Coordinated Primary Frequency Control among Non-Synchronous Systems Connected by a Multi-Terminal HVDC Grid

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1 Coordnated Prmary Frequency Control among Non-Synchronous Systems Connected by a Mult-Termnal HVDC Grd J. Da a, Y. Phulpn a, A. Sarlette b, D. Ernst b a Department of Power & Energy Systems, Supélec, Plateau de Moulon, 3, rue Jolot Cure, Gf-sur-Yvette, France b Department of Electrcal Engneerng and Computer Scence, Insttut Montefore, Unversty of Lège, Sart-Tlman B28, B4000 Lège, Belgum Abstract We consder a power system composed of several non-synchronous AC areas connected by a mult-termnal HVDC grd. In ths context, we propose a dstrbuted control scheme that modfes the power njectons from the dfferent AC areas nto the DC grd so as to make the system collectvely react to load mbalances. Ths collectve reacton allows each ndvdual AC area to downscale ts prmary reserves. The scheme s nspred by algorthms for the consensus problem extensvely studed by the control theory communty. It modfes the power njectons based on frequency devatons of the AC areas so as to make them stay close to each other. A stablty analyss of the closed-loop system s reported as well as smulaton results on a benchmark power system wth fve AC areas. These results show that wth proper tunng, the control scheme makes the frequency devatons converge rapdly to a common value followng a load mbalance n an area. Keywords: power system control, frequency control, mult-termnal HVDC system, consensus algorthm, prmary reserve. 1. Introducton The frequency of an AC power system vares when there s an mbalance between the produced and the consumed amounts of power. To mantan the frequency of a power system close to ts nomnal value, system operators have developed frequency control schemes, whch are usually classfed accordng to the tme scale of ther actons [1]. The actons correspondng to the shortest tme scale are usually referred to as prmary frequency control. It conssts of automatc adjustment, wthn a few seconds after a power mbalance, of the Emal addresses: jng.da@supelec.fr (J. Da), yannck.phulpn@supelec.fr (Y. Phulpn), alan.sarlette@ulg.ac.be (A. Sarlette), dernst@ulg.ac.be (D. Ernst) Preprnt submtted to Internatonal Journal of Electrcal Power & Energy SystemsJuly 16, 2010

2 generators power output based on locally measured frequency varatons. To make t possble, the generators must have so-called prmary frequency control reserves,.e., power margns. The frequency averaged over a few seconds can be consdered dentcal n any part of a synchronous area. Wth a common frequency, every generator partcpatng n prmary frequency control adjusts ts generaton output n response to frequency excursons n the synchronous area, regardless of the locaton of the power mbalances. As the efforts of these generators sum up wthn a synchronous area, larger systems usually experence smaller frequency devatons. In addton, larger systems also have lower costs per MWh produced assocated to the prmary reserves. Ths has been a sgnfcant motvaton for nterconnectng regonal and natonal systems to create large-scale power systems, such as the UCTE network. The development of Hgh Voltage Drect Current (HVDC) systems for bulk power transmsson over long dstances [2] and underground cable crossngs opens new perspectves for nterconnectng non-synchronous areas. In ths context, t s generally expected that the power flows through an HVDC system are set at scheduled values, whle the frequences of the AC areas reman ndependent. Ths type of HVDC control scheme may prevent the system from cascadng outages by lmtng the effects of a severe contngency wthn one area [3]. It also makes sense when the nterconnected transmsson utltes cannot agree on a common practce n terms of frequency control. However, ths control scheme prevents the prmary reserves from beng shared among the non-synchronous AC areas, as the generators n one area are nsenstve to frequency excursons n other areas. Snce the supply of prmary reserves represents a sgnfcant part of the operatonal transmsson costs [4], t would be economcally advantageous to share prmary reserves among the non-synchronous areas by makng use of the fast power-trackng capablty of HVDC converters. Hence, the use of HVDC for prmary frequency control has been consdered n [5] and [6] for a system wth two AC areas. In these references, t s suggested to add to the scheduled power flow settng, a term that s proportonal to the frequency dfference between the two areas. Smulaton results provded n both papers show that such frequency feedback control leads to smaller frequency dfferences between the two areas. Furthermore, [5] also mentons that usng HVDC for prmary frequency control yelds reduced costs of prmary reserves. The present paper proposes and studes a dstrbuted control scheme to share the prmary frequency reserves among an arbtrary number of non-synchronous AC areas connected by a mult-termnal HVDC system. Whle the control laws proposed n [5] and [6] were heurstcally desgned, our control scheme s derved from algorthms for the extensvely studed consensus problem, whch appears n the context of coordnaton of mult-agent systems [7]. Wth the proposed scheme, the power njectons from the AC areas nto the DC grd are coordnated n such a way that the frequency devatons of all the AC areas stay close to each other despte power mbalances n one area. Thus, ths control scheme makes t possble to decrease the necessary amount of prmary reserves for the entre system, as f the AC areas were nterconnected through an AC grd. 2

3 The paper s organzed as follows. Secton 2 descrbes prmary frequency control and proposes a model of a mult-termnal HVDC system. Secton 3 formulates the control objectve and proposes a dstrbuted control scheme. Secton 4 analyzes stablty of the controlled system. Secton 5 gathers smulaton results on a fve-area system. Secton 6 concludes. 2. Prmary frequency control n a mult-termnal HVDC system In ths secton, we frst descrbe the mechansm of prmary frequency control of one generator, from whch the noton of prmary reserve arses. Then, a model for studyng frequency devatons n a Mult-Termnal HVDC (MT-HVDC) system s proposed Prmary frequency control and prmary reserve A generator that partcpates n prmary frequency control s equpped wth a speed governor, whch observes the shaft s rotatng speed, and uses a servomotor to control a throttle that determnes the amount of flud sent to the turbne [8]. When the speed governor detects a devaton of the generator rotaton speed, whch s proportonal to the stator electrcal frequency f, ts servomotor adjusts the openng of the throttle valve wthn a few seconds, thereby modfyng the power nput to the generator P m. When the frequency s equal to ts nomnal value f nom, P m equals ts reference value P o m. As P o m s set by secondary frequency control wthn a perod of about 30 seconds [1], we assume n ths paper that P o m remans constant. The dynamcs of prmary frequency control are expressed as T sm dp m = Pm o P m P m max σ f f nom f nom (1) where T sm s the tme constant of the servomotor, σ s the generator droop, and Pm max s the maxmum mechancal power that the turbne can produce. The lmts on P m, denoted by Pm mn and Pm max, are mposed by techncal and economc attrbutes of the generator. The quantty Pm max Pm o s referred to as prmary reserve Mult-termnal HVDC system model To study frequency varatons n an MT-HVDC system, we consder the system represented n Fg. 1. The system has three types of components: a DC grd, N non-synchronous AC areas, and N converters that nterface the AC areas wth the DC grd. 3

4 Converter 1 AC area 1 DC grd Converter 2 AC area 2 Converter N AC area N Fgure 1: A mult-termnal HVDC system connectng N AC areas va N converters DC grd As the electrcal tme constant of a DC grd s of the order of a few mllseconds [9], transent dynamcs of the DC grd s not consdered hereafter. To take nto account the general case where all the nodes are not connected to an AC area, we suppose that there are n total M N nodes n the DC grd and that node s connected to AC area va converter, {1,..., N}. Then, the power transferred from node to node j wthn the DC network, denoted by Pj dc, can be expressed as: j = V dc (V dc Vj dc ) (2) R j where V dc and Vj dc are the voltages at nodes and j respectvely, and R j s the resstance between these two nodes. If nodes and j are not drectly connected, R j s consdered equal to nfnty. Note that there must be ether a drect or an ndrect connecton between any two nodes, otherwse the DC grd would be made of several parts dsconnected from each other. Let P dc denote the power njecton from AC area nto the DC grd. Then, the power balance at node satsfes where By replacng j j j = { P dc for N, 0 for > N. by Eq. (2), we can wrte Eq. (3) n matrx form: (3) = dag(v dc 1,..., V dc M ) A V dc (4) s a vector of length M wth the frst N components equal to 1,..., N and the last M N components equal to 0. 4

5 V dc s a vector contanng the DC voltages V dc 1,..., V dc M. The components of matrx A are defned as 1 for j, R [A] j = j 1 j for = j. R j Gven a connected matrx A, Eq. (4) has a unque soluton once the voltage V dc of a node and the power njectons of the other nodes are fxed AC areas An AC area n a real HVDC system usually contans a large number of generators. In our study on prmary frequency control, whose tme scale s several seconds, the frequency can be consdered dentcal n any part of the AC area. Thus, we use an aggregated model to represent the mechancal dynamcs of all the generators wthn the area. Aggregaton methods such as the one n [10] can be used to fnd the parameters of the aggregated generator. On the one hand, the evoluton of the frequency s determned by the equaton of moton for area as 2πJ df = P m P e 2πf 2πD g (f f nom, ) (5) where f s the frequency of area, and f nom, ts nomnal value; P m and P e are the mechancal power nput and the electrcal power output of the aggregated generator for area, respectvely; J and D g are the moment of nerta and the dampng factor of ths generator. On the other hand, n response to frequency excursons, the speed governor of the aggregated generator adjusts ts mechancal power nput P m accordng to Eq. (1). The loads wthn an area are also aggregated and ther sum, denoted by P l, s represented by a statc load model [11] P l = P o l (1 + D l (f f nom, )) (6) where P o l s the value of P l when f = f nom,, and D l s the frequency senstvty factor. Fnally, the power balance wthn area requres that P e = P l +. (7) Converters Conventonally, n an MT-HVDC system, only one of the converters regulates the DC voltage, whle all the others control the power exchanged between the AC and the DC sdes [12, 13, 14]. In fact, the converter regulatng the DC voltage plays the role of the slack bus that mantans the power balance wthn the DC grd, and the latter can be consdered as a power exchange center between the 5

6 dfferent AC areas. Wthout loss of generalty, we assume that t s the converter connected to area N that regulates the DC voltage. Formally, wth the notatons ntroduced n Secton 2.2.1, P1 dc,..., PN 1 dc can be used as control varables to acheve a control objectve, whereas PN dc s determned by the DC grd load flow to mantan the power balance wthn the DC grd,.e., the value of PN dc corresponds to the sngle value satsfyng Eq. (4) for gven P1 dc,..., PN 1 dc dc, and VN Events leadng to frequency excursons A frequency excurson results from a power mbalance, whch may orgnate from a varaton n demand or generaton. The most common events consst of contnual varatons n the load demand Pl o, whch requre that the generaton utltes takng part n frequency regulaton contnually adjust ther output. For such types of events, the aggregated generator nerta J and the reserve Pm max Pm o of each area are usually consdered unchanged. More rarely, a sudden loss of a generator also results n a power mbalance. In ths case, the nerta and the reserve of the AC areas change. For the sake of smplcty, n the theoretcal and the expermental studes reported later n ths paper, we wll only consder demand-related mbalances. More specfcally, we wll study the effects, on our controlled system dynamcs, of a step change n the load demand of one of the AC areas. Ths step response analyss s suffcent to characterze system behavor n more general condtons (at very least, by the superposton prncple, for the lnearzed system). 3. Dstrbuted control scheme In ths secton, we propose a dstrbuted control scheme that shares prmary reserves among non-synchronous AC areas Control Objectve To make every generator senstve to a system-wde power mbalance, the frequency of every AC area must reflect the overall generaton/demand balance of the entre system. Ths can be acheved by makng the frequency devatons of all areas follow each other at any tme. Thus, our objectve s to desgn a control scheme that makes equal the frequency devatons of all the areas. The control varables are the power njectons P1 dc,..., PN 1 dc. They are modulated under the followng constrants where,mn,mn V mn and,max V dc,max, {1,..., N} (8) (resp. V mn V max, {1,..., M} (9) and V max ). ) are the mnmum and the maxmum acceptable values of P dc (resp. V dc In ths paper, we assume that the communcaton channels between the AC areas are delay-free,.e., f one AC area has access to the system nformaton 6

7 of another area, then ths nformaton s nstantaneously avalable. The effects of tme-delays on the control scheme are nvestgated n another paper by the authors [15] Control scheme We propose a dstrbuted control scheme composed of N 1 subcontrollers, one for each HVDC converter except converter N whch mantans the voltage of the DC grd. The subcontroller assgned to converter {1,..., N 1} modfes the value of P dc such that where d N N = α b j ( f f j ) + β j=1 j=1 ( df b j df ) j f = f f nom, s the frequency devaton of area. (10) α and β are analogous to the ntegral control gan and the proportonal control gan of a PI controller, respectvely. The hgher the gans, the closer the areas frequences stay to each other when one area experences a sudden power mbalance. Gan values are lmted n practce by () the maxmal rate of change of power flows through the converter, and () stablty consderatons of the controlled system, e.g., takng delays nto account (see [15]). b j s are the coeffcents representng the communcaton graph between the AC areas. The value of b j equals 1 f subcontroller receves frequency nformaton of area j, and 0 otherwse. The ntuton behnd the control scheme defned n Eq. (10) s as follows. If the frequency devaton of area s hgher (resp. lower) than the average frequency devaton of the other areas, then more (resp. less) power should be wthdrawn from ths area to drve ts frequency devaton back towards the other areas. The adjustement of ths power (P dc ) s determned by subcontroller, whch s a PI controller whose error sgnal s the sum of the dfferences between ts own frequency devaton and that of the AC areas from whch t gets nformaton. From a global pont of vew, the control scheme composed of all the subcontrollers takes the form of an algorthm for the consensus problem. Ths problem appears n the context of coordnaton of mult-agent systems: the objectve s, for agents performng a collectve task n a dstrbuted way (that s, wthout a supervsor tellng everyone what to do), to exchange and process nformaton n order to reach agreement on quanttes of nterest that are necessary to coordnate ther actons [7, 16]. Ths framework ndeed corresponds to the HVDC nterconnecton settng. Theoretcal results on consensus wll therefore provde useful tools for the stablty study of our control scheme. 7

8 Remark 1. To comply wth Constrants Eqs. (8) and (9), a saturaton functon for control varables should be ntroduced. In most cases, these restrctons do not affect the effectveness of the control scheme. Indeed, V1 dc,..., VN dc are close to each other under normal operatng condtons. However, lmtatons on P1 dc,..., PN dc could ndeed sgnfcantly lmt the degree of prmary reserve sharng. Ths mpact could be compared wth that of the maxmum current lmts on AC te lnes. 4. Stablty analyss of the control scheme Ths secton reports a theoretcal study on the stablty propertes of the control scheme. We frst characterze the unque equlbrum pont of a system subjected to a power mbalance n one AC area. Then, we elaborate on the condtons under whch the system converges to that equlbrum pont, and we prove stablty for the partcular case where all the AC areas have dentcal parameters. The stablty analyss reles on the followng assumptons: Assumpton 1. The losses wthn the DC grd do not vary wth tme,.e., N =1 d = 0. (11) Remark 2. Assumpton 1 s justfable for the reason that the losses n the DC grd are small compared to the power exchanged n the grd, and that the varatons of these losses are stll smaller compared to the total losses when P1 dc,..., PN 1 dc vary accordng to Eq. (10). Assumpton 2. The communcaton graph that represents the frequency nformaton avalablty at dfferent subcontrollers has the followng propertes: The communcaton graph s constant n tme. The communcaton graph s connected,.e., f b j = 0, then there must exst some ntermedate ndces k 1,..., k m such that b k1 = b k1k 2 =... = b kmj = 1. The communcaton graph s undrected,.e., f the subcontroller of one area has access to the frequency nformaton of another area, then the subcontroller of ths second area also has access to the frequency nformaton of the frst one,.e., f b j = 1, then b j = 1,, j {1,..., N 1}. Assumpton 3. The nonlnear equaton resultng from Eqs. (5)-(7) can be lnearzed around f = f nom, as: df 2πJ = P m Pl o P dc 2πD g (f f nom, ). (12) 2πf nom, 8

9 It s mportant to note that, under Assumptons 1 and 2, the dynamcs of N to satsfy Eq. (11) s the same as descrbed by Eq. (10), where b N = b N, {1,..., N}. In the followng, the HVDC system s thus modeled by a lnear system, where the dynamcs of area {1,..., N} s defned by Eqs. (1), (12), and controller (10) Equlbrum pont Proposton 1. Consder that the HVDC system, ntally operatng at ts nomnal equlbrum, s suddenly subjected to a step change n the load demand of one of ts AC areas. Then, under Assumptons 1, 2, and 3, the (lnearzed) HVDC system has a unque equlbrum pont, at whch the frequency devatons of all AC areas are equal. Proof. Pror to the step change n the load demand of one AC area, each area s consdered n steady state wth ts frequency regulated at f nom,. We denote these steady-state values by the varables wth a bar overhead. After the step change n the load, the varables start to change. We ntroduce the followng ncremental varables: x (t) = f (t) f nom,, y (t) = P m (t) P m, u (t) = P dc dc (t) P, v (t) = Pl(t) o P l o. By ntroducng these varables, Eqs. (1), (10), and (12) become dx (t) = a 1 x (t) + a 2 y (t) a 2 u (t) a 2 v (t), (13) dy (t) = a 3 x (t) a 4 y (t), (14) du (t) N N = α b j (x (t) x j (t)) + β j=1 j=1 b j ( dx (t) dx ) j(t), (15) where a 1 = D g /J, a 2 = 1/(4π 2 f nom, J ), a 3 = Pm max /(T sm σ f nom, ), and a 4 = 1/T sm. Note that a 1, a 2, a 3, and a 4 are all postve constants. Equatons (13)-(15) descrbe the closed-loop dynamcs of AC area, {1,..., N}, where the state varables are x (t), y (t), u (t), and the external nput s v (t). Intally, all the state varables are equal to zero, snce they are defned as the ncremental values wth respect to the ntal nomnal equlbrum. At t 0, a step change n load occurs n area m such that: { vm for = m and t > t v (t) = 0, 0 otherwse. (16) We now search for equlbrum ponts of the system followng the step change n v m (t). Let (x e, ye, ue ) characterze the state of area at such an equlbrum 9

10 pont. At ths pont, dx (t) = dy (t) = du (t) Thus, Eqs. (13)-(15) become algebrac equatons = 0. (17) 0 = a 1 x e + a 2 y e a 2 u e a 2 v (t > t 0 ), (18) 0 = a 3 x e a 4 y e, (19) N 0 = α b j (x e x e j). (20) j=1 Equaton (20) can be wrtten n matrx form for the entre HVDC system. Defne the vector x e = [x e 1,..., x e N ]T and let 0 N (resp. 1 N ) denote the column vector of length N wth all components equal to 0 (resp. 1). Then, Eq. (20) becomes 0 N = αlx e (21) where L s the Laplacan matrx of the communcaton graph. It s defned by { bj for j, [L] j = j b (22) j for = j. The Laplacan matrx L of a communcaton graph satsfyng Assumpton 2 s symmetrc postve semdefnte and ts only zero egenvalue corresponds to the egenvector 1 N. Therefore, equlbrum requres that the frequency devatons of all AC areas be equal. Let x e be the value of ths common frequency devaton at the equlbrum pont. From Eqs. (18) and (19), we obtan y e = a 3 x e, (23) a ( 4 u e a1 = + a ) 3 x e v (t > t 0 ) a 2 a 4 ( ) a 1m a = 2m + a3m a 4m x e v m for = m, ) (24) + a3 a 4 x e otherwse. ( a 1 a 2 We see n the above expressons that f x e can be unquely determned, then the equlbrum pont exsts and s unque. In fact, Assumpton 1 mples that N =1 u (t) remans constant, and the ntal condtons yeld that N =1 u (0) = 0. Thus, N u e = 0. (25) =1 From Eq. (24), we see that x e s unquely determned as ( N ) 1 x e a 1 a 4 + a 2 a 3 = v m. (26) a 2 a 4 =1 10

11 Remark 3. We consdered above that the step change n the load occurs n only one AC area. However, for the general case where v (t) changes n more than one area and eventually settles at v dfferent from 0, t s straghtforward to extend the above results to reach a smlar concluson on the exstence of a unque equlbrum pont, wth x e gven by ( N ) ( N x e = v =1 =1 a 1 a 4 + a 2 a 3 a 2 a 4 ) 1. (27) Remark 4. A physcal nterpretaton for the dfferent components of the equlbrum pont can be found. Indeed, from Eqs. (23) and (24), we have where ( N N N N ) y e = v + u e a 1 x e (28) a 2 =1 =1 =1 N =1 ye s the total change n the mechancal power due to prmary frequency control of all areas; =1 N =1 v s the total change n the load demand; N =1 ue s the total change n the losses wthn the DC grd, and t equals 0, see Eq. (25); ( N =1 a 1/a 2 )x e contans the dampng effects of the generators resulted from non-zero frequency devatons. In practce, the order of magntude of ( N =1 a 1/a 2 )x e s much smaller than that of N =1 v, whch means that the fnal varaton n mechancal power nduced by prmary frequency control of all the areas s approxmately equal to the total varaton n the load. Havng computed the equlbrum of the system as a functon of v, we make the fnal change of varables = 1,..., N. Defne the vectors x new, (t) = x (t) x e, y new, (t) = y (t) y e, u new, (t) = u (t) u e, x new (t) = [x new,1 (t),..., x new,n (t)] T, y new (t) = [y new,1 (t),..., y new,n (t)] T, u new (t) = [u new,1 (t),..., u new,n (t)] T, 11

12 and the matrces A = dag(a 1,..., a N ), = 1, 2, 3, 4. Then, the dynamcs of the lnear system for varables x new, y new, u new s gven by d x new y new = A 1 A 2 A 2 A 3 A 4 0 x new =. M x new. u new αl βla 1 βla 2 βla 2 y new u new y new u new The stablty of ths system and thus the convergence property of the system towards equal frequency devatons when subjected to small load varatons can be nferred from the egenvalues of M. It s not dffcult to see that x e M wll always have one zero egenvalue, assocated to the egenvector y e u e + v defned above. Ths strct zero egenvalue corresponds to an equlbrum shft: there s a pror a contnuum of equlbra for M. However, the addton of Assumpton 1 ensures that there s only one possble equlbrum for a gven contngency see the precedng Proposton 1. As a concluson, M has one 0 egenvalue whch s rrelevant gven Assumpton 1, and stablty s dctated by ts 3N 1 remanng egenvalues. For a gven partcular MT-HVDC system, the egenvalues of M can be computed numercally. The system wth our control scheme s then proven to be locally exponentally stable f M has one zero egenvalue (shft of equlbrum pont, see prevous paragraph) and 3N 1 egenvalues wth negatve real part. However, t s not clear a pror for whch values of α and β (f there exst any) ths s acheved. In the next subsecton, we propose a frst step towards a full theoretcal stablty analyss by consderng the specal case where all AC areas have dentcal parameters,.e., where I N s the N N dentty matrx Stablty of the system wth dentcal AC areas (29) A = a I N, = 1, 2, 3, 4 (30) Proposton 2. Consder that all the AC areas of the HVDC system have dentcal parameters, and that Assumptons 1, 2, and 3 are satsfed. Then, the unque equlbrum wth equal frequency devatons gven by Eqs. (23), (24), and (27) s exponentally stable for the MT-HVDC model wth the control scheme defned n Eq. (10) for any α > 0 and β 0. Proof. Under Assumpton 2, the Laplacan L of the communcaton graph s postve semdefnte and has egenvalues such that 0 = λ 1 < λ 2... λ N. Let U be an orthogonal matrx such that U 1 LU = dag(λ 1,..., λ N ). = L. (31) 12

13 Then we have a 1 I N a 2 I N a 2 I N (I 3 U 1 )M(I 3 U) = a 3 I N a 4 I N 0 =. M, (32) α L βa 1 L βa2 L βa2 L and M has the same egenvalues as M. By smple reorderng of rows and columns, M can further be brought nto block-dagonal form where each block takes the form a 1 a 2 a 2 M = a 3 a 4 0, (33) αλ βa 1 λ βa 2 λ βa 2 λ one for each egenvalue of L. For = 1, we have λ 1 = 0, whch gves the zero egenvalue assocated to equlbrum shft, and covered by[ Assumpton ] 1. The two remanng egenvalues a1 a of M1 are those of the matrx 2. Snce the constants a a 3 a 1, a 2, a 3, a 4 4 are all postve as defned earler, the egenvalues assocated to M 1 must have negatve real part. Let ξ k denote an egenvalue of M for > 1. To fnd t, we wrte det(ξ k I 3 M ) = ξ 3 k + (a 1 + a 4 + βa 2 λ )ξ 2 k + (a 1 a 4 + a 2 a 3 + αa 2 λ + βa 2 a 4 λ )ξ k + αa 2 a 4 λ. (34) By the Routh-Hurwtz stablty crteron for negatvty of polynomal roots, all ξ k wll have negatve real part f, for all > 1, (a 1 + a 4 + βa 2 λ ) > 0, (35) (a 1 + a 4 + βa 2 λ )(a 1 a 4 + a 2 a 3 + αa 2 λ + βa 2 a 4 λ ) αa 2 a 4 λ > 0, (36) αa 2 a 4 λ > 0. (37) Gven the postveness of the effcents a 1, a 2, a 3, a 4 defned n Secton 4.1, the above nequaltes are satsfed for all λ > 0 as long as α > 0 and β 0. Ths completes the proof. Remark 5. If we choose β = 0, then the system wll converge at a slow pace dctated by the system s dsspaton. Takng β > 0 ntroduces a dsspatonlke term n the controller and therefore allows much faster and less oscllatory convergence to equal frequency devatons. Remark 6. Note that the controller gans α and β are always multpled by the graph connectvty egenvalue λ n Eq. (34). The slowest convergence rate wll therefore be dctated by the smallest Laplacan egenvalue, λ 2. The latter s an extensvely studed object of graph theory, where t s called the algebrac graph connectvty. Ths should allow evaluatng whch graphs are more or less favorable (.e., requre smaller or larger gans n our controller) for synchronzaton of frequency devatons. 13

14 Table 1: Parameter values for the AC areas. Parameter Area Unt f nom Hz Pm o MW Pm max MW J kg/s 2 D g kw s/rad σ (dmensonless) T sm s Pl o MW D l Hz 1 The study reported hereabove shows that an MT-HVDC system wth dentcal AC areas s exponentally stable when applyng our control scheme. Ths theoretcal study reles on some smplfyng assumptons. However, the obtaned exponental stablty property s robust to small errors n the system dynamcs, whch ndcates that the result should ndeed hold on real systems wth nonlnear dynamcs and non-dentcal AC areas, as long as the dsturbances are small enough and the dssmlartes between the AC areas are small enough. The next secton further llustrates ths fact wth numercal smulatons. 5. Smulatons In ths secton, the control scheme s appled to a benchmark system that has a fve-termnal HVDC system. Frst, we fully descrbe the benchmark system. Then, we report and dscuss smulaton results Benchmark system and contngency The benchmark system s made of fve non-synchronous areas, connected by a fve-termnal HVDC system. The converter of area 5 s chosen to regulate the DC voltage, whose settng s 100kV. The topology of the DC network s represented n Fg. 2, where a crcle represents a node to whch a converter s connected, and an edge between two crcles represents a DC lne. The communcaton graph concdes wth the network topology,.e., each edge n the fgure also represents a b-drectonal communcaton channel between the two areas t connects. The DC resstances between the AC areas are: R 12 = 1.39Ω, R 15 = 4.17Ω, R 23 = 2.78Ω, R 25 = 6.95Ω, R 34 = 2.78Ω, and R 45 = 2.78Ω. In contrast to the stablty analyss n Secton 4.2, the smulatons consder AC areas wth dfferent parameters, as summarzed n Table 1. The AC areas are smulated wth the full nonlnear model descrbed by Eqs.(1), (5), (6), (7) and (10). For the DC grd, we use Eq. (4) to calculate the DC load flow nstead of makng Assumpton 1. 14

15 Fgure 2: DC grd topology and communcaton graph. The contnuous-tme dfferental equatons are ntegrated usng an Euler method wth a tme-dscretzaton step of 1ms. To observe the system s response to a power mbalance, we assume that all the areas ntally operate n steady state at the nomnal frequency. Then at tme t = 2s, a step ncrease by 5% of the value of Pl2 o (see Eq. (6)) s consdered Results Fgure 3a gves the evoluton of the frequences when the controller gans are chosen as α = β = For comparson, we also show n the same fgure the frequency of area 2 when the dstrbuted control scheme s not appled (.e., wth 1,..., 5 kept constant). The smulatons show that wthout the control scheme, the frequency of area 2 undergoes a devaton wth transent maxmum of 0.216Hz and stablzes at Hz. When the control scheme s appled, the maxmum transent devaton of area 2 drops to 0.170Hz, and the frequences of the fve areas converge to each other to fnally settle at Hz. Fgure 3b shows the evoluton of P1 dc,..., P5 dc expressed n MW. P2 dc s the power njecton whch vares the fastest. In partcular, t decreases by 5% n the 2 seconds followng the load step ncrease. The above results show that our control scheme leads to a sgnfcant mprovement n the steady-state frequency devaton of area 2, from 0.090Hz to 0.021Hz. However, the transent performance s qute poor, snce the maxmum transent devaton s just slghtly reduced from 0.216Hz to 0.170Hz. To mprove the transent performance, we ncrease the controller gans. For the sake of smplcty, we mpose that α = β. Fgure 4a shows the frequences when the controller gans are ncreased to α = β = Compared to Fg. 3a, the transent behavor s sgnfcantly mproved wth the larger controller gans: the frequences of all the areas converge to each other more quckly; the maxmal 15

16 50.05 frequency (Hz) f 1,f 3,f 4,f 5 f 2 f ( 2 constant) tme (s) (a) Frequences of the fve AC areas under the control scheme. f 2 when the prmary reserves are not shared s also shown (MW) tme (s) (b) Power njectons from the fve AC areas nto the DC grd. Fgure 3: Frequences and power njectons of the fve AC areas when α = β =

17 transent devaton of area 2 s reduced to 0.078Hz, whle the steady-state equlbrum does not change, as suggested by Proposton 1. The mprovement n the transent behavor comes at the prce of faster varatons of the power njectons, whch are shown n Fg. 4b. In ths case, P2 dc decreases by 10% wthn 0.6 second after the load step ncrease. Nevertheless, even for these larger controller gans, the varatons n P dc reman moderately small. From an engneerng pont of vew, the power njecton varatons shown n the fgures are wthn modern converters power-trackng speed, as shown n other studes such as [17] and [18]. In the smulatons, we have consdered that the prmary reserve of each generator were equal to ts ntal power output. For the case of smaller reserves, addtonal smulatons not reported n the paper show that the frequences stll converge even f some areas have reached ther prmary reserve lmts. However, when the prmary reserves of all areas are depleted, generators are unable to restore the overall power balance and all the frequences would keep decreasng, untl they reach a certan threshold that trggers emergency control actons such as load-sheddng n a real power system. 6. Conclusons and future work In ths paper, we presented a control scheme to share prmary reserves among non-synchronous AC areas connected by a mult-termnal HVDC system. Wth ths control scheme, n response to a power mbalance n one area, all the areas modfy ther power njectons nto the DC grd n a coordnated way so that ther frequency devatons converge to each other. A theoretcal study ndcates that, under some assumptons, the nterconnected system s stable and, followng a power mbalance n one area, converges toward a new equlbrum wth dentcal frequency devatons n all the areas. Smulatons on a benchmark system wth fve non-dentcal AC areas show that wth a proper choce of the controller gans, the frequency devatons of all the areas rapdly converge to each other followng a step change n the load demand. These smulatons, together wth others not reported n ths paper, hghlght that our control scheme can make good use of the fast power-trackng capablty of HVDC converters to coordnate prmary frequency control efforts among non-synchronous AC areas. Ths control scheme can be mproved along several lnes. One way of mprovement could be to choose dfferent gans for each subcontroller, or even let them vary on-lne n accordance wth the operatonal condtons of the areas, so that other factors can be taken nto account. For example, the relatve sze of one area wth respect to others can be consdered, so that n response to a major power mbalance n a larger area, a smaller area wth less prmary reserves can choose to stop sharng ts own reserve f t judges that a further sharng of ts reserve would jeopardze ts own stablty. In ths way, the HVDC system would stll play the role of frewall that prevents cascadng outages across AC areas. Another way of mprovement would be to adapt the control scheme so as to compensate for the tme needed for the subcontroller to gather the frequency nformaton of other areas. Indeed, compensatng for these tme-delays s m- 17

18 50.05 frequency (Hz) f 1,f 3,f 4,f 5 f 2 f ( 2 constant) tme (s) (a) Frequences of the fve AC areas under the control scheme. f 2 when the prmary reserves are not shared s also shown (MW) tme (s) (b) Power njectons from the fve AC areas nto the DC grd. Fgure 4: Frequences and power njectons of the fve AC areas when α = β =

19 portant snce as shown n another paper of the authors [15], they may lead to undamped frequency oscllatons. Acknowledgment Alan Sarlette s a FRS-FNRS postdoctoral research fellow and Damen Ernst s a FRS-FNRS research fellow. They thank the FRS-FNRS for ts fnancal support. They also thank the fnancal support of the Belgan Network DYSCO, an Interunversty Attracton Poles Programme ntated by the Belgan State, Scence Polcy Offce. Alan Sarlette was an nvted researcher at Mnes ParsTech when carryng out ths research. The scentfc responsblty rests wth ts authors. References [1] Y. G. Rebours, D. S. Krschen, M. Trotgnon, S. Rossgnol, A survey of frequency and voltage control ancllary servces part I: Techncal features, IEEE Transactons on Power Systems 22 (1) (2007) [2] R. Grünbaum, B. Halvarsson, A. Wlk-Wlczynsk, FACTS and HVDC Lght for power system nterconnectons, n: Power Delvery Conference, Madrd, Span, [3] M. P. Bahrman, B. K. Johnson, The ABCs of HVDC transmsson technology, n: IEEE Power and Energy Magazne, Vol. 5, 2007, pp [4] K. Papadoganns, N. Hatzargyrou, Optmal allocaton of prmary reserve servces n energy markets, IEEE Transactons on Power Systems 19 (1) (2004) [5] P. de Toledo, J. Pan, K. Srvastava, W. Wang, C. Hong, Case study of a mult-nfeed HVDC system, n: Jont Internatonal Conference on Power System Technology and IEEE Power Inda Conference, POWERCON 2008., 2008, pp [6] G. Fujta, G. Shra, R. Yokoyama, Automatc generaton control for DClnk power system, n: IEEE/PES Transmsson and Dstrbuton Conference and Exhbton 2002: Asa Pacfc., Vol. 3, 2002, pp [7] R. Olfat-Saber, J. Fax, R. Murray, Consensus and cooperaton n networked mult-agent systems, Proceedngs of the IEEE 95 (1) (2007) [8] UCTE operaton handbook (July 2004). [9] P. Kundur, Power System Stablty and Control, McGraw-Hll,

20 [10] M. L. Ourar, L.-A. Dessant, V. Q. Do, Generatng unts aggregaton for dynamc equvalent of large power systems, n: IEEE Power Engneerng Socety General Meetng, Vol. 2, 2004, pp [11] L. L. Grgsby, Power System Stablty And Control, The Electrcal Engneerng Handbook Seres, CRC Press Inc., [12] V. F. Lescale, A. Kumar, L.-E. Juhln, H. Bjorklund, K. Nyberg, Challenges wth mult-termnal UHVDC transmssons, n: Jont Internatonal Conference on Power System Technology and IEEE Power Inda Conference, POWERCON 2008, 2008, pp [13] H. Jang, A. Ekström, Multtermnal HVDC system n urban areas of large ctes, IEEE Transactons on Power Delvery 13 (4) (1998) [14] L. Xu, B. Wllams, L. Yao, Mult-termnal DC transmsson systems for connectng large offshore wnd farms, n: 2008 IEEE Power and Energy Socety General Meetng - Converson and Delvery of Electrcal Energy n the 21st Century, Pttsburgh, PA, 2008, pp [15] J. Da, Y. Phulpn, A. Sarlette, D. Ernst, Impact of delays on a consensusbased prmary frequency control scheme for AC systems connected by a mult-termnal HVDC grd, n: Proceedngs of the 2010 IREP Symposum - Bulk Power Systems Dynamcs and Control - VIII, Buzos, Ro de Janero, Brazl, [16] J. A. Fax, R. M. Murray, Informaton flow and cooperatve control of vehcle formatons, IEEE Transactons on Automatc Control 49 (9) (2004) [17] K. Meah, A. H. M. S. Ula, A new smplfed adaptve control scheme for mult-termnal HVDC transmsson systems, Internatonal Journal of Electrcal Power & Energy Systems 32 (4) (2010) [18] K. R. Padyar, N. Prabhu, Modellng, control desgn and analyss of VSC based HVDC transmsson systems, n: 2004 Internatonal Conference on Power System Technology PowerCon 2004, Vol. 1, Sngapore, 2004, pp