sequence

Let

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).

In this section, we consider a solution method for Problem 1 using the probabilistic model checker PRISM [

As a preparation, the following lemma [

Lemma 1. Consider two binary variables

i)

ii)

iii)

For example,

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 parts: “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

In addition,

01: mdp

02: module RTP1

03: x1: [0..1] init 1;

04: [RTP] u=3 -> 0.1:(x1’=1) + 0.075:(x1’=0) + 0.6:(x1’=x1) + 0.15:(x1’=x2*x3)

+ 0.075:(x1’=x1)

05: [RTP] u=4 -> 0.1:(x1’=1) + 0.1:(x1’=0) + 0.5:(x1’=x1) + 0.2:(x1’=x2*x3)

+ 0.1:(x1’=x1)

06: [RTP] u=5 -> 0.1:(x1’=1) + 0.125:(x1’=0) + 0.4:(x1’=x1) + 0.25:(x1’=x2*x3)

+ 0.125:(x1’=x1)

07: endmodule

08: module RTP2

09: x2:[0..1] init 1;

10: [RTP] u=3 -> 0.1:(x2’=1) + 0.075:(x2’=0) + 0.6:(x2’=x2) + 0.15:(x2’=x1*x3)

+ 0.075:(x2’=x1)

11: [RTP] u=4 -> 0.1:(x2’=1) + ... (omit)

12: [RTP] u=5 -> 0.1:(x2’=1) + ... (omit)

13: endmodule

14: module RTP3

15: x3:[0..1] init 1;

16: [RTP] u=3 -> 0.1:(x3’=1) + 0.075:(x3’=0) + 0.6:(x3’=x3) + 0.15:(x3’=x1*x2)

+ 0.075:(x3’=x1)

17: [RTP] u=4 -> 0.1:(x3’=1) + ... (omit)

18: [RTP] u=5 -> 0.1:(x3’=1) + ... (omit)

19: endmodule

20: module input

21: u:[3..5] init 3;

22: [RTP] u=3 -> (u’=3);

23: [RTP] u=3 -> (u’=4);

24: [RTP] u=3 -> (u’=5);

25: [RTP] u=4 -> (u’=3);

26: [RTP] u=4 -> (u’=4);

27: [RTP] u=4 -> (u’=5);

28: [RTP] u=5 -> (u’=3);

29: [RTP] u=5 -> (u’=4);

30: [RTP] u=5 -> (u’=5);

31: endmodule.

In line 1, it is described that a given system is an MDP, i.e., the control input (in other words, the nondeterministic variable) that must decide is included. In lines 2-7, the dynamics for ^{1}) with the probability 0.15, and

From the above example, we see that the system

Derivation Procedure of PRISM Source Code:

Step 1: Transform each Boolean function into a polynomial with binary variables by using Lemma 1. Let

Step 2: Describe that a given system is an MDP.

Step 3: Compute the probability

Step 4: Describe module RTP i,

module RTP i;

[RTP]

[RTP]

endmodule.

Step 5: Describe the control input u as follows.

module input

u:

[RTP]

[RTP]

[RTP]

[RTP]

endmodule.

The above procedure is the improved version of the procedure proposed in [

Several properties described by PCTL formulas can be verified by using the obtained model on PRISM. We use the “Properties” part in PRISM.

Consider solving Problem 1 (the reachability problem). Then, we use P_{max} prepared in PRISM. Suppose

This implies that find a maximum probability

From the above results, we see that the verification problem can be easily implemented by using PRISM. The control input sequence

On the other hand, the problem of finding

[Procedure of MPC]

Step 1: Set

Step 2: Find the current control input

Step 3: Apply only the control input at t, i.e.,

Step 4: Set

We present a numerical example. For

We remark that for any

In Problem 1, the control time N, the output, and the target output are given by

In this example, we consider the following cases:

・ Case 1: The initial state is given by

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.

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.

In this paper, using a probabilistic Boolean network (PBN), we discussed verification of

Case | P_{max} |
---|---|

Case 1-1 | 0.6248 |

Case 1-2 | 0.6630 |

Case 2-1 | 0.6455 |

Case 2-2 | 0.6828 |

Case 3-1 | 0.7454 |

Case 3-2 | 0.7756 |

real-time 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.

There are several open problems. It is significant to develop the identification method of Boolean functions and parameters

This research was partly supported by JST, CREST and Grant-in-Aid for Scientific Research (C) 26420412.

Kobayashi, K. and Hiraishi, K. (2016) Verification of Real-Time Pricing Systems Based on Probabilistic Boolean Networks. Applied Mathematics, 7, 1734- 1747. http://dx.doi.org/10.4236/am.2016.715146

In classical propositional logic, truth-value of 0 (false) or 1 (true) is time-invariant. Temporal logic is an extension of propositional logic, and deals with time evolution of truth-value. Since a PBN is a discrete-time system, we also consider temporal logic in discrete-time. First, computation tree logic (CTL) is explained as a class of temporal logics. Next, we introduce probabilistic CTL (PCTL) (see [

In CTL, logical operators and temporal operators are used. The logical operators usually consist of

1) Propositional variables and propositional constants (true or false) are state formulas.

2) If f, y are state formulas, then

3) If f is path formula, then Ef and Af are state formulas.

4) If f, y are state formulas, then Xf, Ff, Gf, and fUy are path formulas.

5) All state and path formulas consist of the above formulas, and all CTL formulas consist of state formulas.

Next, suppose that

・ Af: f has to hold on all paths starting from the current state (All).

・ Ef: there exists at least one path starting from the current state where f holds (Exists).

Furthermore, the meaning of each path-specific quantifier is also explained as follows:

・ Ff: f eventually has to hold (somewhere on the subsequent path) (Finally).

・ Gf: f has to hold on the entire subsequent path (Globally).

・ Xf: f has to hold at the next state (neXt).

・ fUy: f has to hold until at some position y holds. This implies that y will be verified in the future.

In PCTL, the notion of probability is added in CTL, that is, for the CTL formula f, consider

Finally, the temporal operator F is improved to F^{£}^{N}. For the propositional variable f, F^{£}^{N}f implies that f eventually has to hold until time N.

^{1}In PRISM, given Boolean functions may be directly used (see http://www.prismmodelchecker.org/ for further details).