Structural Control of Probabilistic Boolean Networks and Its Application to Design of Real-Time Pricing Systems

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1 Preprints of the 9th Word Congress The Internationa Federation of Automatic Contro Structura Contro of Probabiistic Booean Networks and Its Appication to Design of Rea-Time Pricing Systems Koichi Kobayashi Kunihiko Hiraishi Japan Advanced Institute of Science and Technoogy, - Asahidai, Nomi, Ishikawa , Japan (e-mai: k-kobaya@jaistacjp) Abstract: In this paper, a new contro method for a probabiistic Booean network (PBN) is proposed A PBN is widey used as a mode of compe systems such as gene reguatory networks For a PBN, the structura contro probem is newy formuated In this probem, a discrete probabiity distribution appeared in a PBN is controed by the continuous-vaued input In the proposed soution method, using a matri-based representation for a PBN, the probem is approimated by a inear programming probem Furthermore, design of rea-time pricing systems of eectricity is considered as an appication By appropriatey designing reatime pricing systems, eectricity conservation is achieved The effectiveness of the proposed method is presented by a numerica eampe on rea-time pricing systems Keywords: probabiistic Booean networks, rea-time pricing systems, structura contro INTRODUCTION Anaysis and contro of compe systems such as power systems and gene reguatory networks are one of the fundamenta probems in contro theory of arge-scae systems In order to dea with such a compe system, it is one of the appropriate methods to approimate a compe system by a discrete abstract mode (see, eg, Tabuada (29)) On the other hand, human decision making is aso compe, and is modeed by a discrete mode (see, eg, Adomi et a (2)) Thus, in anaysis and contro of compe systems and those with human decision making, a discrete mode pays the important roe Severa discrete modes such as Petri nets, Bayesian networks, automata-based modes, Booean networks have been proposed so far (see, eg, Jong (22)) In this paper, we focus on a Booean network (BN) (Kauffman (969)) In a BN, the state is given by a binary vaue ( or ), and the dynamics are epressed by a set of Booean functions In addition, since the behavior of compe systems is frequenty stochastic by the effects of noise, it is appropriate that a Booean function is randomy decided at each time among the candidates of Booean functions Thus, a probabiistic BN (PBN) has been proposed in Shmuevich et a (22a) In this paper, we adopt a probabiistic Booean network (PBN) as a discrete mode For a given PBN, we consider the structura contro probem (see, eg, Kobayashi and Hiraishi (2); Shmuevich et a (22b); Xiao and Dougherty (27)) In this probem, a discrete probabiity distribution is controed For eampe, in Kobayashi and Hiraishi (2), a discrete probabiity distribution at each time is seected among a given set In this paper, we consider fine contro of a discrete probabiity distribution by using the continuousvaued input For a newy formuated probem, we propose an approimate soution method First, a matribased representation of BNs proposed in Kobayashi and Hiraishi (24) is etended to that of PBNs Net, using the obtained representation, the origina probem is approimated by a inear programming (LP) probem Furthermore, as one of the appications, we consider a design method of rea-time pricing systems (see, eg, Borenstein et a (22); Roozbehani et a (2); Samadi et a (2); Vivekananthan et a (2)) A rea-time pricing system of eectricity is a system that charges different eectricity prices for different hours of the day and for different days, and is effective for reducing the peak and fattening the oad curve In the eisting methods, the price at each time is given by a simpe function with respect to power consumptions and votage deviations and so on (see, eg, Vivekananthan et a (2)) To the best of our knowedge, decision making of customers has not been epicity considered so far Thus, decision making of customers is modeed by a PBN, and the probem of finding the price at each time is formuated as a structura contro probem The price corresponds to the continuousvaued input By a numerica eampe, the effectiveness of the proposed method is presented Notation: For the n-dimensiona vector [ 2 n ] and the inde set I {i,i 2,,i m } {, 2,,n}, define [ i ] i I : [ i i2 im ] For two matrices A and B, eta B denote the Kronecker product of A and B In addition, for q vectors y,y 2,,y q and the inde set J {j,j 2,,j p } {, 2,,q}, define j J y j : y j y j2 y jp 2 PROBABILISTIC BOOLEAN NETWORK First, we epain a (deterministic) Booean network (BN) ABNisdefinedby Copyright 24 IFAC 2442

2 (k +)f () ([ j (k)] j N ()), 2 (k +)f (2) ([ j (k)] j N (2)), n (k +)f (n) ([ j (k)] j N (n)), where : [ 2 n ] {, } n is the state, and k,, 2, is the discrete time The set N {, 2,,n} is a given inde set, and the function f i : {, } N {, } is a given Booean function consisting of ogica operators such as AND ( ), OR ( ), and NOT ( ) If N hods, then i (k +) is uniquey determined as or Net, we epain a probabiistic Booean network (PBN) (see Shmuevich et a (22a) for further detais) In a PBN, the candidates of f are given, and for each i, seecting one Booean function is probabiisticay independent at each time Let ( f [ j (k)] j N ),, 2,,q denote the candidates of f The probabiity that f is seected is defined by ( ) c : Prob f f (2) Then, the foowing reation q () c () must be satisfied Probabiistic distributions are derived from eperimenta resuts Finay, N i, i, 2,,n are defined by N i : q N We show a simpe eampe Eampe Consider the PBN in which Booean functions and probabiities are given by { f () f () (k), c () 8, f () 2 (k), c () 2 2, f (2) f (2) (k) (k), c (2), { f () f () (k) 2 (k), c () 7, f () 2 2 (k), c () 2, where q() 2, q(2) and q() 2 hod, N {}, N 2 {, }, andn {, 2} hod, and we see that the reation () is satisfied Net, consider the state trajectory Then, for () [ ],weobtain Prob ( () [ ] () [ ] ) 8, Prob ( () [ ] () [ ] ) 2 In this eampe, the cardinaity of the finite state set {, } is given by 2 8, and we can obtain the discretetime Markov chain by computing the transition from each state PROBLEM FORMULATION In this section, we formuate the contro probem studied in this paper In the standard contro probem, the contro input is added to a given Booean function For eampe, the contro input is added as foows: f ([ j (k)] j N,u(k)), u(k) {, } In genera, we assume that the vaue of the contro input can be arbitrariy given However, there is a possibiity that there eists no contro input satisfying this assumption In contro of gene reguatory networks, a structura contro (or structura intervention) method for PBNs has been proposed so far (see, eg, Kobayashi and Hiraishi (2); Shmuevich et a (22b)) For eampe, in Kobayashi and Hiraishi (2), the discrete probabiistic distribution is switched at each time In other words, the discrete probabiistic distribution is seected among the set of candidates On the other hand, in compe systems such as gene reguatory networks, power systems, and socia systems, it wi be desirabe to consider a weaker contro method Thus, in this paper, we consider fine contro of probabiities in a discrete probabiistic distribution This contro method can be regarded as a kind of structura contro methods In the structura contro probem formuated here, we assume that the probabiity c in (2) is given by c (k) a + b u i (k), (4) where u : [ u u 2 u n ] [u, u ] [u 2, u 2 ] [u n, u n ] R n is the contro input The set [u i, u i ] epresses the input constraint, and u i, u i R are given in advance Of course, we must find u(k) such that c (k) satisfies () In addition, the dimension of the contro input may be ess than the dimension n of the state Under the above preparation, we consider the foowing probem Probem Suppose that for the PBN with (4), the ower and upper bounds of input constraints u i, u i, and the initia state () are given Then, find a contro input sequence u(),u(),,u(n ) [u, u ] [u 2, u 2 ] [u n, u n ] minimizing the foowing cost function [ N J E {Q(k)+Ru(k)} + Q f (N) k ] (), (5) where Q, Q f R n, R R m are weighting vectors whose eement is a non-negative rea number, and E[ ] denotes a conditiona epected vaue The inear cost function (5) is appropriate from the foowing reason: For a binary variabe δ {, }, thereation δ 2 δ hods That is, in the cost function, the quadratic term such as 2 i (k) is not necessary According to the resut in Kobayashi and Hiraishi (22), Probem can be rewritten as a poynomia optimization probem However, in the case of arge-scae PBNs, it wi be difficut to sove a poynomia optimization probem In this paper, an approimate soution method for Probem isproposed 244

3 Hereafter, the condition () in the conditiona epected vaue is omitted 4 DERIVATION OF APPROXIMATE SOLUTION METHOD In this section, we derive an approimate soution method for Probem First, a matri-based representation for PBNs is derived The obtained representation is an etension of a matri-based representation for BNs proposed in Kobayashi and Hiraishi (24) Net, using the matribased representation, an approimate soution method for Probem is derived 4 Matri-Based Representation for PBN As preparations, some notations are defined Binary variabes i (k) and i (k) are introduced If i(k) hods, then i (k) hods, otherwise i (k) hods If i (k) hods, then i (k) hods, otherwise i (k) hods Then, the equaity i (k) + i (k) is satisfied Using i (k) and i (k), Consider transforming the BN () into a matri-based representation First, we epain the outine of a matri-based representation by using a simpe eampe Eampe 2 Consider the foowing BN: { (k +) 2 (k), 2 (k +) (k), (k +) (k) 2 (k), where N () {2}, N (2) {}, andn () {, 2} Then, we can obtain the truth tabe for each i (k +) See Tabe and Tabe 2 From these truth tabes, we can obtain the foowing matri-based representation: (k +) (k +) 2 (k +) 2(k +) (k +) (k +) A [ () ] A (2) 2 (k), 2(k) (k), (k) (k) 2(k) (k) 2(k) (k) 2(k), (k) 2(k) A () where each eement of A, i, 2, is given by a binary vaue ( or ), and a sum of a eements in each coumn of A is equa to Such a matri-based representation has been proposed in aso Cheng and Qi (29); Cheng et a (2) However, in the representation proposed in Cheng and Qi (29); Cheng et a (2), the matri with the size of 2 n 2 n must be manipuated (n is the dimension of the state) In the matri-based representation proposed in Kobayashi and Hiraishi (24), the matri with the size of 2 2 Ni are manipuated for each i Thus, the proposed representation enabes us to mode a BN using matrices with the smaer size Tabe Truth tabes for i (k +),i, 2 2 (k) (k +) (k) 2 (k +) Tabe 2 Truth tabe for (k +) (k) 2 (k) (k +) Consider a genera case Define ( ) i (k) : i (k) i (k) i (k) i (k) Then, the matri-based representation for i (k+) is given by i (k +)A j N i j (k), (6) where A {, } 2 2 N i and j N i j (k) {, } 2 N i The matri A can be derived from the foowing procedure Procedure for deriving A in (6): Step : Derive a truth tabe for i (k +) Step 2: Based on the obtained truth tabe, assign i (k + ) or i (k + ) for each eement of j N i j (k) Step : Epress the assignment obtained in Step 2 by a row vector Denote the obtained row vector by A {, } 2 N i Step 4: Derive A as [ ] A 2 N i A A Net, consider etending the matri-based representation of BNs to that of PBNs First, using a simpe eampe, we epain the outine Eampe Consider the PBN in Eampe Using the matri-based representation, the epected vaue of i (k + ) can be obtained as ( ) E[ E[ (k +)] (k)] E[, 2(k)] A () A () 2 E[ (k) (k)] E[ (k) (k)] E[ 2 (k +)] A (2) E[ (k) (k)] E[ (k) (k)], (7) 2444

4 ( ) E[ (k +)] 7 + A () A () 2 E[ (k) 2(k)] E[ (k) 2(k)] E[ (k) 2(k)], E[ (k) 2(k)] where the condition () is omitted In this representation, the matrices A () and A () 2 correspond to the Booean functions f () and f () 2, respectivey In a simiar way, A (2), A(),andA() correspond to the Booean functions f (2), f () (),andf 2, respectivey In genera, using the matri-based representation, the epected vaue of i (k + ) can be obtained as q E[ i (k +)] c (k)a E[ j (k)], (8) j N i where A {, } 2 2 N i and j N i j (k) {, } 2 N i The matri A can be derived from the above procedure 42 Reduction to a Linear Programming Probem Using the matri-based representation (8), consider transforming Probem First, Probem can be rewritten as the foowing probem: Probem 2 Suppose that for the PBN with (4), the ower and upper bounds of input constraints u i, u i, and the initia state () are given Then, find u(),u(),, u(n ) minimizing the cost function (5) subject to (), (8), and the input constraint In a simiar way to Probem, Probem 2 is rewritten as a poynomia optimization probem In this paper, we focus on the structure of j N i E[ j (k)], and derive the reaed probem for Probem 2 The reaed probem is a inear programming (LP) probem, and can be soved faster than a poynomia optimization probem First, we show an eampe Eampe 4 Consider the matri-based representation obtained in Eampe We remark that the discrete probabiistic distribution for each i is independent Then, in (7), we can obtain E[ (k) (k)] + E[ (k) (k)] E[ (k)](e[ (k)] + E[ (k)]) E[ (k)] In a simiar way, we can obtain E[ (k) (k)] + E[ (k) (k)] E[ (k)], E[ (k) (k)] + E[ (k) (k)] E[ (k)], E[ (k) (k)] + E[ (k) (k)] E[ (k)] In addition, E[ (k) (k)] + E[ (k) (k)] + E[ (k) (k)] +E[ (k) (k)] hods The obtained equaities can be used as constraints in the reaed probem Net, consider a genera case Define z i (k) : E[ j (k)] [, ] 2 N i j N i Then, (8) can be rewritten as q E[ i (k +)] a q + A b z i (k) A w i (k) (9) where w i (k) : u i (k)z i (k) [, ] 2 N i The reation between E[ i (k)] and z i (k) isgivenby E[ j (k)] C j z i (k), j N i, () where the matri C j {, } 2 2 N i can be derived in a simiar to Eampe 4 Let z (j) i (k) andw (j) i (k) denotethe j-th eement of z i (k) andw i (k), respectivey Then, we can obtain 2 N i j From w i (k) :u i (k)z i (k), we can obtain 2 N i j z (j) i (k) () w (j) i (k) u i (k) (2) In addition, we introduce the foowing constraints: u i z i (k) w i (k) u i z i (k), () z i (k), (4) w i (k) (5) For probabiities, we introduce the foowing constraint: a + b u i (k),, 2,,q (6) Thus, we can obtain the foowing probem as a reaed probem of Probem 2 Probem Suppose that for the PBN with (4), the ower and upper bounds of input constraints u i, u i, and the initia state () are given Then, find u(),u(),, u(n ) minimizing the cost function (5) subject to (), (9) (6) By a simpe cacuation, Probem is reduced to an LP probem By soving Probem, we can evauate the ower bound of the optima vaue of the cost function in Probem In this paper, ony an approimate soution method is provided However, since the contro input is obtained by soving an LP probem, the proposed soution method using the matri-based representation enabes us to sove the structura contro probem for severa casses of PBNs 2445

5 i (k + ) Fig Iustration of rea-time pricing systems 5 APPLICATION TO DESIGN OF REAL-TIME PRICING SYSTEMS In this section, we consider a design method of rea-time pricing systems as an appication of structura contro of PBNs First, the outine of rea-time pricing systems of eectricity is epained Net, the PBN-based mode of rea-time pricing systems is derived Finay, a numerica eampe is presented 5 Outine Fig shows an iustration of rea-time pricing systems studied in this paper This system consists of one controer and mutipe eectric customers such as commercia faciities and homes For an eectric customer, we suppose that each customer can monitor the status of eectricity conservation of other customers In other words, the status of some customer affects that of other customers For eampe, in commercia faciities, we suppose that the status of riva commercia faciities can be checked by ighting, Bog, Twitter, and so on Depending on power consumption, ie, the status of eectricity conservation, the controer determines the price If eectricity conservation is needed, then the price is set to a high vaue Since the economic oad becomes high, customers conserve eectricity Thus, eectricity conservation is achieved The price does not depend on each customer, and is uniquey determined f, c a + b u(k), f2, c2 a2 + b 2 u(k), f i (k), c a + b u(k), f4 g ([j (k)]j Di ), c4 a4 + b4 u(k), f5 (k), c5 a5 + b5 u(k), where u(k) [u, u] R is the contro input correspond ing to the price The Booean functions f and f2 impy that customer i forciby conserves (or does not conserve) eectricity In these cases, time evoution of the state does not depend on the past state The Booean function f impies that the state is not changed The Booean function f4 impies that the state of customer i is changed de pending on the other customers The Booean function f5 impies that the state of customer i is changed depending on the eader Thus, decision making of customers can be modeed by a PBN The above Booean functions are an eampe of modes for decision making Depending on rea situations, we may use other Booean functions For the PBN-based mode obtained, we consider the foowing probem: find a time sequence of the price such that customers conserve eectricity as much as possibe However, it is not desirabe that the price is too high The condition that customers conserve eectricity as much as possibe can be characterized by E[i ] In other words, power consumption is epressed by E[i ] Hence, this probem can be formuated as Probem by appropriatey setting the weights Q and R 5 Numerica Eampe We present a numerica eampe Parameters in the system are given as foows: n 8, D {2, 8}, Di {i, i + }, i 2,,, 7, D8 {, 7}, a, b, a2, b2 25, a 9, b, a4, b4 5, a5, b4 25, u, and u 7 We remark that under the input constraint u(k) [u, u], () and (6) hod The Booean function g is given by 52 Mode g ([j (k)]j Di ) j (k) j2 (k) j Di (k), Consider modeing the set of customers as a PBN The number of customers is given by n We assume that the state of customer i {, 2,, n} is binary, and is denoted by i The state impies $ customer i conserves eectricity, i customer i normay uses eectricity The binary vaue of i is determined by power consumption of customer i Let Di {, 2,, n}, i, 2,, n denote the set of customers, which affect to customer i In addition, we assume that there eists one eader in the oca area The state of a eader is given by Then, for customer i, we consider the foowing PBN: {j, j2,, j Di } Di Parameters in Probem are given as foows: () [ ], N 5, Q [ ], and R 5 Net, we present the computation resut Fig 2 shows trajectories of E[i (k)] Fig shows trajectories of the contro input (the price) From these figures, we see that E[i ] becomes sma by fine adjustment of the contro input In this eampe, the epected vaue of each state converges to 2 In addition, when the obtained contro input is appied to the system, the vaue of the cost function in Probem was 794 In the case of u(k) (ie, the constant 2446

6 The epected vaue of each state Time Fig 2 The epected vaue of the state Some states are indistinguishabe Contro input (price) Time Fig The obtained contro input (price) input), the vaue of the cost function in Probem was 8558 In the case of u(k) 7, the vaue of the cost function in Probem was From these vaues, we see that the obtained contro input is effective than trivia contro inputs Of course, there is a possibiity that the obtained contro input is not optima for Probem Finay, we discuss the computation time for soving Probem The computation time was 6 [sec], where we used IBM ILOG CPLEX as the LP sover Thus, athough Probem is an approimation of the origina probem, Probem can be soved fast It wi be difficut to sove the origina probem (ie, the poynomia optimization probem) for this case by using MATLAB 2bit 6 CONCLUSION In this paper, we discussed the structura contro probem for a probabiistic Booean network (PBN) First, the structura contro probem was formuated Net, an approimate soution method was proposed Finay, as an appication, we considered design of rea-time pricing systems of eectricity The proposed method provides us a new contro method for compe dynamica systems In future work, it is significant to evauate the accuracy of an approimation from the theoretica viewpoint ACKNOWLEDGEMENTS REFERENCES M Adomi, Y Shikauchi, and S Ishii, Hidden Markov mode for human decision process in a partiay observabe environment, Procofthe2thInt ConfonArtificia Neura Networks, LNCS 65, pp 94, 2 S Borenstein, M Jaske, A Rosenfed, Dynamic pricing, advanced metering, and demand response in eectricity markets, Center for the Study of Energy Markets, UC Berkeey, 22 D Cheng, H Qi, Controabiity and observabiity of Booean contro network, Automatica, vo 45, no 7, pp , 29 D Cheng, H Qi, and Z Li, Anaysis and Contro of Booean Network: A Semi-tensor Product Approach, Springer, 2 H D Jong, Modeing and simuation of genetic reguatory systems: A iterature review, Journa of Computationa Bioogy, vo 9, pp 67, 22 S A Kauffman, Metaboic stabiity and epigenesis in randomy constructed genetic nets, Journa of Theoretica Bioogy, vo 22, pp , 969 K Kobayashi and K Hiraishi, Optima contro of probabiistic Booean networks using poynomia optimization, IEICE Transactions on Fundamentas of Eectronics, Communications and Computer Sciences, vo E95- A, no 9, pp 52 57, 22 K Kobayashi and K Hiraishi, Optima contro of gene reguatory networks with effectiveness of mutipe drugs: A Booean network approach, BioMed Research Internationa, Vo 2, Artice ID 24676, pages, 2 K Kobayashi and K Hiraishi, Design of Booean networks based on prescribed singeton attractors, The th European Contro Conference, 24 (accepted) M Roozbehani, M Daheh, and S Mitter, On the stabiity of whoesae eectricity markets under rea-time pricing, Proc of the 49th IEEE Conf on Decision and Contro, pp 9 98, 2 P Samadi, A-H Mohsenian-Rad, R Schober, V WS Wong, and J Jatskevich, Optima rea-time pricing agorithm based on utiity maimization for smart grid, Proc of the First IEEE Int Conf on Smart Grid Communications, pp 45 42, 2 I Shmuevich, E R Dougherty, S Kim, and W Zhang, Probabiistic Booean networks: A rue-based uncertainty mode for gene reguatory networks, Bioinformatics, vo 8, pp , 22 I Shmuevich, E R Dougherty, and W Zhang, Contro of stationary behavior in probabiistic Booean networks by means of structura intervention, Journa of Bioogica Systems, vo, no 4, pp 4 445, 22 P Tabuada, Verification and Contro of Hybrid Systems, Springer, 29 C Vivekananthan, Y Mishra, and G Ledwich, A nove rea time pricing scheme for demand response in residentia distribution systems, Proc of the 8th Annua Conf of the IEEE Industria Eectronics Society, pp , 2 Y Xiao and E R Dougherty, The impact of function perturbations in Booean networks, Bioinformatics, vo 2, pp , 27 This research was party supported by JST, CREST 2447