Analytically Exact Solutions for Distributed Voltage Stability Index based on Power Flow Circles

Analytically Exact Solutions for Distributed Voltage Stability Index based on Power Flow Circles Kishan Prudhvi Guddanti, Student Member, IEEE, Amarsagar Reddy Ramapuram Matavalam, Student Member, IEEE, Yang Weng, Member, IEEE arxiv:90.0303v [cs.sy] 5 Jan 09 Abstract The real-time monitoring of long-term voltage stability is the key to wide-area monitoring, protecting and control applications. Although centralized monitoring methods are exact and accurate, they are unscalable, calling for decentralized methods. Unfortunately, current decentralized algorithms are accompanied with approximation errors. Solving this problem, we propose a new distributed voltage stability index VSI) pertaining both accuracy and scalability. Specifically, the proposed VSI is calculated at each bus locally only using communication between phasor measurement units PMUs) at the neighboring buses. In addition to providing VSI at different loading conditions, the new VSI also features the ability for fair comparison at zero loading condition, so that the system operator can identify the most critical bus induced by network topology and branch admittance values. As a byproduct, the VSI can identify the line outage location. The effectiveness of the proposed index is demonstrated successfully with IEEE test systems. Index Terms Rectangular coordinates, Long-term voltage stability, Distributed algorithm, Phasor measurement units I. INTRODUCTION The advent of phasor measurement units PMUs) in the 980s made it possible to obtain real-time, time-synchronized information of voltage and current phasors with many use cases [], []. These potential applications of PMUs are not only limited to static monitoring but also tracking of the system dynamics [3]. For example, in wide-area monitoring, protection, and control WAMPAC) concept [], the realtime monitoring of long-term voltage stability is needed for system security [3]. Considered as a bifurcation problem [], [5], long-term voltage stability is caused by the inability of the generation and transmission system to provide sufficient power to loads, e.g., caused by infrastructure limitations and generators reaching VAR limits [6], [7]. Due to its importance, the instability needs to be tracked in real-time, currently supported by Thevenin or other sensitivity methods using PMU/SCADA measurements [8], [9]. For Thevenin-based methods, it only requires local measurements at a bus, e.g., using a PMU to calculate a twobus Thevenin equivalent of the power system. In such an equivalence model, one node represents the bus of interest while the other accounts for rest of the system. Therefore, we can compute an index value at a bus of interest using maximum power transfer theorem [0]. Therefore, these Thevenin-based methods have a nice property of decentralized computation of the index values [0] [3]. Notably, such property have drawn steadily increased interest in recent literature in different applications. These distributed applications include WAMPAC [], [5], load shedding [6], [7], optimal power flow [8], [9], economic dispatch [0], transfer capability assessment [], and control techniques in microgrids [3], []. While the Thevenin-based method has a natural decentralized property, the direct application of such a method has an unavoidable error, due to the approximation error in linearizing the system and the ideal assumption of constant Thevenin impedance during a small interval of time, in which there is a sufficiently small change in operating conditions [5]. While the latter issue is solved for a simple two-bus Thevenin equivalent of the general power system [6]. However, two-bus Thevenin equivalent of the power system is too conservative to solve multi-load power systems for more practical use of the calculated index [6], [7]. This is addressed by introducing the concept of coupled single-port models that can account for the impact of other branch currents using coupling voltage [7], [8]. However, the original problem of linearizing a non-linear system persists even though [7] [9] improved the accuracy with a better approximation method. Fundamentally, due to the approximation in Thevenin methods, it is difficult to provide a formal proof that an index for saddle-node bifurcation point SNBP), calculated using two-port network equivalent, indicates the SNBP of the actual system. Fortunately, there are methods to solve this exactness problem. Such a solution is built on the observation that Thevenin methods involve network reduction, which made them inaccurate. Later, it is shown that the singularity of Jacobian is used to indicate SNBP [30], [3]. This leads to an approximation-free analysis, namely the sensitivity method. Such sensitivity methods are based on the concept of the singularity of the Jacobian, which is calculated using either state estimation results or full observable system measurements. When the minimum singular value is zero, it indicates the SNBP. However, when this sensitivity index is non-zero, it does not have the concept of distance to SNBP. Furthermore, the sensitivity methods strictly require a centralized architecture [3], [33], requiring different entities to cooperate in a deregulated environment and may need data to be transmitted over long distances raising cyber-security concerns. While there are works claiming to provide local calculation with the sensitivity method [5], it reintroduces errors and new drawbacks. For example, it assumes that eigenvalues of the Jacobian are non-negative while calculating the VSI, not true for highly compensated networks, e.g., IEEE 300-bus network. Hence, there is a tradeoff between the Thevenin method and the sensitivity-based method on distributed capability and the exactness of the solution. Therefore, in this paper, we aim at answering the following question: can we obtain a VSI that is ) locally calculated like the Thevenin-based methods), ) without approximation error like sensitivity methods), and retains the sense of distance to voltage collapse point unlike

the sensitivity methods)? We solve these problems by looking at a general power flow equations at each bus as circles, instead of looking at two bus equivalent of the power system [3]. Since there are no approximations when utilizing the n-bus power flow equations, our index derivation proves that: when SNBP at a bus is reached then we have SNBP of the system for sure. This leads to a new voltage stability index that is exact approximation free) and calculated locally. It also provides a sense of distance when it is non-zero while exactly indicating the collapse point when its value is zero. The proposed VSI is tested on various IEEE systems such as the IEEE 30, 8, and 300-bus systems. Satisfactory observations are made accordingly. In summary, the key contributions of the paper are: The proposed VSI can identify the SNBP exactly with a distributed communication framework between neighboring buses. An exactness proof is provided. The proposed VSI is linear to loading level and can indicate the topological effect to the loadability. VAR limits of a generator can be incorporated in the proposed VSI for a PV bus seamlessly using circles. Also, the proposed VSI can identify the line outage location. The paper is organized as follows: Section II compares polar, rectangular formulations of power flow equations and describes the tracking of PV curve using power flow circles. Section III introduces a new representation of rectangular power flow equations for ease of deriving the VSI and presents the VSI equation. Section IV shows the efficacy of the proposed scheme and VSI via simulations on different IEEE systems. Section V concludes the paper. II. MATHEMATICAL MODELING FOR CIRCLE-BASED TRACKING The inaccuracies in Thevenin-based VSIs are due to the approximation of the system as a two-port equivalent network. In this section, we show why to move from polar coordinatebased power flow equations to rectangular coordinate-based equations to reduce mathematical complexity. Let p d and q d be the active and reactive power injections at bus d. Let g k,d +j b k,d is k,d) th be the element of the admittance matrix Y. For bus d s neighboring bus set, we use Nd) for such a representation. A. Coordinate Selection In polar formulation of power flow equations, the complex voltage phasor ˆv d at bus d is represented by polar coordinates ˆv d = ˆv d θ d. The power flow equations in polar coordinates are given by p d = ˆv d ˆv k g k,d cosθ d θ k )+b k,d sinθ d θ k ))), q d = ˆv d k Nd) k Nd) ) ˆv k g k,d sinθ d θ k ) b k,d cosθ d θ k ))). In a distributed framework, the neighboring voltage phasors are constant values obtained from the PMUs as measurements. Thus, ) and ) can be abstractly represented by f ˆv d,θ d ) and hˆv d,θ d ). For power flow analysis, such as this paper, ) the intersection of f ˆv d,θ d ) and hˆv d,θ d ) for a given p d and q d is the key, leading to voltage solutions) at bus d. However, such an intersection is not easy to represented a) Polar formulation. b) Rectangular formulation. Fig. : Voltage solutions at d for different formulations of power flow equations. analytically, leading to the trouble in existing solutions for distributed voltage stability index. For example, f ˆv d,θ d ) and hˆv d,θ d ) in Fig. a shows one illustration, where the polar coordinate based intersection is complex to illustrate. This is due to the fact that f ˆv d,θ d ) and hˆv d,θ d ) are functions of two functionals, namely a multiplication of trigonometric functionals and polynomial functionals. Differently, ) and ) are reduced into quadratic form in case of rectangular with ˆv d = v d,r +j v d,i, leading to p d = t d, v d,r +t d, v d,r +t d, v d,i +t d,3 v d,i, 3) q d = t d, v d,r t d,3 v d,r +t d, v d,i +t d, v d,i. ) The parameters t d,,t d,,t d,3,t d, are given by t d, = g k,d, t d, = v k,r g k,d v k,i b k,d ), 5) k Nd) t d,3 = k Nd) k Nd) v k,r b k,d +v k,i g k,d ), t d, = k Nd) b k,d, 6) Such a representation is good for distributed framework and mathematically more tractable. This may be the reason, why the index is observed to be linear in the numerical section. Specifically, 3) and ) are homogeneous quadratic equations of two variablesv d,r, v d,i that can be represented in a standard geometric form: circles as shown in Fig. b. Unlike polar formulation, the voltage solution at bus d can be understood in a simple and intuitive manner as the intersection of real and reactive power circles at bus d [3], [35]. The real power equation 3) at bus d can be represented as a standard circle with center C p and radius r p. The reactive power equation ) at bus d can be represented as a standard circle with center C q and radius r q. These centers and radii are given by td,, t ) d,3 td,3, C q =, t ) d,, 7) t d, t d, t d, t d, C p = r p = r q = p d + t d,) +t d,3 ) t d, t d, q d + t d,3) +t d, ) t d, t d,, 8). 9)

3 a) P and Q at bus 3. b) Voltage solution: intersection of power flow circles zoomed version of Fig. a). c) PV curve for a certain load increase direction. Fig. : PV curve tracking using power flow circles. Remark. In general, PQ buses have real and reactive power circles. However, in case of a PV bus, we have constant real power and voltage conditions. Hence PV buses also have two circles, ) real power circle 3) and ) voltage circle centered at origin with radius V specified. Later we show that the proposed VSI derivation can also identify the voltage collapse point for PV buses due to VAR limit violations. B. PV Curve Tracking Using Power Flow Circles This interpretation of power flow equations as circles for a -bus system exists in the literature [3]. However, a similar understanding of power flow circles for an n-bus system is used in this paper to derive an analytic equation for VSI that is approximation free. In this subsection, we describe the advantage of its simplified form to visualize power flow circles 3),) and thereby track the PV/QV curve exactly in a distributed fashion. Specifically and in a distributed framework, all the neighboring bus voltages are measurements obtained from the PMUs and hence they are constant values. This makes it possible to write 3), ) as two circles and their intersection provides the voltage solution at d for distributed applications. This circle representation of n-bus power flow equations in a distributed setting provides an intuitive approach to track the PV curve and its SNBP at bus d. Geometrically, two arbitrary circles can have two, one or no points of intersections. Hence, in this per bus analysis at noded, the two common points between the power flow circles indicate the high and low voltage solution, one common point between the power flow circles indicate the SNBP voltage solution and no common points represent the in-feasibility of operating conditions. For illustration purpose, we use a completely connected 3- bus system with connecting branch admittance being j 0.5. The behavior of the power system at bus 3 is observed by varying the real power at this bus with a constant power factor of 0.9. A load with the real power of 0.5 p.u. is connected to bus. Specifically, the real power demand at bus 3 is varied from 0. p.u. to 0.5 p.u. until the power flow circles at bus 3 had only one common point. The corresponding power flow circles intersection is plotted as shown in Fig. a. In Fig. b, the solid and dashed circles indicate the reactive and real power circles respectively. The common points A, B, C, D and E in Fig. b, indicate the higher magnitude voltage solutions corresponding to a real power load of 0., 0., 0.3, 0. and 0.5 p.u. respectively at bus 3. Similarly, lower magnitude voltage solutions are also present in Fig. b. The intersection of black power flow circles has only one common point E due to the critical loading at bus 3. This common point E represents the saddle-node bifurcation point SNBP) and Fig. c shows the PV curve for a certain load increase direction. More importantly, it can be observed in Fig. b that the distance between power flow circles decreased to zero SNBP) as the real power demand increased to critical load value. A metric to define the closeness of the current operating point to SNBP can be obtained by formulating an equation that indicates the distance between power flow circles. When this defined indicator is zero, it exactly represents the SNBP on PV curve. III. DISTRIBUTED VOLTAGE STABILITY INDEX From the last section, we know that the n-bus power flow equations in rectangular coordinates enable tracking the PV curve without any approximations in a distributed fashion. From Fig. b, it is also observed that the distance analogously margin) between power flow circles decreased to zero as the load increases. Calculation of such distance to the voltage collapse point from current operating conditions using 3), ) is not straightforward because of their quadratic form. Hence in this section, an improved representation of power flow equations are used to understand and derive the VSI. A detailed derivation can be found in Appendix. A. A. Representation of Power Flow Equations In general, the power flow equations with circle form are represented in two different forms: ) vectorized tuple form [35] and ) Hermitian matrix form. Therefore, we start with obtaining a vectorized tuple form, which is subsequently converted into a hermitian matrix form for compactness. ) 3-Tuple Vector Representation A circle is a set of points x R in the real Euclidean plane that satisfy equations of the form shown below a x x)+b x+c = 0. 0) For ease of computation, the real and reactive power equations 3), ) can be represented as 3-tuples ), ) as shown

below { [ ] } td, t T d,3 {a p,b p,c p } =,, p d, ) t d, t d, t d, { [ ] } td,3 t T d, {a q,b q,c q } =,, q d. ) t d, t d, t d, From these tuple representation, a real power circle at bus d maybe represented by scalars a and c, and a vector b. The centers 7) and radii 8), 9) of the power flow circles are represented in vector forms as shown below using the tuple representations ) and ). C p = b p,c q = b q, 3) r p = bp b p ) ) c p,rq = b q c q. ) ) Hermitian Matrix Representation [36] presents a circle equation 5) in form of a matrix and it is observed that the circle equations always form Hermitian matrices 6), where Z is the complex conjugate of Z. C Z,Z ) = A Z Z +B Z +C Z +D = 0, 5) [ ] A B =, 6) C D where A 0. A and D are always real, B and C are always complex conjugates for a circle. Since the rectangular power flow equations are circles, they can also be represented as Hermitian matrices. Here onwards, a general hermitian representation of a circle is presented below and then power flow equations are substituted into the generalized hermitian representation. General Hermitian Matrix Representation of a Circle: Let Z = x + j y in complex plane be the locus of a circle C Z,Z ) with radius ρ and center γ = α + j β then the equation of this circle is given by 7). 8) is compared with 5) to determine the elements of hermitian matrix 6) where A =, B = γ, C = γ and D = γ γ ρ. x α) +y β) = ρ, 7) Z γ = ρ, Z γ) Z γ ) = ρ, Z Z γ Z γ Z +γ γ ρ = 0. 8) Hermitian Matrix Representation of Power Flow Equations: The elements of real power hermitian matrix C p are calculated using 3) and ) as shown below A p =,B p = C p bp bp = =, ) ) bp bp C p = C p ) = =, D p = C p ) C p rp = c p. 9) Similarly, the elements of reactive power hermitian matrix are also derived. C p and C q are the hermitian representation of power flow equations. C p = ) bp bp c p,c q = ) c q. B. Voltage Stability Index In this section, an indicator is designed to indicate the distance between the current operating point and SNBP. This is achieved through circle geometry and representing the rectangular power flow equations 3), ) as Hermitian matrices C p, C q. First, two arbitrary circles C and C are considered to intersect each other at a non-zero angle ω then a general distance metric is shown below that indicates the span for circles C and C to make an angle ω = 0 with each other. When ω = 0, the circles C and C touch each other either externally or internally. Second, this obtained general distance metric is translated into power systems domain. General Conditions: The power flow problem of calculating voltage at bus d solves the intersection of two circles 3) and ). In such a scenario, from the proof in the Appendix, an indicator is derived. The general condition for the metric that indicate the distance between two circles is presented below. Fig. 3: Intersection of circles C and C. Let there be two circles 0) and ) whose centers are γ, γ and radii ρ,ρ respectively. C Z,Z ) = A Z Z +B Z +C Z +D = 0, 0) C Z,Z ) = A Z Z +B Z +C Z +D = 0. ) The distance between the two circles can be indicated by. The three conditions ), 3) and ) represent the behavior of family of circles depending on the signed magnitude of. = > 0, ) = = 0, 3) = < 0, )

5 where = A ρ, 5) = A ρ, 6) = A ) A δ ρ ρ, 7) δ is the distance between centers of the circles C and C, ) indicates the scenario when the circles 0), ) have two common points. 3) indicates when there is only one common point i.e., external and internal touching circles. ) indicates no common points between the circles C and C. Application to Power Flow Equations: In power systems domain, these circles 0), ) represent the real and reactive power equations 3), ). These power flow equations are represented using Hermitian matrices C p, C q. corresponding to power flow circles C p, C q is determined by calculating the three terms, and. These three terms for power flow equations are determined by calculating the centers and radii for real and reactive power circles as shown below. ρ = r p = bp b p ) ) c p, ρ = r q = b q c q, γ = C p = b p, γ = C q = b q, δ = γ γ = C p C q. Thus, the VSI represented by of the power flow circles is determined by calculating, and for power flow equations using the centers and radii. ) ) = c p b p c q b q b p c p ) b q The normalized form of is given by normalized = base-case 8 b p b q c q )). 8) no-load where no-load is the value of proposed VSI with zero load and voltage of p.u. at all buses of the network, base-case is the value of proposed VSI at base case loading. Remark. VSI extension for PV Buses): 8) is the distributed VSI for a PQ bus and a similar VSI can be derived for a PV bus. The voltage solution is determined by the intersection of voltage and real power circles for PV buses. When there is no intersection between these two circles then it indicates that there is no feasible solution as V specified cannot be maintained anymore at that bus. IV. SIMULATIONS The proposed VSI is tested on different IEEE test systems such as IEEE-6, 30, 8, and 300 bus systems under several scenarios to validate and test its accuracy. The performances are similar. For illustration purpose, we will show detailed results on the first two systems. Simulations are performed using continuation power flow CPF) from Matpower [37] to generate the voltage measurements that are used to calculate, the proposed VSI. Unless otherwise specified the reactive power limits of generators are not considered. A. Proportional Load & Gen Increase for IEEE-6 Bus and IEEE-30 Bus Systems The loads and generations in the test systems are increased from no load to critical loading in a loading direction proportional to the base loading and the voltage measurements to calculate the proposed distributed VSI. Fig. and Fig. 5 shows the un-normalized and normalized VSI vs the scaling factor λ for the IEEE-6 bus system. The first observation is that at the critical λ value corresponding to λ = 3.), the VSI of bus 5 goes to 0, implying that this is the critical bus in the system and this fact is also verified using sensitivities. The next observation is that the proposed VSI varies in a nearly linear manner with the load scaling parameter at all the load buses buses,5 & 6). This useful property of VSI allows for reasonable extrapolation of margin prediction and this property will be explored in the future. Fig. : Un-normalized VSI for IEEE-6 bus system. Fig. 5: Normalized VSI for IEEE-6 bus system. It can also be observed From Fig. that the VSI corresponding to scaling parameter λ = 0 indicates the no-load conditions with voltages throughout the system being p.u. Thus, the proposed VSI can capture the effect of topology on the voltage stability at each node due to the network structure and branch admittance values of its neighbors. At no-load, the least VSI among all the buses in IEEE-6 bus system occurs at Bus. This indicates that Bus is closer to voltage collapse than any other bus in the network under no-load and it remains a second critical bus at voltage collapse point. The different starting values of the VSI at various buses in the un-normalized

6 case make it hard to easily identify the critical bus as the VSI curves can have different slopes and can intersect, as seen in Fig.. This is why normalizing helps as it ensures that the VSI value is at no-load, making it easier to identify the critical bus and minimizing the possibilities for the intersection. It can be seen from Fig. 5 that the normalized index is able to identify the weakest node to be bus 5 that is prone to voltage collapse starting from λ = 0. In a similar manner, Fig. 6 plots the normalized VSI at PV buses buses,5 & 8) and PQ buses buses,6,9 & 30) in the IEEE-30 bus system as the system loading increases. It can be seen that the VSI at load buses varies almost linearly versus the load scaling, just as in the 6-bus system. Also, bus 30 has the lowest value of the VSI implying that this is the weakest bus in the network with respect to voltage stability and this fact is verified using sensitivities. These results validate the proposed VSI and demonstrate the utility of the VSI to monitor voltage stability using measurements in a distributed manner. Fig. 6: VSI for IEEE-30 bus system without VAR limits. Remark 3. A min-consensus approach can be used for global knowledge of worst-case index and location of the critical bus. In the interest of space, preliminary results for a few test systems are presented in this submission. The derivation of VSI is also not introduced in the first 8 pages of the paper. B. Different Load Increase Direction With Gen Q-Limits In this scenario for IEEE-30 bus system, the loads at buses 7-30 are increased proportionally and all the generator injections are increased proportionally. The reactive power limit of generator- is enforced, leading to a critical λ = 0.63, and the normalized VSI at buses, 6, 9 & 30 for this scenario are plotted in Fig. 7. It can be observed from Fig. 7 that VSI behavior is reasonably well approximated as piecewise linear functions. The slope of the VSI changes at the instant a generator reactive power limit is reached, due to the PV to PQ switching. The VSI calculation does not explicitly change due to PV-PQ switching and is indirectly incorporated due to the change in the measurements. Thus, the measurement based VSI can keep still monitor the voltage stability when the reactive limits of generators are enforced. Fig. 7: VSI for IEEE-30 bus system with VAR limits. C. Effect of Measurement Noise on VSI One of the key practical challenges for any measurement based methodology is the measurement noise that is inherently present in the devices. To understand the impact of noise on the proposed methodology, additive Gaussian noise with zero mean and a standard deviation of 0.00 p.u. on voltage magnitudes and 0.0 on phase angles are introduced in the measurements. To demonstrate the robustness of the proposed VSI with regards to noise, it is compared to the conventional local Thevenin index LTI) [3], [6], [30]. Both the proposed VSI and the LTI lie in the region 0 and so the impact of noise on their value can be compared. Fig. 8 shows the effect of noisy measurements on the proposed VSI when compared to the conventional LTI. The proposed VSI is observed to have a standard deviation of 0.0066 while the LTI has a standard deviation of 0.068, roughly four times more. Thus, the proposed methodology is more robust to noise, ensuring that the alarms triggered using this VSI will have lesser false positive rates. It is also important to note that as the measurements or the system becomes noisier due to renewable penetration), the LTI is more affected than the proposed VSI. To the best of our knowledge, there is no distributed sensitivity index [5] that is normalized within a fixed range of values and is analytically accurate for all systems. Fig. 8: Effect of noise in measurements on proposed VSI versus LTI at node 30 in IEEE-30 bus system. D. Effect of Line Outage on VSI Network reconfiguration due to line outage either due to faults or for maintenance) is a frequent occurrence in the power system and so the proposed VSI calculation methodol-

7 ogy should effectively handle this scenario. In order to study the effect of topology change on the VSI, the line between buses 5 and 3 in the IEEE 30-bus system is taken out of service as the load is increasing. Fig. 9 plots the VSI at buses,5,8 & 9 and it is observed that VSI drops the moment the line outage occurs, indicating that the margin has reduced. In addition, it can also be observed that the index at bus closest to the line outage reduces the most and this fact can be used to identify the fault location in an unsupervised manner. The conventional Thevenin index is unable able to localize the fault location just by using the measurements. Finally, it is also observed from Fig. 9 that the VSI calculated at each bus without updating the admittance matrix Ybus) is almost the same as the VSI calculated by using the new Ybus. Thus the distributed VSI calculation methodology provides a reasonable estimate of the VSI for the operator/relay to activate controls/alarms in the time it takes to correct the Ybus, thus improving the situational awareness of the grid. Fig. 9: Line outage at line-30 in IEEE-30 bus system and effect of Ybus on VSI V. CONCLUSION This paper proposes a PMU measurement based voltage stability index that can accurately identify the saddle-node bifurcation point of the system. The key novelty of this work is the mathematical proof of the accuracy of the index and the distributed nature of the index. The index is derived by analyzing the power flow equations in the rectangular form as circles and at the saddle-node point, the value of the index at the critical bus is exactly 0. Furthermore, the index can be calculated using a distributed communication framework between neighboring buses, making this a scalable approach for assimilation and analysis of PMU measurements in the grid. In the various test scenarios, the index behaved as expected and detected the critical bus under various test cases such as different load increase directions, generator reactive limit enforcement, line outage, etc. The wide area nature of the proposed index makes it robust to measurement and system noise that adversely effects similar techniques such as Thevenin methods. Finally, The proposed index can localize the line outage locations in an unsupervised manner purely using PMU measurements making it a promising method for event detection and localization of other events as well. Similar results were observed on the IEEE-8 and 300 bus systems for various scenarios. The proposed index behavior is also compared with distributed and decentralized methods and has more reliable performance. These results are omitted in the interest of space. 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DeRose, A geometric characterization of parametric cubic curves, ACM Transactions on Graphics TOG), vol. 8, no. 3, pp. 7 63, 989. APPENDIX A. Proof/Derivation of Distributed Voltage Stability Index ) Overview of the Problem C p and C q are the real and reactive power equations represented as hermitian matrices. These two circles intersect with each other at two points for the existence of a feasible voltage solution and as the load increases, eventually, there exists only one feasible voltage solution externally or internally touching circles) at SNBP. It is desired that the proposed index indicate some sort of non-zero distance from the current operating point to SNBP when the two circles intersect and zero distance when they touch each other. ) Mathematical Concept Behind the Proposed VSI In this subsection, a mathematical concept related to the family of circles is utilized to derive the proposed VSI. This family of circles C is generated by two real variables λ and λ. It is important to note that the family of circles C λ, λ is hermitian since C p and C q are hermitian themselves. In order to obtain the desired indicator, we use determinants and their two important properties. ) In analytical geometry, determinant indicates the signed area of a parallelogram and this can be easily applied to higher dimensions [38]. ) The determinant of a complex Hermitian matrix is always real. C = λ C p +λ C q 9) For the family of circles C, its determinant gives the signed area. The sign here indicates whether symmetricalness achirality) across the radical axis of C is preserved or not. For example, a positive sign preserves the symmetricalness across radical axis implying the existence of the family of circles C λ, λ intersecting circles). Similarly, a negative sign does not preserve the symmetricalness across radical axis implying the non-existence of family of circles C λ, λ. In the power flow problem, a negative sign indicates the non-feasibility of voltage solution non-intersecting circles). While the absolute value of signed area provides a sense of distance to SNBP from the current operating point. Briefly, a positive signed area represents a feasible solution with its magnitude indicating a sense of distance to SNBP. Similarly, a negative signed area represents a non-feasible solution with its magnitude indicating a sense of distance to feasible SNBP. 3) Derivation of VSI In power systems domain, the role of real variablesλ,λ in 9) is similar to scaling load/generation and inducing network topology changes that directly impact the radii and centers of power flow circles. The non-existence of feasible solution represented by negative signed area indicates that λ, λ, the power flow circles formed by the PMU measurements have no common points. bp C = λ ) bp +λ ) c p c q bp λ +λ λ +λ C = ) ) bp λ +λ λ c p +λ c q bp λ +λ λ +λ C = ) ) bp λ +λ λ c p +λ c q,,,

9 Upon calculating the determinant and simplifying C into a quadratic form of real variables λ, λ and real coefficients, we have C = p λ + q λ + pq λ λ, 30) where ) p = c p b p, ) q = c q b q, pq = 8 b p b q b q c q )). b p c p ) 30) is a standard bivariate quadratic form which can be written as shown below C = [ [ ] [ ] p λ λ ] pq λ = λ T M λ. 3) pq q λ From 3), the determinant of C is same as that of M as shown below. For 3) and ) to be real circles, their radius should be positive. This implies that p and q should be less than zero. = p q pq. 3) According to Sylvester s criterion, the quadratic form 30) maybe a negative definite/indefinite or a complete square since p < 0 and q < 0. Now a set of three general conditions can be obtained depending on the value of as shown below. ) When > 0 then the quadratic form elliptic) is negative definite. This implies that the value of C is always positive λ & λ. This shows that the signed area is always positive indicating that the real and reactive power circles intersect at two points. ) When < 0 then the quadratic form hyperbolic) is indefinite. This implies that the value of C is always negative λ & λ. This shows that the signed area is always negative indicating that the real and reactive power circles have no intersection. In such a scenario, when < 0, then the current operating point is beyond the SNBP with a distance of. Thus, theory presented here can measure the distance to SNBP when the given operating condition values have no feasible solution as well. 3) When = 0 then the quadratic form parabolic) is a complete square. This implies that the value of C is always zero λ & λ. This shows that the signed area is always zero indicating that the real and reactive power circles intersect at only one point. In such scenario, when = 0, then the current operating point is exactly at SNBP.