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

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1 Hindawi Pubishing Corporation Mathematica Probems in Engineering Voume 05, Artice ID 950, 0 pages Research Artice Optima Contro of Probabiistic Logic Networks and Its Appication to Rea-Time Pricing of Eectricity Koichi Kobayashi and Kunihiko Hiraishi Graduate Schoo of Information Science and Technoogy, Hokkaido University, Sapporo , Japan Schoo of Information Science, Japan Advanced Institute of Science and Technoogy, Ishikawa 9-9, Japan Correspondence shoud be addressed to Koichi Kobayashi; k-kobaya@ssiisthokudaiacjp Received Juy 05; Revised 8 October 05; Accepted 6 November 05 Academic Editor: Qinging Zhang Copyright 05 K Kobayashi and K Hiraishi This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited In anaysis and contro of arge-scae compex systems, a discrete mode pays an important roe In this paper, a probabiistic ogic network (PLN) is considered as a discrete mode A PLN is a mathematica mode where mutivaued ogic functions are randomy switched For a PLN with two kinds of contro inputs, the optima contro probem is formuated, and an approximate soution method for this probem is proposed In the proposed method, using a matrix-based representation for a PLN, this probem is approximated by a mixed integer inear programming probem In appication, rea-time pricing of eectricity is studied In reatime pricing, eectricity conservation is achieved by setting a high eectricity price A numerica exampe is presented to show the effectiveness of the proposed method Introduction In the ast decade, contro of arge-scae compex systems has attracted considerabe attention, where compex noninear dynamics are taken into account In such cases, one of the natura approaches is to approximatey sove the contro probem of such compex systems using simpification/abstraction techniques (see, eg, ) Specificay, a discrete abstract mode pays an important roe in anaysis and contro of argescae compex systems Severadiscretemodeshavebeenproposedsofar(see, eg, ) Petri nets, Bayesian networks, and automata-based modes are frequenty adopted as a mathematica mode in anaysis of arge-scae compex systems Athough Booean networks (BNs) are frequenty adopted as a mode for anaysis, BNs are recenty adopted as a mathematica mode in contro of arge-scae compex systems In a BN, the state takes a binary vaue (0 or ), and the dynamics such as interactions of states are modeed by a set of Booean functions In 4, it ispointedoutthatsinceabnistoosimpe,thebehaviorof arge-scae compex systems such as gene reguatory networks cannot be appropriatey modeed One of the methods to overcome this criticism is to extend Booean functions to mutivaued ogic functions From this viewpoint, severa modeing methods using mutivaued ogic functions have been studied so far (see, eg, 5 8) Furthermore, since the behavior of arge-scae compex systems is frequenty stochastic by the effects of noise, a probabiistic ogic network (PLN), which is an extension of a probabiistic BN (PBN) 9, has been studied so far (see, eg, 0, ) In a PLN, a mutivaued ogic function is randomy seected from the candidates of functions We adopt a PLN as a discrete mode In this paper, first, we formuate the optima contro probem for the PLN with two kinds of contro inputs One of contro inputs is a conventiona contro input, which is incuded in mutivaued ogic functions The other is caed here a structura contro input, which is incuded in a given discrete probabiity distribution By the structura contro input, a discrete probabiity distribution is controed Such a contro input has been considered in, for exampe, 5 Next, an approximate soution method for the optima contro probem is proposed Soution methods have been studied in 0 However, arge sized matrices must be handed From this fact, it is difficut to directy appy the existing methods to arge-scae systems Hence, it is important to consider a new soution method To derive

2 Mathematica Probems in Engineering an approximate soution method, a matrix-based representation of PLNs is proposed The authors have proposed a matrix-based representation of BNs 6 and that of probabiistic BNs However, mutivaued ogic functions have not been studied so far Using a matrix-based representation, the origina probem is approximated by a mixed integer inear programming (MILP) probem In the case where there is no conventiona contro input, the origina probem is approximated by a inear programming probem In 7, it has been proven that the optima contro probem of PBNs is Σ p -hard probem, which is much harder than an NPcompete probem Hence, approximating the optima contro probem of PLNs by an MILP probem is usefu for soving that of a wider cass of PLNs A PLN has severa appications such as gene reguatory networks and power systems As aready expained, BNs and PBNs are we known as a mode of gene reguatory networks Logica modes with mutivaued variabes are aso used in theoretica anaysis of gene reguatory networks A PLN can be regarded as a generaized mode of these mathematica modes In this paper, as one of the other appications, we consider rea-time pricing of eectricity (see, eg, 8 ) 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 Rea-time pricing is effective for reducingthepeakandfatteningtheoadcurveingenera, a rea-time pricing system consists of one controer deciding thepriceateachtimeandmutipeeectricconsumerssuch as commercia faciities and homes If eectricity conservation is needed, then the price is set to a high vaue Since the economic oad becomes high, consumers conserve eectricity Thus, eectricity conservation is achieved In the existing methods, the price at each time is given by a simpe function with respect to power consumptions and votage deviations andsoon(see,eg,)theauthorshaveproposedina method to mode decision making of consumers using a PBN However, the state of a consumer is given by a binary vaue, that is, 0 (a consumer conserves eectricity) or (a consumer normay uses eectricity) A PLN is appropriate as a more precise mode of consumers Notation For the nonnegative integer i, define the finite set D i f {0,,,i }Forthen-dimensiona vector x= x x x n and the index set I = {i,i,,i m } {,,,n}, define x i i I f x i x i x im Fortwo matrices A and B, et A B denote the Kronecker product of A and B In addition, for q vectors y,y,,y q and the index set J ={j,j,,j p } {,,,q}, define j J f y j y j y jp For exampe, for q two-dimensiona vectors z,z,,z q and J ={,5},wecanobtain j J z j =z z 5 = z() z () z() 5 z () 5 = z () z() 5 z () z() 5 z () z() 5 z () z() 5, () where z (i) j is the ith eement of z j Let m n denote the m n matrix whose eements are a one Let I n and 0 m n denote the n nidentitymatrixandthe m nzero matrix, respectivey For simpicity of notation, we sometimes use the symbo 0 instead of 0 m n,andthesymboi instead of I n Probabiistic Logic Network First, we expain a mutivaued ogic network (MVLN) This system is defined by x (k+) =f () (x j (k) j N,u j (k) j M ), x (k+) =f () (x j (k) j N,u j (k) j M ), x n (k+) =f (n) (x j (k) j Nn,u j (k) j Mn ), where x i D r(i), i =,,,n, x f x x x n is the state, and u i D s(i), i =,,,m, u f u u u m, is the contro input In addition, r(i) and s(i) are given, and k {0,,,}is the discrete time The sets N i {,,,n} and M i {,,,m}are given index sets, and the function () f (i) : j N i D r(j) j M i D s(j) D r(i) () is a given mutivaued ogic function Next, we expain a probabiistic ogic network (PLN) (see 0, for further detais) A PLN is a generaized version of a probabiistic Booean network proposed in 9 In a PLN, the candidates of f (i) are given, and, for each x i,seecting one Booean function is probabiisticay independent at each time Let f (i) (x j (k) j Ni,u j (k) j Mi ), =,,, (4) denote the candidates of f (i) Theprobabiitythatf (i) is seected is defined by c (i) f Prob (f (i) =f (i) ) (5) Then, the reation = c(i) =mustbe satisfied Probabiistic distributions are derived from experimenta resuts We present a simpe exampe Exampe Consider the PLN with two states x D, x D (r() =, r() = ) and no contro inputs Suppose that q() = q() = hods Functions f (), f() and f (), f() are given by the truth tabe in Tabe Probabiities are given by c () = 07, c () =0and c () = 08, c () =0Weseethat the reation = c(i) =is satisfied In this case, N = N =

3 Mathematica Probems in Engineering Tabe : Truth tabe x x f () f () f () f () {, } hods Next, consider the state trajectory For exampe, x() for x(0) = 0 0 can be obtained as Prob (x () =0 0 x(0) =0 0 ) = 04, Prob (x () =0 x(0) =0 0 ) = 056, Prob (x () = 0 x(0) =0 0 ) = 006, Prob (x () = x(0) =0 0 ) = 04 This PLN can be equivaenty transformed into the discretetime Markov chain with six states in which the transition probabiity matrix is given by (6) P= (7) Using the transition probabiity matrix expressing a given PLN, we can consider reachabiity anaysis and optima contro However, in many cases, the size of the matrix P becomes arge Remark In this paper, we consider the PLN with no time deay Using additiona states, we can easiy consider time deays Probem Formuation In this section, the optima contro probem for the PLN is formuated We assume that the vaue of the contro input can be arbitrariy set However, there is a possibiity that there exists no contro input satisfying this assumption In such a case, it is desirabe to use the structura contro input, which is used in, for exampe, contro theory of gene reguatory networks (see, eg,, 4) For exampe, in, the structura contro input is regarded as an externa stimuus, and a discrete probabiistic distribution is switched at each time by the structura contro input In other words, thediscreteprobabiisticdistributionisseectedamongthe set of candidates In, the structura contro input is used as the eectricity price in rea-time pricing systems By the structura contro input, probabiities in a discrete probabiity distribution are manipuated Thus, in compex systems, such as gene reguatory networks, power systems, and socia systems, it wi be desirabe to consider a weaker contro method using the structura contro input Hence, we consider both the conventiona contro input and the structura contro input The structura contro input V(k) formuated here is added to the probabiity c (i) in (5) as foows: c (i) (k) = + V (k), (8) where V(k) 0, R (the constraint is normaized) For simpicity of discussion, we consider the one-dimensiona structura contro input, but we may consider the mutidimensiona structura contro input Under the above preparation, we consider the foowing probem Probem Suppose that, for the PLN with the conventiona contro input u and the structura contro input V, theinitia state x(0) = x 0 is given Then, find u(0),u(),,u(n ) {0, } m and V(0), V(),, V(N ) 0, minimizing the foowing cost function: N J=E k=0 (Qx (k) +R u (k) +R V (k)) +Q f x (N) x(0) =x 0, subject to the foowing two constraints on c (i) (k): ( = + V (k)) =, 0 + V (k) (9) (0) In the cost function (9), Q, Q f R n, R R m,andr R are weighting vectors whose eement is a nonnegative rea number, and E denotes a conditiona expected vaue For simpicity of discussion, we consider the inear cost function (9) We remark that x 0, u 0,andV 0 hod The desired state x d, the desired conventiona contro input u d, and the desired structura contro input V d may be introduced Then, in the cost function (9), x(k), u(k), and V(k) mayberepacedwith x(k) x d, u(k) u d,and V(k) V d,respectiveyinthecaseofx i D and u i D (ie, probabiistic Booean networks), Probem can be rewritten as a poynomia optimization probem (see for further detais) However, ony few resuts on optima contro of genera PLNs have been obtained so far (see, eg, 0) Inthispaper,anapproximatesoutionmethodforProbem is proposed Furthermore, Probem is used in mode predictive contro (MPC) See Section 5 for further detais

4 4 Mathematica Probems in Engineering Hereafter, the condition x(0) = x 0 in the conditiona expected vaue is omitted 4 Derivation of Approximate Soution Method In this section, we derive an approximate soution method for Probem First, a matrix-based representation for PLNs is derived The obtained representation is an extension of a matrix-based representation for probabiistic Booean networks proposed in Next, using the matrix-based representation, Probem is approximated by a mixed integer inear programming (MILP) probem 4 Matrix-Based Representation for PLNs As a preparation, the notation is defined Binary variabes x 0 i (k), x i (k),,xr(i) i (k) are introduced If x i (k) = j hods, then x j i (k) = hods; otherwise xj i (k) = 0 hods Then, the equaity r(i) j= xj i (k) = is satisfied In addition, binary variabes u 0 i (k), u i (k),,us(i) i (k) are introduced If u i (k) = j hods, then u j i (k) = hods; otherwise uj i (k) = 0 hods Then, the equaity s(i) j= uj i (k) = is satisfied Using xj i (k), j=,,,r(i),andu j i (k), j=,,,s(i), define x i (k) f u i (k) f x 0 i (k) x i (k) {0, } r(i), (k) x r(i) i u 0 i (k) u i (k) {0, } s(i), (k) u s(i) i i=,,,n, i=,,,m In the case of r(i) =, x i (k) canbeobtainedas () x i (k) = x0 i (k) x i (k) = x i (k) () x i (k) First, consider transforming a given MVLN into a matrixbased representation Then, the matrix-based representation for x i (k+)is given by where x i (k+) =A (i) z i (k), () z i (k) f ( j N i x j (k)) ( j M i u j (k)) {0, } α(i), α (i) f j N i r (j) j M i s (j), A (i) {0, } r(i) α(i) (4) Tabe : Truth tabe for x x 0 (k) x (k) x0 (k) x (k) x (k) x0 (k + ) x (k + ) Tabe : Truth tabe for x x 0 x x 0 x x x 0 (k + ) x (k + ) x (k + ) The matrix A (i) can be derived from the truth tabe for x i (k+) using x j i (k), j=,,,r(i),anduj i (k), j=,,,s(i) We present a simpe exampe Exampe 4 Consider an MVLN with two states and no contro inputs For x D,theogicfunctionf () in Tabe is given For x D, the ogic function f () in Tabe is given Using x 0 (k), x (k) and x0 (k), x (k), x (k), ogic functions f () and f () can be expressed as truth tabes in Tabes and, respectivey From these truth tabes, we can obtain the foowing matrix-based representation of the MVLN: x0 (k+) x (k+) x (k+) x 0 (k) x0 (k) x 0 = (k) x (k) 0 x 0 (k) x (k) x (k) x0 (k), A () x (k) x (k) x (k) x (k) x 0 (k+) 0 0 x (k+) = z (k), x (k+) x (k+) A () z (k) (5) where z (k) = z (k)(= x (k) x (k)) EacheementofA (i), i =,,,isgivenbyabinaryvaue(0 or ), and a sum of a eements in each coumn of A (i) is equa to Such a matrix-based representation has been proposed in aso 6 8 The matrix-based representation for one x i,

5 Mathematica Probems in Engineering 5 in a given PLN is given by A (i) z i (k) Then, the expected vaue of x i (k + ) can be obtained as that is, (), may be derived by the method in 6 8 See aso Remark 9 Next, consider extending the matrix-based representation of an MVLN to that of a PLN Suppose that the matrixbased representation of the ogic function f (i) E x i (k+) = ( = c (i) (k) A (i) ) E z i (k), (6) where A (i) {0, } r(i) α(i), and the condition x(0) = x 0 in the expected vaue is omitted We remark here that the jth eement of Ex i (k + ) is the probabiity that x i (k) = j hods We present a simpe exampe Exampe 5 Consider the PLN in Exampe In this system, there are no contro input and no structura contro input Using the matrix-based representation, the expected vaue of x i (k + ) can be obtained as Ex (k+) = (07A () + 0A () )Ez (k), Ex (k+) = (08A () + 0A () )Ez (k), where z (k) = z (k) = x (k) x (k), A () = , A () = , 0 0 = , A () = A () (7) (8) 4 Reduction to an MILP Probem Using the matrix-based representation (6), consider transforming Probem First, we focus on the structura contro input V(k) 0, Using (8), the system (6) can be rewritten as Ex i (k+) =( = +( = (k) A (i) )Ez i (k) (k) A (i) ) V (k) Ez i (k) (9) Noting that the input constraint is imposed as V(k) 0,, the structura contro input V(k) canberegardedasthe expected vaue of some binary probabiistic variabe Ṽ(k) {0, }Then, V (k) Ez i (k) =Ez i (k) Ṽ (k) =Ez i (k)eṽ (k) (0) hods Here, according to the definitions of x i (k) and u i (k), we define Thus, we can obtain z i (k) f z i (k) Ṽ (k) Ṽ (k) Ex i (k+) =( = z i (k)( Ṽ(k)) z i (k) Ṽ (k) = +0 (k) A (i) )Ez i (k) = (k) A (i) E z i (k) () () Then,Probemcanbeequivaentyrewrittenasthefoowing probem Probem 6 Suppose that, for the PLN with the conventiona contro input u and the structura contro input V, theinitia state x(0) = x 0 is given Then, find u(0),u(),,u(n ) {0, } m and V(0), V(),, V(N ) 0, minimizing the cost function (9) subject to (0) and () Byasimpecacuation,Probem6canberewritten as a poynomia optimization probem However, it wi be difficut to sove a poynomia optimization probem for argescae PLNs In this paper, we focus on the reations among x i (k), u i (k), andz i (k) of (4) and z i (k) of () Using these reations, we derive the reaxed probem for Probem 6 The reaxedprobemisanmilpprobemandcanbesovedfaster than a poynomia optimization probem First, we present a simpe exampe Exampe 7 Consider Ez (k) in Exampe 5 Then, noting that Ex 0 (k) + Ex (k) + Ex (k) = hods, we can obtain Ex 0 (k)ex0 (k) Ex 0 (k)ex (k) Ex (k)ex (k) Ex (k)ex0 (k) Ex (k)ex (k) Ex (k)ex (k) =Ex 0 (k) Ez (k) ()

6 6 Mathematica Probems in Engineering In a simiar way, we can obtain Ex (k) Ex Ez (k) = (k) Ex (k) (4) Ex (k) In addition, a sum of a eements in Ez (k) is equa to Equaities obtained can be used as constraints in the reaxed probem Next, consider a genera case Noting that Eu j (k) = u j (k) hods, we can obtain the foowing constraints for Ez i (k): Ex jx (k) =C jx Ez i (k), j x N i ={j x,j x,,j x Ni }, j x <j x < <j x Ni, u ju (k) =D ju Ez i (k), j u M i ={j u,j u,,j u Mi }, j u <j u < <j u Mi, where (5) C jx = α(i)r(jx ) 0 d {0, } r(jx) α(i), 0 α(i)r(jx ) C jx =C j x C j x {0, } r(j x) α(i), α(i)(r(jx )r(j x )) 0 C j x = α(i)(r(jx )r(j x )) d {0, } r(j x) α(i)(r(j x )r(j x )), 0 α(i)(r(jx )r(j x )) C jx Ni =C j x Ni C j x Ni {0, }r(j x N i ) α(i), C j x Ni = α(i)(r(jx )r(j x ) r(j x Ni )) 0 d {0, } r(j x) α(i)(r(j x )r(j x ) r(j x Ni )), 0 α(i)(r(jx )r(j x ) r(j x Ni )) D ju =D j u D j u {0, } r(j u) α(i), (6) D j u = α(i)(r(jx ) r(j x Ni )r(j u )) 0 d {0, } r(j x) α(i)(r(j x ) r(j x Ni )r(j u )), 0 α(i)(r(jx ) r(j x Ni )r(j u )) D ju Mi =D j u Mi D j u Mi {0, }r(j u M i ) α(i), D j u Mi = α(i)(r(jx ) r(j x Ni )r(j u ) r(j u Mi )) 0 d 0 α(i)(r(jx ) r(j x Ni )r(j u ) r(j u Mi )) {0, } r(j x) α(i)(r(j x ) r(j x Ni )r(j u ) r(j u Mi ))

7 Mathematica Probems in Engineering 7 In a simiar way, we can obtain the foowing constraints for E z i (k): Ex jx (k) = C jx E z i (k), j x N i ={j x,j x,,j x Ni }, j x <j x < <j x Ni, u ju (k) = D j u E z i (k), j u M i ={j u,j u,,j u Ni }, j u <j u < <j u Mi, V (k) = D V V (k) E z i (k) (7) From the definition of z i (k),thatis,(),thematrices C jx, D j u can be obtained by repacing α(i) with α(i) in the matrices C jx,d ju Thematrix D V can be obtained as D V =I I {0, } α(i) (8) Thus, we can obtain the foowing probem as a reaxed probem of Probem 6 Probem 8 Suppose that, for the PLN with the conventiona contro input u and the structura contro input V, theinitia state x(0) = x 0 is given Then, find u(0),u(),,u(n ) {0, } m and V(0), V(),, V(N ) 0, minimizing the cost function (9) subject to (0) and () (7) By a simpe cacuation, Probem 8 can be equivaenty rewritten as an MILP probem In this MILP probem, continuous decision variabes are V(k), Ez i (k), E z i (k), k= 0,,,N, and binary decision variabes are u i (k), k= 0,,,N By soving the MILP probem, we can evauate the ower bound of the optima vaue of the cost function in Probem In this paper, we provide ony an approximate soutionmethodhowever,sincethecontroinputisobtained by soving an MILP probem, the proposed soution method enabes us to sove the optima contro probem for severa casses of PLNs In the case where there is no conventiona contro input, Probem 8 with ony the structura contro input is equivaent to a inear programing probem Remark 9 In anaysis/contro methods in 6 8, the matrixbased representation for x, whichcanbeobtainedfromthat for one x i,isusedasaresut,thematrixwiththesizeof n i= r(i) ( n i= r(i) m i=s(i)) must be manipuated In the proposed method, matrix A (i) with the size of r(i) α(i) is used (α(i) = j Ni r(j) j Ni s(j)) Thus, the proposed method enabes us to use ony matrices with the smaer size Remark 0 The difference between Probems and 8 can be evauated using nu spaces of matrices C j,d j, C j, D j,and D j Detais are one of future works If Ez i (k) and E z i (k) can be uniquey derived from (5) and (7), then Probems and 8areequivaent 5 Appication to Rea-Time Pricing of Eectricity In this section, we consider rea-time pricing of eectricity as an appication of optima contro of PLNs First, the outine of rea-time pricing is expained Next, decision making of consumers is modeed by a PLN Finay, a numerica exampe is presented 5 Outine The outine of rea-time pricing systems studied inthispaperisiustratedinfigurethissystemiscomposed of one controer (the independent system operator, ISO) and mutipe eectric consumers If eectricity conservation is needed, then the ISO decides to set the price to a high vaue To decrease the economic oad, consumers conserve eectricity As a resut, eectricity conservation is achieved Thus, rea-time pricing pays an important roe in energy management systems In the existing methods of rea-time pricing, the price at each time is given by a simpe function with respect to power consumptions and votage deviations and so on However, it is important to determine the eectricity price basedonpredictionusingamathematicamodeofeectricity consumption From this viewpoint, in, 4, probabiistic modes for rea-time pricing have been proposed On the other hand, in, a method to mode decision making of consumers by a PBN has been proposed However, the state of a consumer is given by a binary vaue, that is, 0 (a consumer conserves eectricity) or (a consumer normay uses eectricity) In this paper, we consider extending the method in to the method using a PLN As a mathematica mode of consumers, a PLN is more appropriate than a PBN It is one of the future efforts to discuss the reation between probabiisticmodesin,4andapln 5 Mode Consider modeing the set of consumers as a PLN The number of consumers is given by n Wesuppose that the state x i for consumer i {,,,n} is d-vaued; that is, x i D d (for simpicity, d does not depend on i) The state impies x i 0 consumer i conserves eectricity, {, = { { d consumer i maximay uses eectricity (9) The vaue of x i is determined from power consumption of consumer ifor consumeri, we consider the foowingpln: x i (k+) f (i) =x i (k) +, c (i) { = f (i) =x i (k), c (i) { { f (i) =x i (k), c (i) (k) =a(i) +b(i) V (k) (x i (k) =d ), (k) =a(i) +b(i) V (k), (k) =a(i) +b(i) V (k) (x i (k) =0) (0)

8 8 Mathematica Probems in Engineering 4 Controer Price Power consumption Consumers (eg, homes and commercia faciities) Average trajectory Figure : Iustration of rea-time pricing systems In this mode, there is no conventiona contro input, and the structura contro input V(k) corresponds to the eectricity price The function f (i) impies that power consumption of consumer i increases The function f (i) impies that power consumption of consumer i is not changed The function f (i) impies that power consumption of consumer i decreases Thus, decision making of consumers can be modeed by a PLN In, 4, it has been shown that the behavior of power consumption is probabiistic based on experimenta data Hence, it is appropriate to adopt a PLN as a mode of consumers We remark that the above functions are an exampe of modes for decision making Depending on rea situations, we may use other ogic functions Using the PLN-based mode obtained, we consider the foowing probem: (i) find a time sequence of the price such that consumers conserve eectricity as much as possibe However, it is not desirabe that the price is too high The condition that consumers conserve eectricity as much as possibe can be characterized by Ex i In other words, power consumption is expressed by Ex i Hence,thisprobemcan be formuated as Probem by appropriatey setting the weights Q and R 5 Numerica Exampe We present a numerica exampe Here, Probem 8 is repeatedy soved according to the foowing procedure of mode predictive contro (MPC) Procedure of MPC Step Set t=0,andgivex(0) = x 0 Step Derive V(t), V(t+),, V(t+N ) by soving Probem 8 Step Appy ony V(t) to the system Step 4 According to Ex i (k+) (ie, the discrete probabiity distribution for x i (k+)), obtained by using V(t), derive x i (k+ ) D i randomy Set t+ t,andreturntostep Time Figure : Average state trajectory of 00 sampes Consider the case of n=0and d=5 See the Appendix for parameters and ForV(k), theconstraintv(k) 0, 06 is imposed Under this constraint, (0) are satisfied Parameters in Probem are given as foows: x (0) = , N=0, Q= 0, R = () Next, we present the computation resut According to the above procedure of MPC, 00 sampes of the state trajectory in the time interva 0, 0 were generated Figure shows the average trajectory of 00 sampes Figure shows one sampe of the price (the structura contro input) obtained From thesefigures,weseethatthestatebecomessmabymanipuating the price In this exampe, 000 LP probems were soved Then, the worst computation time was 0548 (sec), and the mean computation time was 097 (sec), where we used Gurobi Optimizer 56 as an LP sover Thus, Probem 8canbesovedfast Finay, we discuss the optimaity Consider soving Probem 8 once under the above setting When the obtained contro input is appied to the system, the vaue of the cost function in Probem was 548InthecaseofV(k) = 0 (ie, the constant input), the vaue of the cost function in Probem was669 InthecaseofV(k) = 06, the vaue of the cost function in Probem was 556 From these vaues,we see thattheobtainedcontroinputismoreeffectivethantrivia contro inputs 6 Concusion In this paper, we discussed optima contro of probabiistic ogic networks (PLNs) and its appication to rea-time pricing of eectricity First, in the probem formuation, both

9 Mathematica Probems in Engineering 9 Price = 07, = 05, = 0, = 05, =0, =, i {, 7,, 7}, Time Figure:Onesampeofthepriceobtained the conventiona contro input and the structura contro input were introduced Next, an approximate soution method was proposed This method is a sophisticated and generaizedversionofthemethodproposedinfinay, we considered rea-time pricing of eectricity as an appication, and we presented a numerica exampe The proposed method provides us a new contro method for arge-scae compex systems There are severa open probems It is significant to deveop the identification method of PLNs It is aso significant to evauate the accuracy of an approximation from the theoretica viewpoint Appendix Parameters in Numerica Exampe Parameters foows: and = 05, = 05, = 05, = 05, =0, =, in the numerica exampe are given as i {, 6,, 6}, = 04, = 05, = 06, = 05, =0, =, = 05, = 04, = 05, = 06, =0, =, = 0, = 0, = 07, = 07, =0, =, i {, 8,, 8}, i {4, 9, 4, 9}, i {5, 0, 5, 0} (A) These parameters in the numerica exampe are given artificiay Athough finding these parameters from rea data is important, this topic is one of the future efforts

10 0 Mathematica Probems in Engineering Confict of Interests The authors decare that there is no confict of interests regarding the pubication of this paper Acknowedgment ThisresearchwaspartysupportedbyJST,CREST References P Tabuada, Verification and Contro of Hybrid Systems,Springer, New York, NY, USA, 009 H D Jong, Modeing and simuation of genetic reguatory systems: a iterature review, Journa of Computationa Bioogy, vo9,no,pp67 0,00 S A Kauffman, Metaboic stabiity and epigenesis in randomy constructed genetic nets, Journa of Theoretica Bioogy, vo, no, pp , A Mochizuki, An anaytica study of the number of steady states in gene reguatory networks, Journa of Theoretica Bioogy,vo6,no,pp9 0,005 5 C Chaouiya, A Nadi, E Remy, and D Thieffry, Petri net representation of muti-vaued ogica reguatory graphs, Natura Computing,vo0,no,pp77 750,0 6DCheng,HQi,andZLi,Anaysis and Contro of Booean Networks: A Semi-Tensor Product Approach, Springer, London, UK, 0 7 D Cheng and X Xu, Bi-decomposition of muti-vaued ogica functions and its appications, Automatica, vo 49, no 7, pp , 0 8 Y Zhao, Z Li, and D Cheng, Optima contro of ogica contro networks, IEEE Transactions on Automatic Contro,vo56,no 8, pp , 0 9 I Shmuevich, E R Dougherty, S Kim, and W Zhang, Probabiistic Booean networks: a rue-based uncertainty mode for gene reguatory networks, Bioinformatics,vo8,no,pp6 74, 00 0 Z B Liu, Y Z Wang, and H T Li, Two kinds of optima contros for probabiistic mix-vaued ogica dynamic networks, Science China Information Sciences, vo57,no5,pp 0, 04 Y Zhao and D Cheng, Controabiity and stabiizabiity of probabiistic ogica contro networks, in Proceedings of the 5st IEEE Conference on Decision and Contro (CDC ), pp , Maui, Hawaii, USA, December 0 K Kobayashi and K Hiraishi, Optima contro of gene reguatory networks with effectiveness of mutipe drugs: a Booean network approach, BioMed Research Internationa, vo0, Artice ID 4676, pages, 0 KKobayashiandKHiraishi, Aprobabiisticapproachtocontro of compex systems and its appication to rea-time pricing, Mathematica Probems in Engineering, vo04,articeid 90677, 8 pages, 04 4 I Shmuevich, E R Dougherty, and W Zhang, Contro of stationary behavior in probabiistic Booean networks by means of structura intervention, JournaofBioogicaSystems, vo 0, no 4, pp 4 445, 00 5 Y Xiao and E R Dougherty, The impact of function perturbations in Booean networks, Bioinformatics,vo, no0,pp 65 7, K Kobayashi and K Hiraishi, ILP/SMT-based method for design of Booean networks based on singeton attractors, IEEE/ACM Transactions on Computationa Bioogy and Bioinformatics,vo,no6,pp5 59,04 7 X Chen, T Akutsu, T Tamura, and W-K Ching, Finding optima contro poicy in probabiistic Booean networks with hard constraints by using integer programming and dynamic programming, in Proceedings of the IEEE Internationa Conference on Bioinformatics and Biomedicine (BIBM 0),pp40 46, Hong Kong, December 00 8 S Borenstein, M Jaske, and A Rosenfed, Dynamic Pricing, Advanced Metering, and Demand Response in Eectricity Markets, Center for the Study of Energy Markets, UC Berkeey, 00 9 M Roozbehani, M Daheh, and S Mitter, On the stabiity of whoesae eectricity markets under rea-time pricing, in Proceedings of the 49th IEEE Conference on Decision and Contro (CDC 0), pp 9 98, IEEE, Atanta, Ga, USA, December 00 0 P Samadi, A-H Mohsenian-Rad, R Schober, V W S Wong, and J Jatskevich, Optima rea-time pricing agorithm based on utiity maximization for smart grid, in Proceedings of the st IEEE Internationa Conference on Smart Grid Communications (SmartGridComm 0), pp 45 40, Gaithersburg, Md, USA, October 00 C Vivekananthan, Y Mishra, and G Ledwich, A nove rea time pricing scheme for demand response in residentia distribution systems, in Proceedings of the 9th Annua Conference of the IEEE Industria Eectronics Society (IECON ), pp956 96, Vienna, Austria, November 0 K Kobayashi and K Hiraishi, Optima contro of probabiistic Booean networks using poynomia optimization, IEICE Transactions on Fundamentas of Eectronics, Communications and Computer Sciences,vo95,no9,pp5 57,0 K Kobayashi and K Hiraishi, Agorithm for optima rea-time pricing based on switched Markov chain modes, in Proceedings oftheieeepower&energysocietyinnovativesmartgrid Technoogies Conference (ISGT 5), pp 5, Washington, DC, USA, Feburary 05 4 I Maruta and Y Takarada, Modeing of dynamics in demand response for rea-time pricing, in Proceedings of the 5th IEEE Internationa Conference on Smart Grid Communications (SmartGridComm 4), pp 806 8, Venice, Itay, November 04

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