Optimal power spatch in networks of high-mensional moels of synchronous machines Tjerk Stegink an Clauo De Persis an Arjan van er Schaft the physical network as well as for the stribute controller esign. Since energy is the main quantity of interest, the port- Hamiltonian framework is a natural approach to eal with the problem. Moreover, the port-hamiltonian framework lens itself to eal with complex large-scale nonlinear systems like power networks 5], 12], 13]. The emphasis in the present paper lies on the moeling an control of (networke) synchronous machines as they play an important role in the power network since they are the most flexible an have to compensate for the increase fluctuation of power supply an eman. However, the full-orer moel of the synchronous machine as erive in many power engineering books like 2], 6], 8] is fficult to analyze, see e.g. 5] for a port-hamiltonian approach, especially when consiering multi-machine networks 4], 9]. Moreover, it is not necessary to consier the full-orer moel when stuying electromechanical ynamics 8]. On the other han of the spectrum, many of the recent optimal controllers in power gris that eal with optimal power spatch problems rely on the secon-orer (non)linear swing equations as the moel for the power network 7], 11], 16], 17], or the thir-orer moel as e.g. in 14]. However, the swing equations are inaccurate an only vali on a specific time scale up to the orer of a few secons so that asymptotic stability results are often invali for the actual system 2], 6], 8]. Hence, it is appropriate to make simplifying assumptions for the full-orer moel an to focus on multi-machine moels with intermeate complexity which provie a more accurate escription of the network compare to the seconan thir-orer moels 2], 6], 8]. However, for the resulting intermeate-orer multi-machine moels the stability analysis is often carrie out for the linearize system, see 1], 6], 8]. Consequently, the stability results are only vali aroun a specific operating point. Our approach is fferent as the nonlinear nature of the power network is preserve. More specifically, in this paper we consier a nonlinear sixth-orer reuce moel of the synchronous machine that enables a quite accurate escription of the power network while allowing us to perform a rigorous analysis. In particular, we show that the port-hamiltonian framework is very convenient when representing the ynamics of the multi-machine network an for the stability analysis. Base on the physical energy store in the generators an the transmission lines, a port-hamiltonian representation of the multi-machine power network can be erive. More specifically, while the system ynamics is complex, the inarxiv:1603.06688v1 math.oc] 22 Mar 2016 Abstract This paper investigates the problem of optimal frequency regulation of multi-machine power networks where each synchronous machine is escribe by a sixth orer moel. By analyzing the physical energy store in the network an the generators, a port-hamiltonian representation of the multimachine system is obtaine. Moreover, it is shown that the openloop system is passive with respect to its steay states which implies that passive controllers can be use to control the multimachine network. As a special case, a stribute consensus base controller is esigne that regulates the frequency an minimizes a global quaratic generation cost in the presence of a constant unknown eman. In ation, the propose controller allows freeom in choosing any esire connecte unrecte weighte communication graph. I. INTRODUCTION The control of power networks has become increasingly challenging over the last ecaes. As renewable energy sources penetrate the gri, the conventional power plants have more fficulty in keeping the frequency aroun the nominal value, e.g. 50 Hz, leang to an increase chance of a network failure of even a blackout. The current evelopments require that more avance moels for the power network must be establishe as the gri is operating more often near its capacity constraints. Consiering high-orer moels of, for example, synchronous machines, that better approximate the reality allows us to establish results on the control an stability of power networks that are more reliable an accurate. At the same time, incorporating economic consierations in the power gri has become more fficult. As the scale of the gri expans, computing the optimal power prouction allocation in a centralize manner as conventionally is one is computationally expensive, making stribute control far more esirable compare to centralize control. In ation, often exact knowlege of the power eman is require for computing the optimal power spatch, which is unrealistic in practical applications. As a result, there is an increase esire for stribute real-time controllers which are able to compensate for the uncertainty of the eman. In this paper, we propose an energy-base approach for the moeling, analysis an control of the power gri, both for This work is supporte by the NWO (Netherlans Organisation for Scientific Research) programme Uncertainty Reuction in Smart Energy Systems (URSES) uner the auspices of the project ENBARK. T.W. Stegink an C. De Persis are with the Engineering an Technology institute Groningen (ENTEG), University of Groningen, the Netherlans. {t.w.stegink, c.e.persis}@rug.nl A.J. van er Schaft is with the Johann Bernoulli Institute for Mathematics an Computer Science, University of Groningen, Nijenborgh 9, 9747 AG Groningen, the Netherlans. a.j.van.er.schaft@rug.nl
terconnection an amping structure of the port-hamiltonian system is sparse an, importantly, state-inepenent. The latter property implies shifte passivity of the system 15] which respect to its steay states which allows the usage of passive controllers that steer the system to a esire steay state. As a specific case, we esign a stribute realtime controller that regulates the frequency an minimizes the global generation cost without requiring any information about the unknown eman. In ation, the propose controller esign allows us to choose any esire unrecte weighte communication graph as long as the unerlying topology is connecte. The main contribution of this paper is to combine stribute optimal frequency controllers with a high-orer nonlinear moel of the power network, which is much more accurate compare to the existing literature, to prove asymptotic stability to the set of optimal points by using Lyapunov function base techniques. The rest of the paper is organize as follows. In Section II the preliminaries are state an a sixth orer moel of a single synchronous machine is given. Next, the multimachine moel is erive in Section III. Then the energy functions of the system are erive in Section IV, which are use to represent the multi-machine system in port- Hamiltonian form, see Section V. In Section VI the esign of the stribute controller is given an asymptotic stability to the set of optimal points is proven. Finally, the conclusions an possibilities for future research are scusse in Section VII. II. PRELIMINARIES Consier a power gri consisting of n buses. The network is represente by a connecte an unrecte graph G = (V, E), where the set of noes, V = {1,..., n}, is the set of buses an the set of eges, E = {1,..., m} V V, is the set of transmission lines connecting the buses. The ens of ege l E are arbitrary labele with a + an a -, so that the incience matrix D of the network is given by +1 if i is the positive en of l D il = 1 if i is the negative en of l (1) 0 otherwise. Each bus represents a synchronous machine an is assume to have controllable mechanical power injection an a constant unknown power loa. The ynamics of each synchronous machine i V is assume to be given by 8] δ i ω i P mi P M i X qi, X X qi, X X, X qi E fi E qi, E E qi, E V qi, E I qi, I T qi, T T qi, T rotor angle w.r.t. synchronous reference frame frequency eviation mechanical power injection power eman moment of inertia synchronous reactances transient reactances subtransient reactances exciter emf/voltage internal bus transient emfs/voltages internal bus subtransient emfs/voltages external bus voltages generator currents open-loop transient time-scales open-loop subtransient time-scales TABLE I MODEL PARAMETERS AND VARIABLES. Assumption 1: When using moel (2), we make the following simplifying assumptions 8]: The frequency of each machine is operating aroun the synchronous frequency. The stator winng resistances are zero. The excitation voltage E fi is constant for all i V. The subtransient saliency is negligible, i.e. X =, i V. X qi The latter assumption is vali for synchronous machines with amper winngs in both the an q axes, which is the case for most synchronous machines 8]. It is stanar in the power system literature to represent the equivalent synchronous machine circuits along the qaxes as in Figure 1, 6], 8]. Here we use the conventional phasor notation E i = E qi+e = E qi +je where E qi := E qi, E := je, an the phasors I i, V i are efine likewise 8], 10]. Remark that internal voltages E q, E, E q, E as epicte in Figure 1 are not necessarily at steay state but are governe by (2), where it shoul be note that, by efinition, the reactances of a roun rotor synchronous machine satisfy X > X > X > 0, X qi > X qi > X qi > 0 for all i V 6], 8]. By Assumption 1 the stator winng resistances are negligible so that synchronous machine i can be represente by a subtransient emf behin a subtransient reactance, see Figure 2 6], 8]. As illustrate in this figure, the internal an external voltages are relate to each other by 8] E i = V i + jx I i, i V. (3) see also Table I. M i ω i = P mi P V I V qi I qi δ i = ω i T Ė qi = E fi E qi + (X X )I T qiė = E (X qi X qi)i qi T Ė qi = E qi E qi + (X X )I T qiė = E E (X qi X qi)i qi, (2) III. MULTI-MACHINE MODEL Consier n synchronous machines which are interconnecte by RL-transmission lines an assume that the network is operating at steay state. As the currents an voltages of each synchronous machine is expresse w.r.t. its local qreference frame, the network equations are written as 10] I = ag(e jδi )Y ag(e jδi )E. (4)
E f j(x X ) T j(x X ) E q T E q jx j(x q X q) j(x q X q ) jx q T q E T E Fig. 1: Generator equivalent circuits for both q-axes 8]. For aesthetic reasons the subscript i is roppe. E i jx q I i V i I V q I q V Fig. 2: Subtransient emf behin a subtransient reactance. Here the amittance matrix 1 Y := D(R+jX) 1 D T satisfies Y ik = G ik jb ik an Y ii = G ii + jb ii = G ik + j B ik where G enotes the conuctance an B R n n 0 enotes the susceptance of the network 10]. In ation, N i enotes the set of neighbors of noe i. Remark 1: As the electrical circuit epicte in Figure 2 is in steay state (3), the reactance X can also be consiere as part of the network (an ational inuctive line) an is therefore implicitly inclue into the network amittance matrix Y, see also Figure 3. 1 Recall that D is the incience matrix of the network efine by (1). jx jx T I l jx k E i V i V k E k Fig. 3: Interconnection of two synchronous machines by a purely inuctive transmission line with reactance X T. To simplify the analysis further, we assume that the network resistances are negligible so that G = 0. By equating the real an imaginary part of (4) we obtain the following expressions for the q-currents entering generator i V: I = B ii E qi I qi = B ii E Bik (E k sin δ ik + E qk cos δ ik ) ], Bik (E qk sin δ ik E k cos δ ik ) ], (5) where δ ik := δ i δ k. By substituting (5) an (3) into (2) we obtain after some rewriting a sixth-orer multi-machine moel given by equation (6), illustrate at the top of the next page. Remark 2: Since the transmission lines are purely inuctive by assumption, there are no energy losses in the transmission lines implying that the following energy conservation law hols: i V P ei = 0 where P ei = Re(E i I i ) = E I + E qi I qi is the electrical power prouce by synchronous machine i. IV. ENERGY FUNCTIONS When analyzing the stability of the multi-machine system one often searches for a suitable Lyapunov function. A natural starting point is to consier the physical energy as a canate Lyapunov function. Moreover, when we have an expression for the energy, a port-hamiltonian representation of the associate multi-machine moel (6) can be erive, see Section V. Remark 3: It is convenient in the efinition of the Hamiltonian to multiply the energy store in the synchronous machine an the transmission lines by the synchronous appears in each of the energy functions. As a result, the Hamiltonian has the mension of power instea of energy. Nevertheless, we still refer to the Hamiltonian as the energy function in the sequel. In the remainer of this section we will first ientify the electrical an mechanical energy store in each synchronous machine. Next, we ientify the energy store in the transmission lines. frequency ω s since a factor ω 1 s A. Synchronous Machine 1) Electrical Energy: Note that, at steay state, the energy (see Remark 3) store in the first two reactances 2 of generator i as illustrate in Figure 1 is given by ( H e = 1 (E qi E fi ) 2 2 X X + (E qi ) E qi )2 X X ( ) H eqi = 1 (E )2 2 X qi X qi + (E (7) E )2 X qi. X qi Remark 4: The energy store in the thir (subtransient) reactance will be consiere as part of the energy store in the transmission lines, see also Remark 1 an Section IV-B. 2 In both the - an the q-axes.
M i ω i = P mi P + B ik (E E k + E qie qk) sin δ ik + (E E qk E qie k) cos δ ik ] δ i = ω i T Ė qi = E fi E qi + (X X )(B ii E qi T qiė = E + (X qi X qi)(b ii E T Ė qi = E qi E qi + (X X )(B ii E qi Bik (E k sin δ ik + E qk cos δ ik ) ] ) Bik (E k cos δ ik E qk sin δ ik ) ] ) T qiė = E E + (X qi X qi)(b ii E Bik (E k sin δ ik + E qk cos δ ik ) ] ) Bik (E k cos δ ik E qk sin δ ik ) ] ) (6) 2) Mechanical Energy: The kinetic energy of synchronous machine i is given by H mi = 1 2 M iω 2 i = 1 2 M 1 i p 2 i, where p i = M i ω i is the angular momentum of synchronous machine i with respect to the synchronous rotating reference frame. B. Inuctive Transmission Lines Consier an interconnection between two SG s with a purely inuctive transmission line (with reactance X T ) at steay state, see Figure 3. When expresse in the local qreference frame of generator i, we observe from Figure 3 that at steay state one obtains 3 jx l I l = E i e jδ ik E k, (8) where the total reactance between the internal buses of generator i an k is given by X l := X + X T + X k. Note that at steay state the mofie energy of the inuctive transmission line l between noes i an k is given by H l = 1 2 X li l I l, which by (8) can be rewritten as H l = 1 ( 2 B ik 2 ( E E qk E ke qi) sin δik 2 ( E E k + E qie qk) cos δik +E 2 + E k 2 + E qi 2 + E qk 2 ), where the line susceptance satisfies B ik = 1 X l < 0 10]. C. Total Energy The total physical energy of the multi-machine system is equal to the sum of the inviual energy functions: H p = i V (9) (H ei + H qei + H mi ) + l E H l. (10) V. PORT-HAMILTONIAN REPRESENTATION Using the energy functions from the previous section, the multi-machine moel (6) can be put into a port-hamiltonian form. To this en, we erive expressions for the graent of each energy function. 3 The mapping from q-reference frame k to q-reference frame i in the phasor omain is one by multiplication of e jδ ik 10]. A. Transmission Line Energy Recall that the energy store in transmission line l between internal buses i an k is given by (9). It can be verifie that the graent of the total energy H L := l E H l store in the transmission lines takes the form H L δ i H L E qi H L E = E I + E I I qi qi I qi P ei = I, I qi where I, I qi are given by (5). Here it is use that the selfsusceptances satisfy B ii = B ik for all i V. 1) State transformation: In the sequel, it is more convenient to consier a fferent set of variable escribing the voltage angle fferences. Define for each ege l E η l := δ ik where i, k are respectively the positive an negative ens of l. In vector form we obtain η = D T δ R m, an observe that this implies D H p η B. Electrical Energy SG = D H L η = P e. Further, notice that the electrical energy store in the equivalent circuits along the - an q-axis of generator i is given by (7) an satisfies X X X X 0 X X Xqi X qi X qi X qi 0 X qi X qi ] H e ] ] E qi E H e = qi E fi E qi E qi E qi ] H eqi ] = E H eqi E E E E ]. By the previous observations, an by aggregating the states, the ynamics of the multi-machine system can now be written in the form (11) where the Hamiltonian is given by (10) an ˆX := X X, ˆX := X X, ˆX = ag i V { ˆX } an ˆX, ˆX q, ˆX q are efine likewise. In ation, T = ag i V{T } an T, T q, T q are efine similarly. Observe that the multi-machine system (11) is of the form ẋ = (J R) H(x) + gu y = g T H(x) (12)
ẋ p = ṗ η Ė q Ė Ė q Ė 0 D 0 0 0 0 D T 0 0 0 0 0 = 0 0 (T ) 1 ˆX 0 (T ) 1 ˆX 0 0 0 0 (T q) 1 ˆXq 0 (T q) 1 ˆXq H p + g(p m P ), 0 0 0 0 (T ) 1 ˆX 0 0 0 0 0 0 (T q ) 1 ˆX q y = g T H p, g = I 0 0 0 0 0 ] T. (11) where J = J T, R = R T are respectively the antisymmetric an symmetric part of the matrix epicte in (11). Notice that the ssipation matrix of the electrical part is positive efinite (which implies R 0) if 2 X X T X X T X X T 2 X X T > 0, i V, which, by invoking the Schur complement, hols if an only if 4(X X )T (X X )T > 0, i V. (13) Note that a similar contion hols for the q-axis. Proposition 1: Suppose that for all i V the following hols: 4(X X )T (X X )T > 0 4(X qi X qi)t qi (X qi X qi)t qi > 0. (14) Then (11) is a port-hamiltonian representation of the multimachine network (6). It shoul be stresse that (14) is not a restrictive assumption as it hols for a typical generator since T T, T qi T qi, see also Table 4.2 of 6] an Table 4.3 of 8]. Because the interconnection an amping structure J R of (11) is state-inepenent, the shifte Hamiltonian H(x) = H(x) (x x) T H( x) H( x) (15) acts as a local storage function for proving passivity in a neighborhoo of a steay state x of (12), provie that the Hessian of H evaluate at x (enote as 2 H( x)) is positive efinite 4. Proposition 2: Let ū be a constant input an suppose there exists a corresponng steay state x to (12) such that 2 H( x) > 0. Then the system (12) is passive in a neighborhoo of x with respect to the shifte external portvariables ũ := u ū, ỹ := y ȳ where ȳ := g T H( x). Proof: Define the shifte Hamiltonian by (15), then we obtain ẋ = (J R) H(x) + gu = (J R)( H(x) + H( x)) + gu = (J R) H(x) + g(u ū) = (J R) H(x) + gũ ỹ = y ȳ = g T ( H(x) H( x)) = g T H(x). 4 Observe that 2 H(x) = 2 H(x) for all x. (16) As 2 H( x) > 0 we have that H( x) = 0 an H(x) > 0 for all x x in a sufficiently small neighborhoo aroun x. Hence, by (16) the passivity property automatically follows where H acts as a local storage function. VI. MINIMIZING GENERATION COSTS The objective is to minimize the total quaratic generation cost while achieving zero frequency eviation. By analyzing the steay states of (6), it follows that a necessary contion for zero frequency eviation is 1 T P m = 1 T P, i.e., the total supply must match the total eman. Therefore, consier the following convex minimization problem: 1 min P m 2 P mqp T m s.t. 1 T P m = 1 T P, (17) where Q = Q T > 0 an P is a constant unknown power loa. Remark 5: Note that that minimization problem (17) is easily extene to quaratic cost functions of the form 1 2 P mqp T m + b T P m for some b R n. Due to space limitations, this extension is omitte. As the minimization problem (17) is convex, it follows that P m is an optimal solution if an only if the Karush- Kuhn-Tucker contions are satisfie 3]. Hence, the optimal points of (17) are characterize by P m = Q 1 1λ, λ = 1T P 1 T Q 1 1 (18) Next, base on the esign of 14], consier a stribute controller of the form T θ = L c θ Q 1 ω P m = Q 1 θ Kω (19) where T = ag i V {T i } > 0, K = ag i V {k i } > 0 are controller parameters. In ation, L c is the Laplacian matrix of some connecte unrecte weighte communication graph. The controller (19) consists of three parts. Firstly, the term Kω correspons to a primary controller an as amping into the system. The term Q 1 ω correspons to seconary control for guaranteeing zero frequency eviation on longer time-scales. Finally, the term L c θ correspons to tertiary control for achieving optimal prouction allocation over the network.
Note that (19) amits the port-hamiltonian representation ϑ = L c H c Q 1 ω P m = Q 1 H c Kω, H c = 1 2 ϑt T 1 ϑ, (20) where ϑ := T θ. By interconnecting the controller (20) with (11), the close-loop system amounts to ] ] ] ẋṗ J R RK G = T g H P ϑ G L c 0 G = Q 1 0 0 0 0 0 ] (21) where J R is given as in (11), H := H p + H c, an R K = blockag(0, K, 0, 0, 0, 0). Define the set of steay states of (21) by Ω an observe that any x := (x p, ϑ) Ω satisfies the optimality contions (18) an ω = 0. Assumption 2: Ω an there exists x Ω such that 2 H( x) > 0. Remark 6: While the Hessian contion of Assumption 2 is require for proving local asymptotic stability of (21), guaranteeing that this contion hols can be bothersome. However, while we omit the etails, it can be shown that 2 H( x) > 0 if the generator reactances are small compare to the transmission line reactances. the subtransient voltage fferences are small. the rotor angle fferences are small. Theorem 1: Suppose P is constant an there exists x Ω such that Assumption 2 is satisfie. Then the trajectories of the close-loop system (21) initialize in a sufficiently small neighborhoo aroun x converge to the set of optimal points Ω. Proof: Observe by (16) that the shifte Hamiltonian efine by (15) satisfies H = ( H) T blockag(r + R K, L c ) H 0 where equality hols if an only if ω = 0, T 1 ϑ = θ = 1θ for some θ R, an E H(x) = E H(x) = 0. Here E H(x) is the graent of H with respect to the internal voltages E q, E, E q, E. By Assumption 2 there exists a compact neighborhoo Υ aroun x which is forwar invariant. By invoking LaSalle s invariance principle, trajectories initialize in Υ converge to the largest invariant set where H = 0. On this set ω, η, θ, E q, E, E q, E are constant an, more specifically, ω = 0, θ = 1λ = 1 1T P 1 T Q 1 1 correspons to an optimal point of (17) as P m = Q 1 1λ where λ is efine in (18). We conclue that the trajectories of the close-loop system (21) initialize in a sufficiently small neighborhoo aroun x converge to the set of optimal points Ω. Remark 7: While by Theorem 1 the trajectories of the close-loop system (21) converge to the set of optimal points, it may not necessarily converge to a unique steay state as the close-loop system (21) may have multiple (isolate) steay states. VII. CONCLUSIONS We have shown that a much more avance multi-machine moel than conventionally use can be analyze using the port-hamiltonian framework. Base on the energy functions of the system, a port-hamiltonian representation of the moel is obtaine. Moreover, the system is proven to be incrementally passive which allows the use of a passive controller that regulates the frequency in an optimal manner, even in the presence of an unknown constant eman. The results establishe in this paper can be extene in many possible ways. 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