Dynamic Pricing Using H Control with Uncertain Behavior in Electricity Market Trading

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1 SICE Journal of Control, Measurement, and System Integraton, Vol. 9, No. 5, pp , September 216 Dynamc Prcng Usng H Control wth Uncertan Behavor n Electrcty Maret Tradng Yoshhro OKAWA, and Toru NAMERIKAWA, Abstract : Ths paper deals wth a dynamc prcng usng the H control wth uncertan behavor n electrcty maret tradng. The dynamc prcng s a decson procedure of an electrcty prce based on power supply and demand balance n power grds. However, n a future deregulated electrcty maret, both power consumers and generators determne ther own power demand and supply selfshly accordng to the prce nformaton. Moreover, uncertantes are also ncluded n ther behavor. For ths problem, we propose a novel prce decson method usng the locatonal prce-updatng H controllers. Ths paper shows a desgn process of ths prce-updatng H controller to evaluate the followng performance to reduce supply-demand mbalances n power grds and the robustness aganst uncertantes n electrcty maret partcpants behavor, respectvely. In addton, ths paper also shows the effectveness of our proposed prce decson method usng the desgned H controller through numercal smulaton results. Key Words : smart grd, dynamc prcng, H control, AC model of power grd. 1. Introducton As the world populaton and energy consumpton ncrease, energy problems become more serous n the world. In partcular, effcent use of the lmted energy resources s strongly requred. Under such world affars, electrcty deregulaton has an mportant role to solve ths crucal problem. In the deregulated electrcty maret, power consumers as well as power generators partcpate n maret tradng. As a result, we can acheve to use or generate energy more effcently [1]. However, power balance between ts supply and demand n power grds should be ept at any tme to ensure stable operaton of power systems. Therefore, dstrbuted energy management systems are requred to realze a power networ wth ths deregulated maret. For ths problem, dynamc prcng s one of the useful methods to manage the deregulated power networ n a dstrbuted manner. In the dynamc prcng, electrcty power prces are changed by the short tme nterval accordng to the power supply and demand balance n power grds. Because of ths property, the dynamc prcng becomes a qute effectve method to acheve both effcent and economcal use of energy [2]. On the other hand, some useful methods usng electrcty prces for power grd management have already been proposed n the optmal power flow problem (OPF) or locatonal margnal prce problem (LMP) [3],[4]. These methods derved the optmal power supply of each generator whch mnmzes the total generatng cost n power grds. In addton, recent studes of the dynamc prcng proposed prce decson methods to maxmze socal welfare of the entre power networ by usng utlty func- School of Integrated Desgn Engneerng, Graduate School of Scence and Technology, Keo Unversty, Hyosh, Kohou-u, Yoohama, Kanagawa , Japan JST CREST, Honcho, Kawaguch, Satama , Japan E-mal: yoshhroo@nl.sd.eo.ac.jp, namerawa@sd.eo.ac. jp (Receved October 15, 215) (Revsed Aprl 11, 216) tons of consumers [5] [8]. In partcular, the papers [6],[7] proposed dstrbuted prce decson methods consderng power flow wth maret tradng by usng the dual decomposton. However, most of these algorthms n the dynamc prcng are based on the optmal power generatng or consumng actons of maret partcpants. In other words, these methods do not consder uncertantes n ther behavor. For ths problem, the paper [9] proposed a power supply-demand management method consderng uncertantes n the power supply from renewable energy generators. Ths method acheved to cover the power shortage of renewable generators by usng electrcty prces n a real-tme maret. Also, the paper [1] dscussed a power adjustment method wth negatve-watt tradng between power consumers and the maret manager. However, most of the power management methods proposed n these lteratures are based on the gradent method, and thus they requre the utlty and cost functons to be convex or concave functons, whch represent maret partcpants behavor. Here, n a future power grd, smart grd, smart meters enable us to communcate the data of our power consumng or generatng results n real tme. Therefore, n order to realze fast demand response based on these real-tme nformaton, ts prce decson method s requred to have the robustness aganst uncertantes n behavor of maret partcpants and the followng performance to reduce power mbalance n a power grd at the same tme even f t uses some nformaton on ther utlty and cost functons. In ths paper, we present a dynamc prce decson method usng the H control consderng uncertan behavor of maret partcpants n electrcty maret tradng. Untl now, some lteratures have dscussed the way n whch electrcty prces are used as feedbac control sgnals to stablze power systems [11] [13]. As compared wth these control methods, the H control used n our study s a robust control method aganst uncertantes and we can desgn ts controller to satsfy our desred control specfcaton wth a generalzed plant ncludng approprate weghtng functons. Based on these characterstcs JCMSI 5/16/ c 215 SICE

2 SICE JCMSI, Vol. 9, No. 5, September of the H control, the goal of our study s to desgn a locatonal electrcty prce-updatng H controller for dynamc prcng, whch evaluates the followng performance to reduce power mbalance n power grds and the robustness aganst uncertantes n maret partcpants behavor at the same tme. As a result of usng these desgned controllers n the locatonal electrcty prce updatng, our proposed method becomes a qute effectve prce decson method n the real-tme dynamc prcng problem wth uncertantes n maret partcpants behavor. The paper s organzed as follows. Frst, we show the behavor models of selfsh power consumers and generators wth uncertantes n Secton 2. Ths secton also shows an AC power grd model used n ths paper. Next, Secton 3 shows the state space representaton of the prce decson problem wth uncertantes n maret partcpants behavor. In addton, we show the generalzed plant of ths prce decson problem and derve the H controller for locatonal electrcty prce updatng based on power mbalance n power grds. Fnally, the Secton 4 presents the numercal smulaton results wth IEEJ EAST 3-machne power grd model to verfy our proposed method. 2. Problem Formulaton Fgure 1 shows a power grd model wth an electrcty maret consdered n ths paper. Ths model has multple areas and power flow occurs between these areas. Also, n ths paper, suppose that there are three nds of maret partcpants: power consumers, power generators, and an ndependent system operator (ISO). Among these partcpants, the ISO s a nonproft organzaton who s responsble for operatng the electrcty maret and organzng power grds. Then, n ths electrcty maret, power consumers and generators determne ther own power demand and supply selfshly accordng to the electrcty prces nformed from the ISO, whle the ISO updates these prces to eep power balance n power grds. where v (x), c (x) and λ are the utlty functon of consumers [5], the cost functon of generators and the electrcty prce per unt n Area, respectvely. However, because of varous problems n ther daly lves, the behavor of these power consumers and generators do not always follow the above equatons. Moreover, especally n the case of the consumers, ther utlty functons nclude uncertantes n therselves snce these functons depend on uncertan human behavor. Then, n order to consder these uncertantes n the above maret partcpants behavor, ths paper represents the actual power demand and supply n each area, ˆd and ŝ wth the followng equatons, respectvely. ˆd = (1 +Δ d w d )d, ŝ = (1 +Δ s w s )s, A. (3) In the above equatons, Δ d w d and Δ s w s are coeffcents whch represent uncertantes n the power consumng and generatng actons by maret partcpants n Area. Now the followng assumpton s ntroduced for the utlty functons v (x) and the cost functons c (x) used n ths paper. Assumpton 1 Utlty functons v ( ) and cost functons c ( ) are n C 2 [, )forall A. In general, utlty functons are represented by usng logarthmc or square root functons [6],[7] and cost functons are by usng quadratc functons [3]. Therefore, ths assumpton s satsfed n most cases of the dynamc prcng problem. 2.2 Power Grd Model wth AC Generators The power grd model consdered n ths paper s shown n Fg. 2. As shown n ths fgure, power grds have a large number of nodes, generators, transmsson lnes and the other electrcal machnery and apparatus. Ths paper dvdes such a huge power grd model nto L small areas. In addton, let n be the number of nodes n Area A. Furthermore, we ntroduce Fg. 1 Power grd model wth an electrcty maret. 2.1 Behavor Models of Power Consumers and Generators Ths subsecton shows behavor models of power consumers and generators n electrcty maret tradng. As mentoned above, power consumers and generators determne ther own power demand and supply selfshly accordng to an electrcty prce of each area. Now let L be the number of areas n the power grd model. Then, accordng to [5] [7], the optmal actve power demand of consumers and the optmal actve power supply of generators n Area, d and s, A, A := {1, 2,, L}, aregvenby Fg. 2 Power grd model wth L areas. d = arg max v (x) λ x, x (1) s = arg max λ x c (x), x (2) Fg. 3 Area connecton between Areas and j.

3 194 SICE JCMSI, Vol. 9, No. 5, September 216 Fg. 4 Structure of a dstrbuted dynamc prce decson problem wth multple areas n power grds. the followng assumptons to ths power grd model n order to smplfy power flow equatons among areas. Assumpton 2 The power grd s assumed to satsfy the followng propertes: ) Resstance loss n the transmsson grd s neglgble. ) The voltage of each node approxmately equals to 1 [p.u.]. ) The voltage phase dfference between each node s suffcently small. Wth these assumptons, the DC approxmaton can be appled to the power grd model, and then the lnearzed actve power flow from Area to j becomes as follows [14]. P j = B j l (θ θ jl ),, j A, (4) (, j l ) N j where θ and θ jl ( =1, 2,...,n, l=1, 2,...,n j ) are the voltage phase angles of node and j l,andb j l s the susceptance of the transmsson lne between these nodes, respectvely. Also, N j s the set of nodes whch are connected wth lnes between Areas and j. For nstance, N j becomes N j = {( 3, j 2 ), ( 4, j 3 )} n the power grd model shown n Fg. 3. As a result, the actve power flow equaton n Area becomesasfollows: ŝ ˆd = j A P j = j A (, j l ) N j B j l (θ θ jl ), A, (5) where ˆd and ŝ are the power demand and supply n Area whch are defned n (3) and A s the set of neghborng areas of Area, respectvely. By summarzng the above equatons n each area, the actve power flow equaton of the entre power networ becomes ŝ + Bθ = ˆ d, (6) where ŝ = [ŝ 1 ŝ L ] T R L, ˆ d = [ ˆd 1 ˆd L ] T R L and θ = [θ T 1 θt L ]T R N (N := L =1 n ), θ := [θ 1 θ n ] T R n, A,and B R L N s gven by B := B 11 B 1L , (7) B L1 B LL where [ ] B 1 B n R 1 n ( j = ) [ ] B j := B j1 B jn j R 1 n j ( j, j A ), (8) 1 n j ( j, j A ) := B j l ( = 1, 2,...,n ), (9) B B jl := j A j l N j j l N j B j l (l = 1, 2,...,n j ). (1) In (9) and (1), N j s the set of nodes n Area j whch are connected wth node n Area. Ths set becomes as follows n the power grd model n Fg. 3. { j 2 } ( = 3) N j = { j 3 } ( = 4). (11) φ (otherwse) Furthermore, let λ = [λ 1 λ L ] T R L be the vector whch summarzes locatonal electrcty prces and f (θ )bethecost functon to change the voltage phase angle of node n Area A. Then, the ISO determnes the voltage phase angles θ by solvng the cost mnmzaton problem: θ = arg mn θ n =1 f (θ ) L λ j B j θ, A. (12) Snce ths optmzaton problem s solved by the ISO, we assume that there s no uncertanty n ths problem. In addton, we ntroduce the followng assumpton. Assumpton 3 The cost functons f (θ )arenc 2 ( π, π) for all Aand {1, 2,...,n }. j=1

4 SICE JCMSI, Vol. 9, No. 5, September Dynamc Prcng Usng the H Control Ths secton presents a dynamc prce decson method usng the H control. Here, the H control s one of the robust control methods aganst uncertan systems and many sgnfcant results have been obtaned by usng ths control method n varous nds of problems whch nclude uncertantes n ther systems. Then, ths paper consders descrbng the prce decson problem ncludng uncertantes n maret partcpants behavor wth a state space representaton and desgnng the H controller to update locatonal electrcty prces. Fgure 4 shows a model of the dstrbuted prce decson problem consdered n ths paper. As descrbed n the prevous secton, power consumers and generators n each area determne ther own demand or supply ncludng uncertantes accordng to the locatonal electrcty prce, λ, A. In addton, the ISO also determnes the voltage phase angles n each area based on the prces n that and the other neghborng areas. By usng these values and power flow from the other neghborng areas, our proposed method updates the locatonal electrcty prces n a dstrbuted manner n each area. 3.1 State Space Representaton of the Prce Decson Problem In order to desgn an H controller to update locatonal electrcty prces, frst, ths subsecton presents the state space representaton of the prce decson problem consdered n ths paper. Let us defne λ as a state, whch s the prce n Area, u as an nput for the prce updatng n Area and y as a measured output whch s the power mbalance n Area. Then, the update equaton of the electrcty prce n Area Acan be represented wth the followng dscrete-tme equatons: λ +1 = λ + u, (13) y = ŝ ˆd j A P j = ŝ ˆd + B θ + B j θ j. (14) j A Note that the optmal power supply and demand of generators or consumers n each area are determned accordng to the prce n ts area as descrbed n (1) and (2). Therefore, these two values, s and d, Acan be represented by s = ċ 1 (λ ), d = v 1 (λ ). (15) On the other hand, from (12), we obtan the followng equaton regardng the voltage phase angle n Area, θ, A: n L f l (θ l ) λ θ l j B j θ l=1 j=1 = n L f l (θ l ) λ θ l B θ λ j B j θ l=1 = f l (θ l ) λ B l = f l (θ l ) λ B l L j=1, j j A λ j λ j B jl j=1, j B jl. (16) Ths equaton becomes when the voltage phase angle n node l s equal to the optmal soluton of the cost mnmzaton problem (12). Therefore, the optmal voltage phase angle of node l accordng to the prces λ := [ λ 1 T L] λ s gven by = f 1 l B l λ + B jl λ j. (17) θ l Now let θ := j A [ θ 1 θ n ] T R n be the optmal voltage phase angles n Area A. Then, from the above equaton, θ becomes as follows: θ = F (λ ), (18) where F (λ ) = [ F 1 (λ ) F n (λ )] T, (19) F l (λ ):= f 1 l B l λ + B jl λ j. (2) j A Usng (15) and (18), the output equaton (14) becomes y = H g (λ ) + w + v, (21) g (λ ):= [ F (λ )T ċ 1 (λ ) v 1 (λ )] T R n +2, (22) H := [ B 1 1 ] R 1 (n+2). (23) In the above equaton, w R 1 and v R 1 are, respectvely, defned as follows: w := Δ s w s s Δ d w d d, (24) v := B j θ j (25) j A 3.2 Generalzed Plant for the Prce Decson Problem Next, n ths subsecton, we consder constructng a generalzed plant wth the descrbed state space representaton of the prce decson problem and weghtng functons so that the desgned H controller satsfes the desred control specfcaton. From the prevous subsecton, the state space representaton of the prce decson problem wth uncertantes s gven by λ +1 = λ + u, (26) y = H g (λ ) + w + v. (27) Lnearzaton usng the Taylor expanson In the above equaton, the functon g (λ ) ncludes the nverse functons of the utlty and cost functons of maret partcpants as shown n (22), and thus t sometmes becomes a nonlnear functon. Then, n the proposed method, we apply the Taylor expanson to the functon g (λ ) around the ntal prce λ, A. The lnearzed functon G R n+2 s gven by G := g λ λ =λ = F (λ )T ċ 1 (λ ) v 1 (λ ) T. (28) λ λ λ λ =λ As a result, the lnearzed state equaton of the prce decson problem becomes as follows: λ +1 = λ + u, (29) y = C λ + w + v, (3) where C := H G, v := v C λ + H g (λ ).

5 196 SICE JCMSI, Vol. 9, No. 5, September Weghtng functon As descrbed n the ntroducton of ths paper, we can desgn an H controller to satsfy desred control specfcaton by choosng weghtng functons approprately [15]. Accordng to ths characterstc of the H control, we regard the prce decson problem consdered n ths paper as a mxed senstvty problem so that the derved prces reduce power mbalance n power grds wth the robustness aganst uncertantes n maret partcpants behavor. Let z 1 (z) be the evaluated output regardng the followng performance to reduce power mbalance and z 2 (z) be the evaluated output regardng the robustness aganst uncertantes n maret partcpants behavor n Area A, respectvely. Then, the transfer functons regardng the weghtng functons W s (z)and W t (z) are, respectvely, gven by z 1 (z) = W s (z)u ws (z) (31) W s (z) = C ws (z A ws ) 1 B ws + D ws, z 2 (z) = W t (z)u wt (z) (32) W t (z) = C wt (z A wt ) 1 B wt + D wt, where u ws and u wt are nput values to these functons as shown n Fg. 5. Then, the dscrete-tme state equatons of the above two weghtng functons become xws +1 = A ws xws +B ws u ws, (33) z 1 =C ws xws +D ws u ws xwt +1 = A wt xwt +B wt u wt, (34) z 2 =C wt xwt +D wt u wt where xws R 1 and xwt R 1 are the states of the above two weghtng functons W s and W t, respectvely. Furthermore, n order to remove the steady-state devaton caused by the lnearzaton descrbed n the prevous subsecton, we ntroduce the followng weghtng functon M (z) [15]. Smlar to (31) and (32), the transfer functon of M (z) sgven by ȳ (z)= M (z)u m (z), (35) M (z)=c m (z A m ) 1 B m +D m s. t. M (z)w s (z)= KT s z 1, (36) where u m s an nput value to ths weghtng functon as shown n Fg. 5. Moreover, n the above equaton, K and T s are a control gan and a samplng tme, respectvely. Now let xm R 1 be the state of the weghtng functon M, and then the dscretetme state equaton of ths functon becomes xm +1 = A m xm + B m u m. (37) ȳ = C m xm + D m u m Generalzed plant of prce decson problem Ths subsecton dscusses how to construct the generalzed plant of the prce decson problem wth uncertan behavor of power consumers and generators n electrcty maret tradng. Fgure 5 shows the model of the generalzed plant regardng the prce decson problem consdered n ths paper. Accordng to ths model, the nputs for each weghtng functon, u ws, u wt and u m, can be represented as follows: Fg. 5 Generalzed plant of prce decson problem. u ws = D m C λ + C m xm + D m w, u wt = C λ, (38) u m = C λ + w. Then, wth the dscrete-tme state equatons of the lnearzed prce decson problem n (29) and (3), those of the weghtng functons n (33), (34) and (37), and the nputs n (38), the state space equaton of the generalzed plant regardng the prce decson problem consdered n ths paper becomes x +1 = A x + B 1 w + B 2 u, (39) z = C 1 x + D 11 w + D 12 u, (4) ȳ = C 2 x + D 21 w, (41) where 1 B A := ws D m C A ws B ws C m R 4 4, B wt C A wt B m C A m 1 B B 1 := ws D m R 4 1, B 2 := R 4 1, B m [ ] Dws D C 1 := m C C ws D ws C m R 2 4, D wt C C wt [ ] [ ] Dws D D 11 := m R 2 1, D 12 := R 2 1, C 2 := [ ] D m C C m R 1 4, D 21 := D m R 1. The state x := [ ] T λ xws xwt xm R 4 contans the prce of each area λ R 1 and the states of the weghtng functons W s (z), W t (z) andm (z). In addton, ȳ R 1 and z := [ ] T z 1 z 2 R 2 are the measured output and the evaluaton output, respectvely. Now we let G (z) andk (z), A be the transfer functon matrces of the generalzed plant (39) (41) and ts dynamc feedbac controller n Fg. 5, respectvely. In addton, we defne ts closed loop system Φ (z) as follows: Φ (z) = LFT (G (z); K (z)), A. Then, the followng assumpton s ntroduced to the above generalzed plant, whch ensures the exstence of the stablzng controller for the closed loop system Φ (z) [15]. Assumpton 4 (A, B 2 ) s stablzable and (C 2, A )sdetectable, A.

6 SICE JCMSI, Vol. 9, No. 5, September Note that the coeffcent matrces n the above generalzed plant n (39) (41) consst of the coeffcents of ther weghtng functons n (33), (34) and (37). Therefore, by choosng these weghtng functons approprately, we can satsfy ths assumpton and mae the closed loop system Φ (z) nternally stable. 3.3 Desgn of H Feedbac Controller Wth the generalzed plant of a dynamc prcng problem descrbed n the prevous subsecton, ths subsecton consders desgnng the H controller to update locatonal electrcty prces based on power mbalance n a power grd. Frst, we defne the dynamc output feedbac controller used n our problem as follows: x +1 c = A c x c + B c ỹ, (42) u = C c x c + D c ỹ, (43) where x c R 4 s the state of the controller and ỹ R 1 s defned as ỹ := ȳ + v. Furthermore, we denote a system matrx of the above controller by K as follows: [ ] Ac B K := c. (44) C c D c Here, our control objectve s to adjust power mbalance n a whole power networ by reducng the nfluence of the uncertantes n maret partcpants behavor n a dynamc prcng problem. Then, as we descrbed n the prevous subsecton, Φ (z) s a closed-loop transfer functon consstng of the generalzed plant G (z) and ts stablzng controller K (z), whch represents the nfluence of the uncertantes n maret partcpants behavor on the followng performance to reduce power mbalance and the robustness aganst ther uncertantes n each area. Therefore, the H control problem of the prce decson problem consdered n ths paper s to fnd the controller K (z), A whch satsfes the followng condton for a gven γ >, A: Φ (z) <γ. (45) Note that a more effectve H controllers would be obtaned from the H cost for a whole power networ nstead of the local H cost n (45). However, when we try to desgn such an H controller, the dmenson of ts controller ncreases as the number of areas n the networ ncreases. In order to avod ths problem, ths paper sets the local H cost functon for each area and desgns ts local H controller to update locatonal electrcty prces as we descrbed n ths subsecton. 3.4 Electrcty Prce Decson Algorthm Usng H Control Through the prevous subsectons, we present a desgn process of the locatonal prce-updatng H controller consderng uncertan behavor n electrcty maret tradng. Then, the electrcty prce decson maret algorthm usng ths prce-updatng H controller becomes as follows. Algorthm: Dynamc prcng algorthm usng H control Step : Controller desgn The ISO desgns the H controller by usng the pre-reported utlty and cost functons of power consumers and generators wth approprate weghtng functons and derves the system matrx of the controller K, A. Step 1: Decson of the ntal prce The ISO arbtrary determnes the ntal prces λ, A and nforms them to the consumers and generators n each area. Step 2: Decson of demand, supply and voltage phase angle Based on the nformed prce λ ( ), the power consumers and generators n each area determne ther power demand ˆd or supply ŝ, respectvely. Furthermore, the ISO determnes the voltage phase angles of the nodes n each area θ, Aby solvng the followng cost mnmzaton problem: θ = arg mn θ n =1 f (θ ) L j=1 λ j B j θ. (46) Step 3: Update of the prce Accordng to the measured power mbalance n each area r := ŝ ˆd + B θ + j A B j θ j v, the ISO calculates the measured output n each area ỹ, Aas follows: A m ỹ 1 +D m r ỹ := +(C m B m D m A m )r 1 + v ( 1). (47) r + v ( = ) Then, usng ths measured output ỹ, the ISO updates the state of the controller x c and derves the nput for prce update u as the followng: x +1 c = A c x c + B c ỹ, (48) u = C c x c + D c ỹ. (49) Fnally, the ISO updates the prce n each area accordng to the followng equaton wth the nput u derved from (49) and nforms updated prces λ +1 to the power consumers and generators n each area: λ +1 = λ + u. (5) Step 4: Repetton Change the teraton number to +1 and go bac to Step Smulaton Verfcaton In ths secton, we verfy the effectveness of the proposed prce decson method usng the H control through the results of a numercal smulaton. 4.1 Smulaton Condton Fgure 6 shows the power grd model used n ths smulaton. We use the IEEJ EAST 3-machne model [16] and dvde ths model nto four areas. In addton, the pea load of each area n Table 1 and the other physcal parameters of ths power grd model are determned based on the parameters gven n [16]. Table 1 Grd condtons n Areas 1 4. Area 1 Area 2 Area 3 Area 4 pea load (MW) number of node neghborng areas {2} {1, 3, 4} {2} {2}

7 198 SICE JCMSI, Vol. 9, No. 5, September 216 ( ) v 1,t (λ a,t ) = μ 1 d,t + μ 2 1, ċ 1 (λ ) = 1 λ, λ 2b f 1 ( B λ ) = B λ, A. (54) 2ζ Fg. 6 IEEJ EAST 3-machne system wth 4 areas. Moreover, we assume that there are three nds of consumers; resdental, commercal, and ndustral consumers. In ths smulaton, suppose that each maret partcpant has demand response equpment and t automatcally determnes the power demand or supply n every mnute accordng to the electrcty prces nformed from the ISO. Furthermore, we use the followng functons as a utlty functon of consumers v,t (d ) and a cost functon of generators c (s ), A, respectvely: ( ) d μ 1 d,t v,t (d ) = a,t μ 2 log + 1, c (s )=b s 2 μ. (51) 2 In the above equaton, μ 1 d,t denotes the nelastc power demand of consumers n each area between tme t and t+1 (h). By usng such tme-varyng utlty functons, we can smulate the stuaton n whch consumers have tme-varyng tendences n ther power consumng behavor. Moreover, n order to represent the locatonal power consumng tendences, d,t s determned accordng to the followng equaton: d,t = { R R(t)+ C C(t)+ I I(t)}d pea, (52) s. t. R + C + I =1, X 1, X {R, C, I }, where d pea s the pea load n each area shown n Table 1, R, C, and I are the ratos for each nd of consumers n each area, and R(t), C(t), and I(t) are ther tme-varyng power consumng tendences, respectvely. Ths smulaton sets these ratos to ( R, C, I ) = (.5,.4,.1), (.3,.6,.1), (.6,.2,.2), (.1,.3,.6) n Areas 1 4, and R(t), C(t), and I(t) to the parameters gven n [16]. The coeffcents of the utlty functon and cost functon, a,t and b, are determned by usng a fxed electrcty prce λ f. Specfcally, they are gven by ( ) d μ 1 d,t a,t =λ f +1, b = λ f, A. (53) μ 2 2d Among the above functons, v 1,t ( ) s a nonlnear functon regardng λ. Then, we appled the Taylor expanson to ths functon as descrbed n Secton 3. As a result, the lnearzed observaton functon G n (28) s gven by G = GT f 1 2b [ B 1 μ 2a,t λ 2 T R n +2, A, (55) ] B T n where G f := 2ζ 1 2ζ n R n, A. From the above equaton, C, Abecomesasfollows: C = H G = n =1 B 2 2ζ μ 2a,t. (56) 2b λ 2 Next, we show the weghtng functons used n ths smulaton. In order to desgn the H controller to satsfy our desred control specfcaton, we let the weghtng functons, W s (z)and W t (z), have a role as a low-pass or a hgh-pass flter, respectvely. Specfcally, we use the followng weghtng functons wth the samplng tme T s = 6 s, wth whch the magntude of ts senstvty functon and complementary senstvty functon are reduced n a hgh or low frequency, respectvely. T s W s (z)=, W t (z)= T h(z 1 + δ), A, T h (z 1) + T s T h (z 1) + T s (57) where T h = 75 and δ = Moreover, we determne the weghtng functon M (z) to satsfy the condton n (36) wth the above weghtng functons and K = Ts 1. Fnally,wedervetheH controller n each area K (z), A by solvng the LMI wth Robust Control Toolbox n MAT- LAB R214a. Each controller has sngle nput and sngle output and ts dmenson s four. The H condton of each controller becomes γ = 1.36, A. The bode dagram of the controller K (z) := K (z)m (z) s shown n Fg. 7. From ths fgure, we can confrm that each controller has a smlar form n ther magntude plots. However, these magntude are dfferent from each other accordng to areas. Ths result shows that we can desgn the approprate prce-updatng H controller by usng the proposed controller desgn method. Also, the cost functons to change the voltage phase angle f (θ )ssetto f (θ )=ζ θ 2 where ζ = , A. The other parameters used n ths smulaton are μ 1 =.8 and μ 2 =.2, Aand the uncertantes of power consumers and generators n Area are represented by Δ d = Δ s = 1, w d N(,.1 2 )andw s N(,.1 2 ), A. 4.2 H Controller Desgn Ths subsecton shows the desgn process of the output feedbac H controller used n ths smulaton. Wth the utlty and cost functons n (51) and the cost functons regardng voltage phase angles, f (θ ), we obtan the followng equatons: Fg. 7 Bode dagram of H controller.

8 SICE JCMSI, Vol. 9, No. 5, September Fg. 8 Locatonal electrcty prces between 9:-12: (h) va the proposed prce decson method usng H control. Fg. 9 Locatonal power balances between 9:-12: (h) va the proposed prce decson method usng H control. Fg. 1 Prces n Area 2 between 9:-12: (h) wth dfferent scales of uncertantes n maret partcpants behavor. (a): the proposed prce decson method usng H control, (b): the conventonal gradent method (proportonal control). 4.3 Smulaton Results The results of ths smulaton are shown n Fgs Frst, Fgs. 8 and 9 show the electrcty prces and the power supplydemand balance ncludng power flow n Areas 1-4 between 9: 12: (h) obtaned va our proposed prcng method usng the H control, respectvely. Fg. 9 confrms that the power mbalance occurs n the begnnng of each hour n every area, especally at 9 (h) n Area 2. These power mbalances are caused by the tme-varyng utlty functons of consumers whch are shown n the prevous subsecton. Accordng to the nformaton of these power mbalances, our proposed method updates the locatonal electrcty prces wth the desgned H controller as shown n Fg. 8. Ths fgure shows that those electrcty prces are changed by the tme n each area. As a result, the power mbalances are reduced as the prces are updated n every area as shown n Fg. 9. Therefore, we can conclude that our proposed prce-updatng H controller has a followng performance to reduce power mbalance n power grds. Next, we verfy the robustness of our proposed prcng method aganst uncertantes n maret partcpants behavor. Fgure 1 shows the results of the prce change n Area 2 between 9: 12: (h) when power consumers and generators have dfferent scales of uncertantes n ther behavor; 1w, 5w and w where w := (w s, w d ). In addton, fgures (a) and (b) show the results va the proposed method usng the H control and va the conventonal gradent method [5], respectvely. Note that the gradent n the conventonal method s represented by the power mbalance of each area. Therefore, those results are equal to the results usng the proportonal control (P control). Moreover, we use the same proportonal gan n the conventonal method whch mnmzes power mbalance when the scale of uncertantes s w n each stuaton. By comparng these two fgures n Fg. 1, we can confrm that the range of the prce change becomes small by usng our proposed method even f maret partcpants have a large scale of uncertantes n ther behavor. In addton, Fg. 11 shows the results of power devaton n Area 2 when the prces wth 5w or 1w n Fg. 1 are used n our problem. From the results n ths fgure, t s shown that our proposed method reduces power devaton as compared wth the conventonal prcng method Fg. 11 Power devaton n Area 2 between 9:-12: (h) wth 5w n (a) and wth 1w n (b). Table 2 Root mean square of the power devaton n Area 2 between 9:- 12: (h). 5w 1w Conventonal (MW) Proposed (MW) based on the proportonal control. In order to evaluate ths result n detal, the results of the root mean square of the power devaton obtaned by usng each prcng method are shown n Table 2. From ths table, we can confrm that the proposed prcng method becomes more effectve to reduce power devaton when maret partcpants have a large scale of uncertantes n ther behavor. Therefore, t s shown that our proposed method s a robust prce decson method aganst uncertantes n behavor of electrcty maret partcpants. 5. Concluson Ths paper deals wth a dynamc prce decson problem usng the H control consderng uncertan behavor of maret partcpants n electrcty maret tradng. The proposed prce decson method updates locatonal electrcty prces wth the H controller desgned by usng the dscrete-tme model of the dynamc prcng problem ncludng uncertan behavor of power consumers and generators. As a result, our proposed

9 2 SICE JCMSI, Vol. 9, No. 5, September 216 method becomes a qute effectve prce decson method n electrcty maret tradng snce t has both robustness aganst uncertantes n electrcty maret partcpants behavor and the followng performance to reduce power mbalance n power grds. Smulaton results confrmed the effectveness of our proposed prce decson method usng the H control. These results showed that our proposed method acheves to reduce power mbalance n power grds whle t has robustness aganst uncertan behavor of maret partcpants. Acnowledgments Ths wor was partally supported by JSPS KAKENHI Grant Numbers 15J5585 and References [1] J. Hansen, J. Knudsen, A. Kan, A. Annaswamy, and J. Stoustrup: A dynamc maret mechansm for marets wth shftable demand response, Proc. 19th IFAC World Congress, pp , 214. [2] S. Borensten, M. Jase, and A. Rosenfeld: Dynamc Prcng, Advanced Meterng and Demand Response n Electrcty Marets, UC Bereley, 22. [3] J. Warrngton, P. Goulart, S. Maréthoz, and M. Morar: A maret mechansm for solvng mult-perod optmal power flow exactly on AC networs wth mxed partcpants, Proc. Amercan Control Conference, pp , 212. [4] G.C. Chaspars, A. Rantzer, and K. Jörnsten: A decomposton approach to mult-regon optmal power flow n electrcty networs, Proc. European Control Conference, pp , 213. [5] P. Samad, A. Mohsenan-Rad, R. Schober, V. Wong, and J. Jatsevch: Optmal real-tme prcng algorthm based on utlty maxmzaton for smart grd, Proc. IEEE Internatonal Conference on Smart Grd Communcatons, pp , 21. [6] M. Roozbehan, M. Dahleh, and S. Mtter: Dynamc prcng and stablzaton of supply and demand n modern electrc power grds, Proc. IEEE Internatonal Conference on Smart Grd Communcatons, pp , 21. [7] Y. Oawa and T. Namerawa: Dstrbuted dynamc prcng based on demand-supply balance and voltage phase dfference n power grd, Control Theory and Technology, pp. 9 1, 215. [8] T. Namerawa, N. Oubo, R. Sato, Y. Oawa, and M. Ono: Real-tme prcng mechansm for electrcty maret wth bultn ncentve for partcpaton, IEEE Trans. Smart Grd, Vol. 6, No. 6, pp , 215. [9] L. Jang and S. Low: Mult-perod optmal energy procurement and demand response n smart grd wth uncertan supply, Proc. IEEE Conference on Decson and Control and European Control Conference, pp , 211. [1] Y. Oawa and T. Namerawa: Regonal demand supply management based on dynamc prcng n mult-perod energy maret, Proc. 54th IEEE Conference on Decson and Control, pp , 215. [11] F.L. Alvarado, J. Meng, C.L. DeMarco, and W.S. Mota: Stablty analyss of nterconnected power systems coupled wth maret dynamcs, IEEE Trans. Power Systems, Vol. 16, No. 4, pp , 21. [12] A. Joc, M. Lazar, and P.P.J. van den Bosch: Real-tme control of power systems usng nodal prces, Internatonal Journal of Electrcal Power and Energy Systems, Vol. 31, No. 9, pp , 29. [13] K. Ma, G. Hu, and C.J. Spanos: Dstrbuted energy consumpton control va real-tme prcng feedbac n smart grd, IEEE Trans. Control System Technology, Vol. 22, No. 5, pp , 214. [14] P. Kundur: Power System Stablty and Control, McGraw-Hll, [15] K. Zbou, J.C. Doyle, and K. Glover: Robust and Optmal Control, Prentce Hall, [16] IEEJ: Japanese Power System model, model/englsh/ ndex.html. Yoshhro OKAWA (Student Member) He receved the BS and MS of Engneerng from Keo Unversty, Toyo, Japan, n 212 and 213. He s currently pursung the Ph.D. degree from the Department of System Desgn Engneerng, Keo Unversty. Hs research nterests nclude robust control, control systems, and real-tme prcng n power networ systems. Toru NAMERIKAWA (Member) He receved the B.E., M.E., and Ph.D. n electrcal and computer engneerng from Kanazawa Unversty, Japan, n 1991, 1993, and 1997, respectvely. From 1994 untl 22, he was wth Kanazawa Unversty as an Assstant Professor. From 22 untl 25, he was wth the Nagaoa Unversty of Technology as an Assocate Professor, Ngata, Japan. From 26 untl 29, he was wth Kanazawa Unversty agan. In Aprl 29, he joned Keo Unversty, where he s currently a Professor at Department of System Desgn Engneerng, Keo Unversty, Yoohama, Japan. He held vstng postons at Swss Federal Insttute of Technology n Zurch n 1998, Unversty of Calforna, Santa Barbara n 21, Unversty of Stuttgart n 28 and Lund Unversty n 21. Hs man research nterests are robust control, dstrbuted and cooperatve control and ther applcaton to power networ systems.