WIND and solar power account for almost half of newly

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1 Computing Saddle-Node and Limit-Induced Bifurcation Manifolds for Subtransmission and Transmission Wind Generation Sina S. Baghsorkhi, Student Member, IEEE Department of Electrical Engineering and Computer Science University of Michigan - Ann Arbor sinasb@umich.edu Abstract This paper introduces an exact and efficient method for computing and visualizing the power flow feasibility boundary of large networks composed of () saddle-node bifurcation manifolds where the Jacobian is singular and () limit-induced bifurcation manifolds where the network loses structural stability as the limit is encountered. This method is applied to the DTE/ITC system in Eastern Michigan where significant wind development has taken place at both transmission (-345kV) and subtransmission (4kV) levels. It is shown that reactive compensation capability of subtransmission wind farms, in voltage control mode, can shrink the feasibility boundary and hence negatively impact the voltage stability margin of the system. I. INTRODUCTION WIND and solar power account for almost half of newly installed electricity generation capacity worldwide. Due to falling technology costs, this trend is expected to continue despite the global economic turmoil and uncertainty over policy incentives for these fledgling sectors []. A sizable portion of this capacity is connected to subtransmission networks characterized by medium voltage levels (4-kV) and resistive lines (X/R 4). This is either due to the unavailability of high voltage transmission lines or the cost of connection assets (mainly transformers) which rises sharply with voltage levels and makes it uneconomical to connect wind and solar farms below a certain capacity directly to higher voltage networks even when the option exists. The resistivity of subtransmission networks creates a strong coupling between active power flows and voltage magnitudes that is atypical in high-voltage transmission systems. In the absence of any voltage control scheme, voltage fluctuations caused by generation variability pose serious challenges for resistive networks. As the size of subtransmission-connected variable generation increases, new grid codes are adopted that require the voltage at the point of interconnection (POI) of wind or solar farms to be strictly regulated. This mode of voltage control could require significant absorption of reactive power at peak generation, turning the POI into a reactive sink. It is not uncommon to have larger-scale wind generation directly connected to transmission systems as in the case of the DTE/ITC system serving Eastern Michigan where smaller wind farms are connected to the DTEowned 4kV subtransmission network and larger wind projects are directly connected to the ITC-owned kv and 345kV transmission loops. At periods of high wind, this reactive requirement of sub-transmission wind generation may strain the reactive reserves of the system. This is at the very same time that large amounts of wind power enters directly into the transmission network pushing the system closer toward its power transfer capability limit. Hence reactive compensation associated with wind and solar generation in resistive networks have potentially adverse consequences for the voltage stability of the power system. Although many criteria and proximity indices for voltage instability have been proposed over the past few decades, all are related to the maximum power transfer capability which is rooted in the feasibility boundary of the power flow algebraic equations. Equating the power flow feasibility boundary with the voltage stability boundary has its origin in the influential works of Zhdanov, Venikov and their colleagues in the former USSR []. Power system dynamics can be modeled mathematically by a system of autonomous nonlinear differential-algebraic equations (DAEs): ẋ = f(x, y, z, λ) (a) = g(x, y, z, λ) (b) = h(z, λ) (c) where x and y are the dynamical and algebraic states associated with electromechanical devices in the network such as synchronous and doubly-fed induction generators and their (AVR, rotor speed, pitch-angle) controllers, z the set of power flow variables and λ the set of parameters. As demonstrated by Sauer and Pai [3], the standard set of power flow equations or its variants represented in (c), can always be solved for the multi-valued z independent of initial conditions of dynamical states and other algebraic variables. Once a particular z is obtained, (a)-(b) can be solved for the corresponding equilibrium point E = (x, y, z ). Assuming λ is deterministic, () can be linearized at E in the following way: ẋ = f x f y f z x g x g y g z y () h z z Note that the singularity of h z, the power flow Jacobian, implies that the obtained solution of z is on the solution space boundary. From a geometric point of view this is a branch point in the parameter space of (c) where at least two algebraic sheets coalesce. As the power flow parameters are perturbed there is a structural change in the set of equilibria containing E where at least two equilibria coalesce into a single equilibrium and disappear. This is typically but not always a saddle-node bifurcation where one equilibrium is a stable node and the other is an unstable saddle. It should be noted that the singularity of h z is only sufficient and

2 not necessary for a structural change in the set of equilibria. Algebraic states in () can be eliminated to obtain the statespace matrix A the spectrum of which determines the stability of the system at E : instantaneous loss of structural stability. Note that the revised Jacobian is no longer singular but has a negative eigenvalue that has already crossed the imaginary axis to the left at the saddle-node bifurcation of Q = Q max trajectory. ẋ = (f x f y g y g x ) x = A x (3) It has been shown that under certain assumptions on the power flow and dynamical models, A is singular if and only if h z is singular [], [3]. However in general A can reach singularity while h z is non-singular. Hence power flow Jacobian singularity does not capture certain rare instability phenomena that can arise from the dynamical interaction of generators and their controllers with the network. With the caveat mentioned above, this paper proposes a method for computing and visualizing the power flow feasibility boundaries with the main purpose of assessing the voltage stability margin in networks characterized by high penetration of variable generation. The paper is organized as follows. Section II begins by contrasting the saddle-node and limit-induced bifurcations and describes succinctly the method used to obtain these manifolds which together form the power flow feasibility boundary. Section III applies this methodology to the DTE/ITC system to obtain the feasibility boundary for subtransmission and transmission wind injection and analyzes the findings. Section IV outlines the ongoing work in this area. Fig.. V V =. θ = Two bus system Q=Q max Stable Branch X = j. Saddle Node Bifurcation Stable Branch V =. Unstable Branch P Limit Induced Bifurcation II. AN ALGORITHM FOR COMPUTING THE SADDLE-NODE AND LIMIT-INDUCED BIFURCATION BOUNDARY The power flow feasibility boundary and its geometry have been explored previously [4], [5]. Reference [4] introduces a method to obtain the saddle-node bifurcation boundary based on continuation method that involves the power flow Hessian. Reference [5] shows that the boundary is generally nonconvex. However, there are very few works that have examined the reactive limit-induced instability [6], [7]. Reference [6] examines the dynamic of voltage collapse when the reactive limit of a generator under study is encountered. Reference [7] examines the relationship between non-smoothness of the boundary and reactive limits. What is missing from the literature is a methodology for efficiently obtaining both the singular and limit-induced segments of the feasibility boundary for large networks. In order to highlight the relationship between saddle-node and limit-induced bifurcations Figure contrasts these two phenomena in the context of the two-bus system of Figure. As active load P increases with no reactive compensation, the voltage magnitude gradually declines until the power transfer capability limit is reached at P = 5.. This is the point where the stable (solid blue) and unstable (dashed blue) branches of the bifurcation diagram meet and the power flow Jacobian is singular. The transfer capability can be extended significantly by regulating the voltage at V =. until the reactive limit of the regulating device is reached at P 9.7. At this point the network loses structural stability and the operating point lies on the unstable (dotted black) branch of the Q = Q max bifurcation diagram. Hence a reactive limit encountered in a heavily compensated network can cause an Unstable Branch Fig.. Loss of structural stability: Saddle-node versus limit-induced bifurcations A. Singular Segment Suppose λ is a subset of parameters that are set free to vary and z is the set of variables. The power flow and the singularity equations can be expressed as: P h(z, ) = (4) det h z (z, ) = (5) Here z R n, R p and h : R n+p R n. Assuming that p =, there are n + equations and n + unknowns and the solution is a -manifold in the parameter space of. To obtain this manifold, the first step is to find an initial point. This is easily done by using the continuation method to reach a turning point where det h z changes sign. Once the initial point is obtained, homotopy methods can be used to solve (4)-(5) for the successive points on the singular manifold. This, however, involves computing the Hessian matrix of h(z, ). Unfortunately, it is impractical to compute and work with the Hessian matrix of large networks. What further complicates the computation of the Hessian is the existence of multiple types of FACTS devices each appearing with its own unique variables in power flow equations. Alternatively the vector that is normal to the singular manifold in the parameter space can be exploited to estimate the successive points with much less computational effort. The following relation can be deduced from (4):

3 h + h z = (6) z Finding the normal vector requires a knowledge of the left eigenvector that corresponds to the zero-approaching eigenvalue of the Jacobian. Multiplying (6) by w, the left eigenvector corresponding to the smallest eigenvalue, i.e. the zero-approaching eigenvalue, gives the following relation (Note w h z = ): w h = (7) Therefore the normal vector, at the initial point on the manifold, is w h which can be used to find the vector that is tangent to the manifold at that point. This gives an initial prediction of (z, ) for the second point on the manifold. The initial prediction of free parameters, λ p is directly given by the tangent vector and z p, the initial prediction for the states, is given as, z p = z f + ( h z ) h(z f, λ p) (8) where z f is the set of states at the first point on the boundary, ( h z ) is the inverse of the Jacobian at the first point and h(z f, λ p) is the mismatch in power flow equations caused by λ p. It should be noted that in the realm of computation, as the boundary is approached the Jacobian can near but never reach singularity. Hence there does not exists a single point in the parameter space of where the Jacobian is truly singular. A close-to-singular Jacobian matrix can still be exploited within the default arithmetic precision of numerical computing environments which is more than sufficient for the type of computation discussed in this section. The initial prediction, λ p, as can be seen in Figure 3, is infeasible if the manifold is convex. However by formulating the problem as a nonlinear least squares estimation, the point on the boundary that is closest (in a norm- sense) to the predicted value λ p can be obtained through a number of iterations. Figure 3 shows the procedure of finding the second point. The sequence λ p, λ p, λ 3 p,... approaches to the second point on the boundary. Given that the initially predicted values of (z, ) are typically very close to the actual values, the second point can be obtained by monitoring the smallest eigenvalue of the Jacobian and with only a few iterations. Figure 3 also highlights the robustness of the method in obtaining the second point on the high curvature region of the boundary with an exaggerated choice of tangent vector length (green arrow). Once the second point is obtained, secant method can be used in the following way to find the successive points: z p = z i + τ (z i z i ) z i z i Most parameters of interest, such as active or reactive power injections and loads can be explicitly expressed as functions of the state variables. Hence can be predicted as k(z p). Fixing at its predicted value and solving for (4) gives an estimate z p where (z p, ) is usually a solution for (4)-(5). In other words, the smallest eigenvalue tends to be less than the tolerance. If the solution to (4) does not meet the tolerance (9) λ Fig. 3. Continuation Curve First Point Normal Vector Feasibility Boundary High Curvature Region Tangent Vector λ The process of obtaining successive points on the feasibility boundary criteria, continuation method can be applied to move, along the normal vector, closer to the singular manifold. Similarly, if = k(z p) is outside the feasibility region, nonlinear least squares estimation techniques can be used to find the nearest point on the boundary. However, in the cases investigated so far, this happens infrequently. The successive points are usually obtained within - iterations. It should be noted that for types of parameters that can not be stated explicitly in terms of power flow variables (i.e. voltage magnitudes and angles) such as line impedances or setpoints quantities, a secant method similar to (9) can be used to obtain. However, this prediction, most likely lies outside the feasible region which requires a more frequent application of nonlinear least square estimation techniques. This makes the process of convergence to the successive points computationally less efficient. For simple three bus systems with no reactive limits the method discussed above obtains the singular manifold more efficiently than the Hessian-based method discussed earlier. It is expected that as the dimension of the network increases, the computational advantage of this method becomes even more apparent. B. Non-singular Segment The singular manifold may encounter limits of regulating devices where, as explained before, the singularity condition (5) no longer holds. The feasibility boundary, then is defined by the -manifold of the regulating device limit in the -parameter space, beyond which (4) has no solution that satisfies the limit. (4)-(5) is replaced by: h(z, ) = () q j (z, ) = L j () where the function q j, associated with a certain network controller, e.g. reactive power injection, is held fixed at its limit L j. Here again there are n + equations and n + unknowns and the solution is a -manifold in the parameter space of. This -manifold can be computed efficiently by the continuation method.

4 Notice that as the feasibility boundary on this manifold is traversed, either the singularity condition is approached again which means that the singular manifold splits from the non-singular (limit-induced) manifold or another limit is encountered. In the first case, all that is needed is to repeat the procedures outlined in the previous subsection. The second case is less straightforward as the structure of the power flow equations changes which creates the following three possibilities: The feasibility boundary continues on the newly encountered limit while the previous limit is enforced concurrently. The feasibility boundary continues on the newly encountered limit while the previous limit is freed. The feasibility boundary continues on the existing limit while the newly encountered limit changes status (either freed or enforced) The spectrum of the Jacobian associated with each of these possibilities need to be investigated to find the true limitinduced segment of the feasibility boundary. III. FEASIBILITY BOUNDARY FOR THE DTE/ITC SUBTRANSMISSION AND TRANSMISSION WIND INJECTIONS For a heavily-loaded heavily-compensated network, visualizing the power flow feasibility boundary in a given parameter space can be very useful in assessing the stability margin. Unfortunately the visualization of this boundary is restricted to three dimensional parameter spaces. Nonetheless, careful selection of parameters can still render the visualization of the boundary instructive for certain types of analysis. One such analysis deals with the complicated interaction of highly variable wind generation at transmission and sub-transmission networks, the reactive requirement of sub-transmission networks and its impact on the voltage stability margin. Figure 4 shows three nodes on the ITC owned kv transmission network where large wind farms are connected. The key issue to be investigated here is that as wind generation reaches its peak across the system how the stability margin, as defined by the distance to the feasibility boundary of the system is affected for different reactive compensation capabilities at the sub-transmission wind injection nodes (i.e. W G, W G and W G 3 ). For this purpose, it is assumed that the transmission wind injection nodes each have a capability of ±5 MVAr and regulate their voltage magnitudes to fixed setpoints in the range of.4-.6 p.u. and the sub-transmission wind injection nodes have the following capabilities: ) No reactive capability ) ±5 MVAr 3) ±3 MVAr 4) ±5 MVAr The POI voltage of subtransmission wind injection is also regulated to fixed setpoints in the range of.-. p.u. However, as the reactive compensation of each node, a combination of the capabilities of wind turbines and ancillary devices such as STATCOMs or SVCs [8], reaches its limit the POI voltage can no longer be regulated to its setpoint and starts to rise. Using the method for computing the feasibility boundary discussed in section II, the parameter space is defined as the transmission generation index λ t and sub-transmission generation index λ s and the boundaries corresponding to these 4 cases of subtransmission wind farms reactive capability are depicted in Figure 5. Note that as λ s increases from, the generation at W G, W G and W G 3 increases with a fixed ratio which here is chosen to be but can be modified to represent the generation capacity ratios at these nodes. The same holds for λ t with respect to W G 4, W G 5 and W G 6. λ s.5.5 ± 5MVAr ± 3MVAr ± 5MVAr No Compensation λ t Fig. 5. Power flow feasibility region for sub-transmission and transmission wind injection for varying reactive capabilities at the DTE s sub-transmission wind generation nodes W G, W G and W G 3. It is clear that at high levels of sub-transmission wind generation, the feasibility region shrinks as the reactive capability is increased. This contrast, with regard to reactive capability, becomes less apparent as transmission wind injection increases. This is due to a combination of factors. First as λ t increases the voltage magnitudes at the transmission system tend to decline. This in combination with reactive power flow into the subtransmission network pushes the OLTCs to their upper limit at which point the voltages across the 4kV network start to decline. This in turn alleviates the voltage rise issue and curbs the reactive requirement of the 4kV sub-transmission network. Hence OLTCs reaching their limits has a stabilizing impact on the DTE/ITC system by relieving the strain on the reactive reserves of the transmission system. Figures 6 and 7 contrast the singular and non-singular segments of the feasibility boundary for the two cases of no compensation and ±5MVAr capability. Notice that the red segments correspond to the capacitive limits of either W G 4, W G 5 or W G 6. IV. ONGOING WORK The focus of the ongoing research is to further enhance the computational efficiency and robustness of the proposed method. A detailed description of the underlying nonlinear least square estimation technique and its parameters along with more examples illustrating the complicated interaction of reactive limits on the feasibility boundary will be presented in a future publication. This will also include a more detailed analysis of the feasibility boundaries in relation to subtransmission and transmission wind injections and potentially interesting patterns of interaction between OLTCs and wind farm reactive limits at both subtransmission and transmission levels.

5 WG T Gen T3 WG WG 5 WG 3 WG 4 T4 WG 6 T T5 Fig. 4. DTE/ITC wind development network: The kv transmission loop (in red) is connected to the meshed 4kV subtransmission network (in black) through T, T, T3, T4 and T5 OLTC transformers. Subtransmission wind injection nodes are highlighted in blue..5.5 λ λ s s λt λt.5 Fig. 6. Power flow feasibility boundary for sub-transmission and transmission wind injections corresponding to zero reactive capability at the DTE s subtransmission wind generation nodes W G, W G and W G3. Fig. 7. Power flow feasibility boundary for sub-transmission and transmission wind injections corresponding to ±5 MVAr capability at the DTE s subtransmission wind generation nodes W G, W G and W G3. R EFERENCES [5] Y.V. Makarov, Z.Y. Dong and D.J. Hill, On Convexity of Power Flow Feasibility Boundary, IEEE Transactions on Power Systems, vol.3, no., pp.8-83, May 8. [6] I.A. Dobson and L. Lu, Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered, IEEE Transactions on Circuits and Systems, vol.39, no.9, pp , Sep 99. [7] Y. Kataoka and Y. Shinoda Voltage stability limit of electric power systems with generator reactive power constraints considered, IEEE Transactions on Power Systems, vol., no., pp [8] E. Camm, et al., Reactive power compensation for wind power plants, Proceedings of the IEEE PES General Meeting, Calgary, Alberta, July 9. [] Global Trends in Renewable Energy Investment 4, Annual Report Published by the Frankfurt School-UNEP Collaborating Centre for Climate & Sustainable Energy Finance (FS-UNEP), April 4 [Online]. [] V.A. Venikov, et al., Estimation of electrical power system steadystate stability in load flow calculations, IEEE Transactions on Power Apparatus and Systems, vol.94, no.3, pp.34-4, May 975. [3] P.W. Sauer and M.A. Pai, Power system steady-state stability and the load-flow Jacobian, IEEE Transactions on Power Systems, vol.5, no.4, pp , Nov. 99. [4] I.A. Hiskens and R.J. Davy, Exploring the power flow solution space boundary, IEEE Transactions on Power Systems, Vol. 6, No. 3, August, pp