NONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION

Transcription

1 NONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION Steven Ball Science Applications International Corporation Columbia, MD Steve Schaer Department o Mathematics New Mexico Institute o Mining and Technology Socorro, NM Ernest Barany Department o Mathematical Sciences New Mexico State University Las Cruces, NM Kevin Wedeward Department o Electrical Engineering New Mexico Institute o Mining and Technology Socorro, NM ABSTRACT A method based on dierential geometric control theory is presented with the intention to provide insight into how the nodes o a power network can aect each other We consider a simple model o a power system derived rom singular perturbation o the power low equations The accessibility properties o the model are investigated, and it is shown that or a simple model with chain topology the network is actually eedback linearizable The results are illustrated using input/output linearization in a standard IEEE test system with 118 busses KEY WORDS Feedback Linearization, Power Networks, Nonlinear Control 1 Introduction Dynamical analysis o large electric power networks is increasingly important as power systems become larger and more interconnected and are operated close to stability limits An important aspect o the dynamical behavior o power systems that is particularly diicult to codiy is that directly related to its network structure The connectivity implies that any change at one bus necessarily aects all busses, so that undamental questions such as how power rom a given bus is distributed in the network are quite subtle In this paper we use geometric control theory to analyze the network-dependent structure o power systems We construct minimally complicated dynamical models o power networks as aine nonlinear control systems and use these to investigate how the inputs o a given node o the network inluence the other nodes 2 Dynamical models o power networks An electric power network can be useully modeled in the context o what are known as coupled cell systems in the nonlinear dynamics and control literature The cells correspond to busses o the power networks, and devices connected to the busses such as generators, loads, voltage control devices or other components deine internal states and dynamics o the cells A characteristic o power networks that simpliies this analysis is when the coupling between nodes occurs solely through the power low equations This is not always strictly true as certain devices such as transormers and static VAR compensators are located between busses, but these devices can oten be absorbed into redeinitions o eective busses In this case the evolution o the internal states o the devices sees only the voltage phasor o the bus to which it is connected, and also that the evolution o the voltage phasor depends only on the bus s own internal states and the voltages o the other busses to which it is connected but not the internal states o those busses The nature o the evolutionary equations depend on the modeling paradigm The simplest representation o an operating power grid is its power low solution In this situation real and reactive power generation and consumption are regarded as constant, and the solution is a set o constant voltage phasors or the nodes In this context there is no dynamics at all Adding internal node dynamics to this picture results in a dierential-algebraic system o equations DAE in which case the equations or the jth node o the bus take the orm z j = j z j, V j 21 0 = P j + iq j Σ k V j V k Y jk 22 where z j is the vector o internal states, V j is the voltage phasor, P j and Q j are the real and reactive power injection at the node which can be positive or negative and may depend on internal states and the voltage phasor Y jk is the admittance o the line connecting node j to node k, which vanishes i the two nodes are not directly connected The assumption o exact solution o the power low equations or all times is an approximation that amounts to regarding the dynamics at the bus as ininitely ast This can be a convenient assumption, but is not really justiied rom irst principles In act, there will be nontrivial dynamics on

2 ast, but inite, time scales that depends in a complicated way on the speciics o the devices connected to the bus The ininitely ast dynamics o the power low equations is oten justiied by arguing that no realistic modeling o the ast dynamics in general is possible While this may be so, there is another solution seen in the literature [1, 2] that has its own advantages, which is to replace 22 with a singularly perturbed version SPDE that relects that the voltage phasor will exhibit ast dynamics This type o approximation cannot be used thoughtlessly see, eg, chapter 8 o [2], but suiciently close to a hyperbolic ixed point there is reason to believe that smooth ast dynamics exist, and a perturbation o this type can be very useul The precise way in which the time evolution o the phasor components is associated with the mismatch o power low is not uniquely determined However, in some common situations [2] the lore o the electrical engineering literature attributes evolution o the phasor angle to the real power mismatch, and that o the voltage amplitude to the reactive power mismatch In this case, the ully dynamical equations or the jth node become ż j = j z j, V j 23 ɛ θ j = P j ReΣ k V j V k Y jk 24 ɛ V j = Q j ImΣ k V j V k Y jk 25 where ɛ << 1 is a representation o the separation o ast and slow time scales The equations have many advantages over rom the point o view o analysis and simulation Simulations o DAEs can be hampered by the need to ind good initial conditions which also aects the ability o such simulations to deal with protection events such as load shedding or line and generator tripping Also, being a smooth dynamical system the tools o nonlinear dynamics such as biurcation theory can be applied more easily Finally, as will be seen below, the methods o dierential geometric control theory can be easily adapted to provide a means o looking at the node inluence problem that is the heart o this paper It should be acknowledged that the nonuniqueness o the association o the evolution o states to the power mismatch is somewhat problematic rom a modeling point o view Mathematically, however, one can think in terms o blowups o singular problems in the sense o Takens [3] This theory guarantees that when the techniques are valid ie, away rom singularities o the DAE [2] and properly interpreted any such desingularization can be used to provide inormation about the associated DAE Experience with these models seems to support these ideas, and we take the position that any stable blowup o the DAE is an equally good representative o the ast dynamics as ar as the large scale, long time structure o the network is concerned Moreover, in the dierential geometric context to be laid out below, many considerations are invariant under dieomorphism, so that a wide class o possible desingularizations are equivalent In act, the controllability results we obtain are in a way independent o this choice, though the details o the dynamical trajectories certainly depend on it 3 Aine control model o power low networks The coupled cell structure allows the ull control problem to be addressed in two stages The undamental quantities at each bus can be taken to be the real and reactive powers, which are unctions o the bus states ie, the voltage phasors and the internal states The separation o internal bus states rom grid states suggests the strategy o using the power injections at each bus as controls at the network level, and later taking up the problem o tracking the desired powers by manipulating the inputs o the internal systems In other words, we assume we can directly manipulate the components o the power at the control bus as inputs In uture work we will take up the problem o achieving these inputs in a more concrete way that necessarily depends on speciic devices attached to each bus This is somewhat similar in concept to the use o kinematic models or mechanical systems where it is oten convenient to assume that velocities are directly accessible as inputs or the purposes o path planning, and inding orce-based dynamical controllers to track the velocity inputs as a secondary problem In practice, the reasoning in the preceding paragraph allows us to ignore the presence o the internal states or the initial controller We regard the powers either as parameters or controls, depending on the role o the bus The starting point o the simulation is a power low solution obtained using standard Newton-Raphson techniques For computational and analytic reasons we adopt rectangular coordinates or the complex phasors V j = x j + iy j or the analytical part o the paper The more amiliar polar orm will be used below in the example that concludes the paper In the rectangular case the power low equations depend only on two types o quadratic polynomials For notational simplicity, we deine D jk = x j x k + y j y k C jk = x j y k x k y j and the equations we consider are ẋ j = P j + Σ k D jk G jk C jk B jk 36 ẏ j = Q j Σ k D jk B jk + C jk G jk 37 where G jk is the real part o Y jk the conductance and B jk is the imaginary part the susceptance We have absorbed the singular parameter ɛ into the deinition o the unit o time which is permissible at this stage since we are assuming no internal slow time dynamics are present As mentioned above, the power components P j and Q j, j = 1, 2,, N are either parameters or controls as described below

3 To construct the aine control system, we assume that we have control o the inputs at some bus, which we label as node 1 That is, we identiy P 1 and Q 1 as control inputs Denoting by x the state vector constructed by concatenating all n = 2N phasor components, where N is the number o busses, the resulting control system is ẋ = x + g 1 u P + g 2 u Q 38 where g 1 = 1, 0, 0,, 0 T and g 2 = 0, 1, 0,, 0 T are the control vector ields associated with the inputs u P and u Q which are the departures o the power inputs rom their equilibrium values x is the rest o 36 and 37 ater the control vectors have been isolated and is reerred to as the drit o the aine control system Putting the system into this orm allows us to address the question o how manipulating the inputs to node 1 can aect the state o the system We use the apparatus o dierential geometric control [4, 5, 6] The problem we consider is a type o input-output control Speciically, we wish to ind control laws or the power inputs at bus 1 that will steer the voltage phasor at another bus to prescribed values We will achieve this by the technique o input-output linearization The theoretical analysis concentrates on the simple example o a network with chain topology o length N That is, we assume the admittance matrix Y jk is zero unless the indices j and k dier by at most one We take as controls the inputs at the endpoint It will be seen that the chain network is particularly well suited to these methods, and that or arbitrary networks, input-output control [6] is achieved by identiying the chain as the shortest path in a connected network that links the control bus to the output bus This will be discussed urther below 4 Accessibility properties o chain networks To begin, we deine the undamental object o study: Deinition 1 The node accessibility algebra o node 1 is the smallest involutive, drit invariant distribution that contains the control vector ields g 1 and g 2 Note that this distribution comprises a Lie algebra A undamental result rom dierential geometric control theory is that i the dimension o this distribution is d, then the system 38 can be dieomorphically put into triangular orm in a neighborhood o a regular point x such that an n d dimensional subspace is uncontrolled, and the reachable set o the control system contains an open subset o the complementary d dimensional subspacethe corollary is that when the distribution has ull rank, the system s reachable set is an open subset o the state space This is a necessary condition or controllability The simpliying eature o the chain topology is that the admittance matrix is band diagonal consisting o the main diagonal and the irst super- and sub-diagonals For simplicity, we assume the conductance G jk vanishes or all links which amounts to assuming losslessness and that the susceptance is the same or all links B jk = 1 X which 1 implies B jj = n j X, where n j is the number o nodes connected to node j [7] It is easy to see that the controllability results are independent o these assumptions By a urther rescaling o time, and deining ˆP j = XP j, we can eliminate the constant X rom the equations and we have a system o the orm 38 where = ˆP 1 + C 12 ˆQ 1 + D 12 D 11 ˆP 2 + C 21 + C 23 ˆQ 2 + D 21 + D 23 2D 22 ˆP j + C j,j 1 + C j,j+1 ˆQ j + D j,j 1 + D j,j+1 2D jj ˆP N + C N,N 1 ˆQ j + D N,N 1 D NN 49 and where g 1 and g 2 are deined as above We proceed to compute the node accessibility algebra Speciically, we prove: Theorem 41 The aine control system corresponding to the chain power network with inputs given by real and reactive power increments o the irst node is accessible Proo: The starting point is the constant distribution spanned by g 1 and g 2, which we denote by 0 This distribution is clearly involutive, but is not drit invariant, so we irst compute the Lie brackets o the drit with the control ields To do so, we compute the jacobian J o The band diagonal structure o the admittance matrix induces a similar block band diagonal structure o the jacobian Using the notation {J rs } N r,s=1 to denote the 2 2 submatrix with rows in subspace o the r th bus and columns in the subspace o the s th bus, it is easy to see rom 49 that yr x J r,r+1 = r de = x x r y r x r r = 1, 2, N 1, r J r,r = J r,r 1 = x r x r r = 2, 3, N, y r 1 + y r+1 x r 1 x r+1, x r 1 + x r+1 4x r y r 1 + y r+1 4y r r = 2, 3, N 1, J 1,1 = y 2 x 2 x 2 2x 1 y 2 2y r,

4 and J N,N = y N 1 x N 1 x N 1 2x N y N 1 2y N Since the control ields are constant, the brackets o g 1 and g 2 with the drit turn out to be the irst and second columns o the jacobian, or [g 1, ] = Jg 1 = y 2, x 2 2x 1, y 2, x 2, 0,, 0 T [g 2, ] = Jg 2 = x 2, y 2 2y 1, x 2, y 2, 0,, 0 T To simpliy the subsequent analysis, we do not take these ields as generators o the accessibility algebra, but instead use g 1 and g 2 to simpliy the ields by eliminating the components in the subspace o bus 1 Thereore, we take as our new distribution where 1 = 0 span{g 3, g 4 } g 3 = 0, 0, y 2, x 2, 0,, 0 T g 4 = 0, 0, x 2, y 2, 0,, 0 T or, using the block index notation g 3 r = x 2 δ r2 g 4 r = x 2 δ r2 The important observation here is that since det J 21 = x 2 2 these new ields span the state space or bus 2 as long as the voltage there does not vanish So by allowing our direct controls to interact with the drit, we extend our ability to inluence the system by a distance o one connection Now, 1 is again involutive but is not drit invariant, so we continue the procedure by computing the brackets o g 3 and g 4 with the drit Since these new vector ields are state dependent, we must consider both terms in the bracket For example, or g 3, we have [g 3, ] = Jg 3 Jg 3 The second term is irrelevant or purposes o accessibility since it always lies in the subspace corresponding to bus 2, which is already accounted or The irst term consists o state dependent linear combinations o the third and ourth columns o the jacobian o the drit Using the previously obtained vector ields to eliminate the components in the subspaces o the irst two busses as beore we can take as our new ields g 5 r = x 2 x 3 + y 2 x 3 δ r3 g 6 r = y 2 x 3 + x 2 x 3 δ r3 Since det g 5 3 g 6 3 = x 2 2 x 3 2, it ollows that the two vectors are independent in the subspace o bus 3 as long as the voltages at bus 3 and bus 2 are nonzero The pattern should now be clear Due to the banddiagonal nature o the jacobian o the drit, each additional layer o bracket will extend the span o the accessibility distribution into the next bus subspace Moreover, it is easy to see that det g 2r 1 r g 2r r = Π r s=1 x s 2 so that ater N brackets the accessibility distribution will have ull rank 5 Controllability by eedback linearization Accessibility is an easily proved property that is the starting point or geometric analysis, but is o limited utility or control synthesis In this section we show that the chain network is in act controllable via the technique o eedback linearization The meaning o linearization in his context is not an approximation procedure, but reers to inding a coordinate change and a eedback controller such that the transormed system is linear The main purpose o this section is to prove: Theorem 51 The aine control system corresponding to the chain power network with inputs given by real and reactive power increments o the irst node is eedback linearizable [5, 4, 6] Proo: The theory o eedback linearizability involves speciication o outputs, though the eedback laws obtained depend on ull state eedback, and not just the outputs For the chain network with inputs given by the power increments at one end, the appropriate outputs are the states at the other end That is, we supplement the aine control system 38 with the outputs h 1 x = x N h 2 x = y N Then the theory o MIMO eedback linearization we ollow the theory in [5] most closely states that i the relative degree vector, r 1, r 2, o the system this will be deined below obeys the condition that r 1 + r 2 = 2N, then there exist nonlinear state eedback laws v 1 = q P x + S P 1 xu P + S P 2 xu Q v 2 = q Q x + S Q1 xu P + S Q2 xu Q 510 and a local dieomorphism z = T x such that the transormed system is o the orm ż 1 = z 2 ż r1+1 = z r1+2 ż 2 = z 3 ż r1+1 = z r1+2 ż r1 1 = z r1 ż 2N 1 = z 2N ż r1 = v 1 ż 2N = v 2

5 where v 1 and v 2 are new inputs This is the Brunovsky orm o the linearized system and allows easy control synthesis to meet local controllability targets To prove the theorem, we need only show the relative degree result A two input-two output system has a relative degree vector r 1, r 2, i L gj L k h i = or i, j = 1, 2 and or all k < r i 1, and i the 2 2 matrix Lg1 L r1 1 Sx = h 1 x L g2 L r1 1 h 1 x L g1 L r2 1 h 2 x L g2 L r h 2 x has rank two Since g 1 and g 2 are constant, the Lie derivative o any unction with respect to either one amounts to taking the partial derivative with respect to x 1 and y 1, respectively Thus, by deinition o h 1 and h 2, 511 is satisied or k = 0 as long as N > 1 For k 1, rom 49, the Lie derivatives o the outputs with respect to the drit are o the orm L k h j = F k j x N, x N 1,, x N k G k j x N, x N 1,, x N k 2 C N k 1,N k + H k j x N, x N 1,, x N k 2 D N k 1,N k or some unctions F k j, G k j and H k j which are unctions o the variables speciied In particular, L k h j depends only on the states o the last k + 1 busses, which means that there will be no dependence on x 1 or y 1 or k + 1 < N Thereore 511 will hold or k N 2 To show Sx has rank two, note that L N 1 h j = F N 1 j x N, x N 1,, x 2 and it ollows that S = + G N 1 j x N, x N 1,, x 3 C 21 + H N 1 j x N, x N 1,, x 3 D 21 y 2 G N x 2 H N 1 1 x 2 G N y 2 H N 1 1 y 2 G N x 2 H N 1 2 x 2 G N y 2 H N 1 2 and dets = x 2 2 H N 1 1 G N 1 2 G N 1 1 H N 1 2 Using 49 one can ind G k j = y N k 2G k 1 j + x N k 2 H k 1 j 514 H k j = x N k 2G k 1 j + y N k 2 H k 1 j 515 rom which it ollows that H k 1 G k 2 G k 1 Hk 2 = 516 x N k 2 2 H k 1 1 G k 1 2 G k 1 1 H k 1 2 Then, since G 0 1 = H 0 2 = 1 and H 0 1 = G 0 2 = 0, we have that H k 1 G k 2 G k 1 Hk 2 = Π k 1 j=1 x N j 1 2 rom which it ollows that dets = Π N 1 j=1 x N j 1 2 = Π N j=2 x j 2 which establishes the relative degree condition as long as none o the voltages vanish 6 Example: Input-output control We illustrate the oregoing methods in this section In order to deal with networks whose topology is more complicated than a chain, we modiy our control goal to that o input-output or partial state linearizationthis means we identiy a pair o busses, one whose power components are speciied as inputs, the other whose states are controlled as outputs When the oregoing methods are applied, one inds that the two components o the relative degree vector will be equal to the number o connections between the selected busses A eedback control law will exist such that the outputs will be governed by a linear system whose controls will be realized by the input powers Since the degree vector will not generally obey the linearization condition r 1 + r 2 = 2N, there will be unobservable and uncontrollable dynamics on some o the state space [4, 6] That this is possible, ollows rom: Assumption 1 We assume that the control and target nodes are connected by a unique path o buses o minimal length r that we label 1, 2,, r Moreover, any other paths connecting any o the nodes 1, 2,, r with each other are o length greater than r This assumption is exactly what is needed to ensure that the proos o Section 5 remain applicable In particular, we have Theorem 61 The states at any bus connected to the control bus have controllability indices r, r, where r is the minimum number o transmission lines that must be crossed in going rom the control bus to the target bus Proo: The only dierence is that the F, G and H unctions will also depend on the variables o the busses that are within r hops o the target bus, but this will not aect the values o the indices We also have Theorem 62 The matrix S is nonsingular

6 Proo: Under the assumption, the recurrences 514 and 515 still hold, so the computations that ollow and the rank condition continue to be true We illustrate this on the 118-bus IEEE test system shown in igure 1 The system model was obtained rom the Power Systems Test Case Archive o the Department o Electrical Engineering o the University o Washington, and details can be ound on line at Figure 2 Response o controlled bus voltage and angle geometric control can be applied to the problem o controlling the states o one bus by adjusting the power inputs o another For theoretical purposes, we irst considered a simple chain network where these tools worked out particularly well due to the act that the system is ully eedback linearizable Then in a more general context, similar methods were applied in the context o input-output decoupling or a 118-bus IEEE test system Reerences Figure 1 IEEE 118-bus test system with control input arrow and controlled star busses indicated Generator and load models in the path between the control and target busses are modeled as constant power sources or sinks, while all other generator and load models are dynamic aggregate models as ound in [7, 8] The components o the relative degree vector or outputs given by the components o the target voltage phasor are equal to the number o connections traversed along the unique path o minimal length between the control and target busses, which is 3 in this case We choose our control goal as ollows We use closed loop eedback to set the voltage and angle at the target bus to speciied values that are arbitrarily chosen to be about 10% dierent than original power low values We do this using a standard PI eedback controller implemented on the linear system That is, we convert the state measurements into the linearizing coordinates, and get the desired controls I d P rom the equations v 1 = K1 z1 z1 + K2 z 1 and I d P v 2 = K2 z4 z4 + K2 z 4 Then we use these to compute the controls as the P, Q increments, and use these in the nonlinear simulation The results are shown in Figure 2 7 Conclusions In this paper we have illustrated a method or analyzing the eects that busses o a power system have on each other due to the network structure The initial goal has been to model the system as simply as possible based on the power low equations and show that the apparatus o dierential [1] Sastry, S and C Doeser, Jump Behavior o Circuits and Systems, IEEE Trans on Circuits and Systems, CAS-28 #12, December, 1981 [2] Ilic, M, and J Zaborsky Dynamics and Control o Large Electric Power Systems, Wiley-Interscience, New York, 2000 [3] Takens, F, Singularities o vector ields, Inst Hautes Etudes Sci Publ Math, 43, pp , 1974 [4] Nijmeijer, H, and A van der Schat, Nonlinear Dynamical Control Systems, Springer-Verlag, New York 1990 [5] Isidori, A, Nonlinear Control Systems, 2nd ed, SpringerVerlag, NY, 1989 [6] Vidyasagar, M, Nonlinear Systems Analysis, 2nd ed, Prentice Hall, Englewood Clis, NJ, 1993 [7] Sauer, P, and M Pai, Power System Dynamics and Stability, Prentice Hall, Englewood Clis, NJ, 1998 [8] Anderson, P and A Fouad, Power System Control and Stability, Iowa State University Press, 1977