Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 952310, 10 pages

http://dx.doi.org/10.1155/2015/952310

## Optimal Control of Probabilistic Logic Networks and Its Application to Real-Time Pricing of Electricity

^{1}Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan^{2}School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan

Received 1 July 2015; Revised 18 October 2015; Accepted 16 November 2015

Academic Editor: Qingling Zhang

Copyright © 2015 Koichi Kobayashi and Kunihiko Hiraishi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In analysis and control of large-scale complex systems, a discrete model plays an important role. In this paper, a probabilistic logic network (PLN) is considered as a discrete model. A PLN is a mathematical model where multivalued logic functions are randomly switched. For a PLN with two kinds of control inputs, the optimal control problem is formulated, and an approximate solution method for this problem is proposed. In the proposed method, using a matrix-based representation for a PLN, this problem is approximated by a mixed integer linear programming problem. In application, real-time pricing of electricity is studied. In real-time pricing, electricity conservation is achieved by setting a high electricity price. A numerical example is presented to show the effectiveness of the proposed method.

#### 1. Introduction

In the last decade, control of large-scale complex systems has attracted considerable attention, where complex nonlinear dynamics are taken into account. In such cases, one of the natural approaches is to approximately solve the control problem of such complex systems using simplification/abstraction techniques (see, e.g., [1]). Specifically, a discrete abstract model plays an important role in analysis and control of large-scale complex systems.

Several discrete models have been proposed so far (see, e.g., [2]). Petri nets, Bayesian networks, and automata-based models are frequently adopted as a mathematical model in analysis of large-scale complex systems. Although Boolean networks (BNs) [3] are frequently adopted as a model for analysis, BNs are recently adopted as a mathematical model in control of large-scale complex systems. In a BN, the state takes a binary value (0 or 1), and the dynamics such as interactions of states are modeled by a set of Boolean functions. In [4], it is pointed out that since a BN is too simple, the behavior of large-scale complex systems such as gene regulatory networks cannot be appropriately modeled. One of the methods to overcome this criticism is to extend Boolean functions to multivalued logic functions. From this viewpoint, several modeling methods using multivalued logic functions have been studied so far (see, e.g., [5–8]). Furthermore, since the behavior of large-scale complex systems is frequently stochastic by the effects of noise, a probabilistic logic network (PLN), which is an extension of a probabilistic BN (PBN) [9], has been studied so far (see, e.g., [10, 11]). In a PLN, a multivalued logic function is randomly selected from the candidates of functions. We adopt a PLN as a discrete model.

In this paper, first, we formulate the optimal control problem for the PLN with two kinds of control inputs. One of control inputs is a conventional control input, which is included in multivalued logic functions. The other is called here a structural control input, which is included in a given discrete probability distribution. By the structural control input, a discrete probability distribution is controlled. Such a control input has been considered in, for example, [12–15].

Next, an approximate solution method for the optimal control problem is proposed. Solution methods have been studied in [10]. However, large sized matrices must be handled. From this fact, it is difficult to directly apply the existing methods to large-scale systems. Hence, it is important to consider a new solution method. To derive an approximate solution method, a matrix-based representation of PLNs is proposed. The authors have proposed a matrix-based representation of BNs [16] and that of probabilistic BNs [13]. However, multivalued logic functions have not been studied so far. Using a matrix-based representation, the original problem is approximated by a mixed integer linear programming (MILP) problem. In the case where there is no conventional control input, the original problem is approximated by a linear programming problem. In [17], it has been proven that the optimal control problem of PBNs is -hard problem, which is much harder than an NP-complete problem. Hence, approximating the optimal control problem of PLNs by an MILP problem is useful for solving that of a wider class of PLNs.

A PLN has several applications such as gene regulatory networks and power systems. As already explained, BNs and PBNs are well known as a model of gene regulatory networks. Logical models with multivalued variables are also used in theoretical analysis of gene regulatory networks. A PLN can be regarded as a generalized model of these mathematical models. In this paper, as one of the other applications, we consider real-time pricing of electricity (see, e.g., [18–21]).

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. Real-time pricing is effective for reducing the peak and flattening the load curve. 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 becomes 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., [21]). The authors have proposed in [13] a method to model decision making of consumers using a PBN. However, the state of a consumer is given by a binary value, that is, 0 (a consumer conserves electricity) or 1 (a consumer normally uses electricity). A PLN is appropriate as a more precise model of consumers.

*Notation*. For the nonnegative integer , define the finite set . For the -dimensional vector and the index set , define . For two matrices and , let denote the Kronecker product of and . In addition, for vectors and the index set , define . For example, for two-dimensional vectors and , we can obtainwhere is the th element of . Let denote the matrix whose elements are all one. Let and denote the identity matrix and the zero matrix, respectively. For simplicity of notation, we sometimes use the symbol instead of , and the symbol instead of .

#### 2. Probabilistic Logic Network

First, we explain a multivalued logic network (MVLN). This system is defined bywhere , , is the state, and , , , is the control input. In addition, and are given, and is the discrete time. The sets and are given index sets, and the functionis a given multivalued logic function.

Next, we explain a probabilistic logic network (PLN) (see [10, 11] for further details). A PLN is a generalized version of a probabilistic Boolean network proposed in [9]. In a PLN, the candidates of are given, and, for each , selecting one Boolean function is probabilistically independent at each time. Letdenote the candidates of . The probability that is selected is defined byThen, the relation must be satisfied. Probabilistic distributions are derived from experimental results.

We present a simple example.

*Example 1. *Consider the PLN with two states , (, ) and no control inputs. Suppose that holds. Functions , and , are given by the truth table in Table 1. Probabilities are given by , and , . We see that the relation is satisfied. In this case, holds. Next, consider the state trajectory. For example, for can be obtained asThis PLN can be equivalently transformed into the discrete-time Markov chain with six states in which the transition probability matrix is given byUsing the transition probability matrix expressing a given PLN, we can consider reachability analysis and optimal control. However, in many cases, the size of the matrix becomes large.