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Identification of weak nodes in power systems Dehnel, Morgan P. 1987

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IDENTIFICATION OF WEAK NODES IN POWER SYSTEMS by MORGAN P. DEHNEL B.A.Sc. THE UNIVERSITY OF BRITISH COLUMBIA, 1986 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER 1987 « MORGAN P. DEHNEL, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. DEPARTMENT OF ELECTRICAL ENGINEERING The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: OCTOBER 1987 Abstract This thesis describes a method for determining weak nodes in power systems which may cause divergence in Newton-Raphson loadflow methods. When divergence occurs in production loadflow programs, information related to the cause of divergence is not available. The "Weak Node Method" of this thesis provides such information by identifying one or more weak nodes. The development of the Weak Node Method required extensive experimentation with the Newton-Raphson method. The behaviour of the vectors and matrices of the Newton-Raphson method during divergence are discussed in an effort to familiarize the reader with observed trends. These trends suggested the techniques which comprise the Weak Node Method. With this method, a "quasi" solution is sought first, and, secondly, three analysis procedures are then used to pinpoint the weak nodes. The method was tested using three test cases which would normally have caused the Newton-Raphson loadflow method to diverge. ii Table of Contents Abstract i i Table of Contents iii List of Tables v List Of Figures vi Acknowledgements vii 1. INTRODUCTION 1 2. NEWTON-RAPHSON METHOD 2 2.1 General Newton-Raphson Method 2 2.2 Divergence of the Newton-Raphson Method 4 2.3 Divergence in Loadflow Studies 6 2.4 Conclusion 6 3. MODIFIED NEWTON-RAPHSON METHODS 7 3.1 Test Case Type 7 3.2 Loadflow Equations 7 3.3 The Minimum Mismatch Solution 9 3.4 Damped Newton-Raphson Method 10 3.5 Observation of Newton-Raphson Behaviour 14 3.6 The Singularity Search Method 21 3.7 Conclusion 22 4. ANALYSIS AT THE MINIMUM MISMATCH DISTANCE SOLUTION 23 4.1 Relationship Between Minimum Mismatch Distance Solution and System State 23 4.2 Loadflow Output Analysis 25 4.3 Nodal Mismatch Analysis 26 4.4 Sensitivity Analysis 27 4.5 Conclusion : 31 iii 5. T E S T C A S E S 32 5.1 Active Power Overload at a Single Node 32 5.2 Reactive Power Overload at a Single Node 36 5.3 System Overload 38 5.4 Conclusion 40 6. S U G G E S T I O N S F O R F U R T H E R R E S E A R C H 41 7. C O N C L U S I O N 42 L I S T O F R E F E R E N C E 43 A P P E N D L X A . Loadflow Solution with the Newton-Raphson Method 44 A P P E N D I X B . Damped Newton-Raphson and Singularity Search Method Comparison . . . . 47 A P P E N D I X C . Twenty Eight Node Test Case Data 48 A P P E N D L X D . Five Node Test Case Data 51 A P P E N D L X E . Routines Required for the Thesis Research 53 iv List of Tables 3.1 Algorithm Performance when Analytical Solution Exists 13 3.2 Algorithm Performance when No Analytical Solution Exists 14 3.3 Inner Product Data 15 3.4 Inner Product Data 16 4.1 Solution Changes as a Function of P 2 24 4.2 Solution Changes as a Function of Q 2 24 4.3 Mismatch Values at the Minimum Mismatch Distance Solution 26 4.4 Sensitivity Results 30 5.1 Solution Changes as a Function of P 2 8 33 5.2 28 Node Case Line Quantities for P 2 8 = -2.05 p.u 34 5.3 Nodal Mismatches 34 5.4 Sensitivity Results 35 5.5 Solution Changes, as a Function of Q 2 8 36 5.6 Nodal Mismatches 37 5.7 Sensitivity Results 37 5.8 Solution Change from Base Case 38 5.9 Nodal Mismatches 39 5.10 Sensitivity Results 40 B. l Damped Newton-Raphson and Singularity Search Method Results 47 C. l Initial Voltage Guess 48 C.2 Load, Generation and Shunt Capacitance 49 C.3 Line Impedances 49 C. 4 28 Node Solution Vector 50 D. l Initial Voltage Guess 51 D.2 Load and Generation .- 52 D.3 5 Node Solution Vector 52 v List of Figures 2.1a One Dimensional Scenario 1 4 2.1b One Dimensional Scenario 2 5 2.2 One Dimensional Scenario 3 5 3.1 p Mismatch Curves 9 3.2 The Eigenvalue of the Jacobian Matrix Closest to Zero Versus Iteration 17 3.3 The Condition Number of the Jacobian Matrix Versus Iteration 18 3.4 The Mismatch Distance Versus Iteration 19 3.5 0 3 Versus Iteration 20 5.1 28 Node Scenario 32 C. l 28 Node Test Case 48 D. l 5 Node Test Case 51 vi Acknowledgements I would like to express my gratitude to Dr. H.W. Dommel, who provided ideal conditions for conducting research work. In particular, Dr. H.W. Dommel provided assistance, accessibility, and reassurance throughout this research project. I would also like to give special thanks to: Yanan Yin and Luis Naredo for providing encouragement, and for assisting in computer related matters. Rob Hannebauer and Lyle King for the use of their micro-computer equipment and for their support. West Kootenay Power for providing assistance throughout my academic career. This company has provided the monetary benefits of summer employment in the past, as well as an education in power system engineering. B .C. Hydro for providing information related to this research project. N.S.E.R.C. for providing funding. my family for enthusiastically supporting all of my endeavours. vii 1. INTRODUCTION This thesis presents a practical method for pinpointing the weak node in power systems which cause the Newton-Raphson loadflow method to diverge. The Newton-Raphson method has become the accepted iteration scheme for solving steady-state loadflow problems in power systems [l]. It is an efficient method in combination with sparsity techniques, and it only diverges occasionally for certain types of loadflow scenarios. When diver-gence occurs, however, knowing the weak node in the system aids in correcting the system data. Under these circumstances, a method for determining the weak node is needed. The method for acquiring this information will be called the Weak Node Method for the purpose of this thesis. To illustrate the development and the usefulness of the Weak Node Method, the following will be discussed: (1) the background theory of the Newton-Raphson method to aid in the analysis of divergence (2) the suggestion of a "quasi" solution for loadflow studies which diverge (3) three analysis techniques for pinpointing the weak node at the "quasi" solution point (4) isolation of the weak node in three test cases (5) suggestions for further research: i) the upgrade of existing code to production code standards ii) investigation of alternate analysis techniques at the "quasi" solution point iii) development of procedures for correcting data at the problem node In summation, this thesis describes the Weak Node Method and the basis for formulating the method given that the weak node is needed when correcting system data which has caused the Newton-Raphson loadflow method to diverge. 1 2. NEWTON-RAPHSON METHOD This chapter outlines the background theory of the Newton-Raphson (N-R) method, and situations in which the method fails. This background information will help to establish the mathematical basis of the Weak Node Method. 2.1 General Newton-Raphson Method The N-R method can be used to find the solution of n nonlinear algebraic equations in n unknowns. In other words, the vector * s = (2-1) such that (*?,..., 4) \ G(xs) = 0 (2.2) is sought. The N-R iterative technique for obtaining X S from an initial guess x° can be established by assuming that one has an approximation x1 to the solution and that the Taylor expansion in the neighbourhood ofx' G(xs + Ax1) « Gfx1) + G'(xi)(xi + Ax1 - x1) (2.3) is acceptable. Thus and G(x> + Ax') « 0 « G(x!) + G V J A X ' G'(xi)Axi = -G(x4) where -i+l _ _i X ' + Ax' (2.4) (2.5) Equations (2.4) and (2.5) are matrix equations whose components are defined below, (i) G '(x') is an n X n matrix of partial derivatives known as the Jacobian matrix, where G'(x) = . it an f'Qn . (2.6) represents the slopes of the tangential hyperplanes to the curves G(x) at the point x. 2 (ii) x' is a vector which is the ith iteration approximation to Xs. (2.7) (rn) G(x') is a vector whose elements are the values of the n equations (git..., gn) at x1. G(x') = (2.8) (iv) Ax1 is the correction vector obtained by solving the system of linear equations (2.4). The iteration procedure of the general N-R method can now be stated: General Newton-Raphson Method 1. An initial guess x° is chosen. 2. The system of linear equations defined by (2.4) is solved for Ax1. 3. The approximate solution of the next iteration step is calculated from (2.5). 4. Termination occurs if one of the following criteria is met: i) The elements of Ax1 are close enough to zero. ii) The elements of G(x') are close enough to zero. iii) The iteration limit has been met. 5. If termination has not occurred, then calculations are continued from (2) above. Appendix A expands the above algorithm to include the details required to write a Newton-Raphson loadflow program. Ax1 = (2.9) It is used to find an improved approximation to Xs from (2.5). 3 2.2 Divergence of the Newton-Raphson Method If a N-R algorithm has been implemented to solve n nonlinear loadflow equations, it usually con-verges rapidly to a solution with a high degree of accuracy. However, the method can diverge or converge to undesired solutions under certain circumstances. The following figures illustrate various scenarios for the N-R method applied to one nonlinear equation with one real variable. Fig. 2.1a One Dimensional Scenario 1 In Fig. 2.1a convergence to the desired solution will occur if the initial guess x° is within region c. Convergence to a physically undesired solution will occur if the initial guess is within region a. Undesired loadflow solutions can be recognized from criteria not contained in the equations, such as very large currents or very low voltages. Divergence would be likely if the initial guess were within region b, since the Jacobian matrix would be nearly singular (tangent slope « 0) and the magnitude of the correction vector Ax° would be large. Thus, the approximate solution x1 of the next iteration step could be in a region where convergence was not possible. In Fig. 2.1b convergence will occur if the initial guess x° is in region b. Divergence will occur if the initial guess is in region a, since the approximate solution vector would oscillate about the point where the Jacobian matrix is singular [2]. 4 Fig. 2.1b below illustrates the oscillatory behaviour of the correction vector. D e s i r e d S o l u t i o n Fig. 2.1b One Dimensional Scenario 2 In some instances divergence is independent of where the initial guess is located. A case in which no analytical solution exists, as indicated in Fig. 2.2, is such an instance. X Fig. 2.2 One Dimensional Scenario S Since the general N-R method is susceptible to divergence, it is expected that the Newton-Raphson loadflow method would have similar behaviour. This is indeed the case, and the ramifi-cations of this behaviour are discussed in the following section. 5 2.3 Divergence in Loadflow Studies If a power system is operational and insensitive to reasonable changes in load, it can be thought of as being stable. This type of system is usually what one studies in practice. Such a system would be represented by equations whose curves would be similar to the one in Fig. 2.1a. If load changes were to move the curve up or down on the g axis, with the voltage on the x axis of Fig. 2.1a, it is apparent that load changes would result in relatively small voltage changes at the desired solution point. The N-R method would have no difficulty obtaining the solution provided that the initial guess was close enough to the desired solution (region c). In practice an initial guess of |Vj| = 1.0 p.u. and 6t = 0.0 radians is used, and if the system is stable and operational, this initial guess is sufficiently close to the solution to achieve convergence. Though divergence is rare in loadflow analysis, it can be encountered when the system being studied is stressed, as in contingency studies in which one or more lines are assumed to be out of service. Occasionally under such conditions, power cannot be properly supplied through the system, and the equations which represent the system would have no solution, analogous to Fig. 2.2, or possibly the curves representing the equations would be contorted enough to be similar to the curve in Fig. 2.1b. Another type of study where divergence may occur is the system study for a future condition 10 or 20 years from now. Once the future loads have been forecast, they are applied to the present system in an effort to determine how the system can be upgraded to most effectively handle future loads. In this type of system study reactive power requirements are very crudely known, and inaccurate modelling of reactive power is the usual source of divergence. Specific factors which affect the convergence of N-R loadflows have been documented in [3]: i) position of the reference slack bus ii) existence of negative reactance iii) certain types of radial systems iv) high ratio of long to short line reactance for lines terminating on the same bus v) choice of acceleration factors Of the causes which are likely to lead to divergence, only overloading of the system was used extensively to test the methods of this thesis, while causes ii, iv, and v were used to a lesser degree. 2.4 Conclusion The concepts and causes of divergence of the Newton-Raphson method were presented as back-ground material from which the Weak Node Method will be formulated. 6 3. MODIFIED NEWTON-RAPHSON METHODS This chapter presents two modified N-R methods which prevent loadflows from diverging by pro-ducing a "quasi" solution which yields useful information about the power system involved. The first modified N-R method, presented in Section 3.4, is a computationally efficient algo-rithm derived from [4], which provides some useful information of practical interest. The second method of Section 3.6, which was briefly discussed in [5], is very expensive computationally, but provides more information than the first method. The type of test cases used, and typical loadflow equation behaviour are discussed first to illustrate the motives for choosing these particular modified N-R methods and the "quasi" solutions they seek. 3.1 Test Case Type The focus of this thesis project was to provide useful information to the user of a N-R loadflow program if the case being studied diverged. This information should enable the user to find problem areas in the power system being studied so that corrections can be made to it. As a result of this focus, a number of loadflow cases which diverged were sought as test cases. Though a number of factors can cause divergence in loadflow studies, as indicated in Section 2.3, stressed systems were almost exclusively analyzed. Contingency studies and future system studies stress systems, and tend to account for most diverging N-R loadflow situations in industry. Therefore, such cases were accumulated for test cases. As verified by B.C. Hydro engineers, divergence due to unusual impedance scenarios is rela-tively rare, and when it arises, the problem is usually identifiable. Since loadflow cases are built up gradually, a problem impedance would usually have been introduced in the preceding system update, and would be easily found and corrected. Thus, this type of test case was not studied. 3.2 Loadflow Equations The equations which are solved by N-R loadflow programs are derived in the following manner, as obtained from [6]: Y V = I (3.1) where T _ Pk-jQk i k - — v * — vk 7 (3-2) and thus E Y * ™ v ™ ~ ? k v > Q k = 0 k = l,...,N or v ; £ Y k i a V l u - ( P k - j Q k ) = 0 k = l,...,N (3.3) ui= i In practice (3.3) must be separated into two real equations with real variables, which are analogous to the in (2.8), because (3.3) is not analytical, which makes the use of complex variables impossible. Then N pk(V) = Re(V*]T Y k m V m ) - P k = 0 (3.4) m = l N qk(V) = - I m ( V * ^ Y k m V m ) - Q k = 0 (3.5) m=i In (3.1) to (3.5), a) Y is the nodal admittance matrix, b) V is the complex node voltage vector solution, c) I is the complex node current vector, d) P k is the prescribed active power load or generation at node k, e) Qk is the prescribed reactive power load or generation at node k, f) k or m subscripting refers to the kth or mth node, g) p k(V) is the mismatch between active power supplied to node k and the prescribed active power load or generation, h) qk(V) is the mismatch between reactive power supplied to node k and the prescribed reactive power load or generation. Note that (3.4) and (3.5) are satisfied only for a specific solution voltage vector: / V ? \ V = If no mathematical solution exists, the V vector would never satisfy (3.4) or (3.5), and pk(V) or °ik(V) would never equal zero. In the following section, a one-dimensional example will be analyzed to derive further infor-mation about diverging loadflows. 8 3.3 The Mmimum Mismatch Solution When the N-R method diverges the N-R correction vector directions tend to oscillate about a region in which the mismatch curves cause the Jacobian matrix to be singular. This oscillatory behaviour wastes computer time, and the successive solution vectors do not , in general, provide enlightening information about the system. Therefore, a means of converging to some "quasi" solution was deemed necessary. Converging to a "quasi" solution would eliminate the oscillatory behaviour ,which would save computer time, and if the "quasi" solution provided some information about the state of the system, the loadflow user would be better prepared to alter the system in an effort to get convergence to a proper solution. Consider a one dimensional system study with a P (active power) load, and ap(V) mismatch governed by a single voltage variable V p(V) = f(V) - P Suppose that initially the p mismatch is analogous to g, and the voltage V is analogous to x in Fig. 2.1a. Under this condition a solution exists. As the load increases (P becomes more negative), the mismatch curve rises, and the voltage solution point moves to the left, as indicated below. Fig. 3.1 p Mismatch Curves Eventually the mismatch curve will only equal zero at one point, and if the load P is increased 9 an infinitesimal amount more, no analytical solution would exist. Note, however, that when no analytical solution exists, the p.t(V) vertex is in the neighbourhood of the Ps(V) solution voltage. The vertex is the minimum mismatch point on p.»(V). Thus, the voltage which minimizes p 4(V) would be the "quasi" solution sought, since this voltage provides similar information to the voltage solution of p 8(V) (i.e. the voltage is low). As a result of these observations, a minimum mismatch solution was sought for multi-dimensional N-R loadflows which diverged. A multi-dimensional mismatch distance was defined, and in the language of chapter 2, the mismatch distance was of the form: f(x) = (rf(x) + ... + rf(x))4 (3.6) or for loadflows F(V) = (p?(V) + • • • + p£(V) + q?(V) + .. • + qi(V))i (3.7) Routines for finding the minimum of the mismatch distance function are discussed in the following sections. Once these routines were implemented, tests were performed on 5 node and 28 node systems (see chapter 4) to verify whether the minimum mismatch distance solution provided insight into the state of the system. 3.4 Damped Newton-Raphson Method The discussion of the damped N-R method is done in terms of the general equations (2.4), (2.5) and (3.6). The method seeks a multiplier A, less than one in absolute value [7], such that xi+1 = x! + A Ax1, (3.8) if x1 appears to be diverging from a solution. To obtain the multiplier A, the mismatch distance f(x) in the direction Ax1 from the reference point x1 was approximated by a quadratic $(z) with z being a scalar real variable for the distances along the vector Ax1 originating from the point x1. The required damping multiplier was defined as A = zm where the superscript m indicates that zm minimizes the quadratic ${z). 10 A formula for A was derived in the following manner: 1. The approximate quadratic is of the form: $(2) = ao + axz + as*2 (3.9) 2. The following equations must be solved to obtain the coefficients ( o o , O i , a?) of $(2). / f ( X l A (1 Zl zf\ (a«\ f(x2) = 1 z, 4 a, (3.10) Vffxs); \l * 4J W 3. The Xj have the form where X j ^ ^ A x ' + x 1 (3.11) O^SllAx'-MlA1-1 anc 22 = 0 (3.13) Z2 is zero so that the evaluation of f(x 2) occurs at the point x'. In calculating zx and 23, care was taken to keep the locations of f(x x) and f(x 3) near to f(x 2) so that the approximation $(2) attempts to match locally and not in regions far from the minimum mismatch distance. The terms 2xAx' and 23Ax1 in (3.11) represent one quarter of the distance of the previous damped N-R correction vector divided by a normalized version of the current iteration's N-R correction vector. Since it is assumed that the damped N-R correction vector of the previous iteration is of a reasonably small magnitude (the method has previously damped correctly), the resulting X j and x 3 are near to x 2. 4. To minimize the quadratic the first derivative — j i - i = Oi + 2022 (3.14) d2 must be set to zero. Thus, one obtains zm 2a 2 5. Solving for ai and 02 from (3.10) and noting that ZQ, = 0 and that zx = —z3 results in the following equation for A [4]: x _ . m _ 2 1 ( f(x 8) - f ( X l ) ) [2f(x 8)-4f(x 2) + 2f(x1)] 11 (3.15) Through experimentation with the method, it was discerned that if no analytical solution existed for the loadflow being studied, the multiplier would be in the range 1.0 > A > 0.0, and A would approach zero as x1 approached the minimum mismatch distance solution. However, if the loadflow being studied had an analytical solution, A was often greater than one or less than zero. Under these circumstances A was set to one, and, essentially, the N-R method of chapter two was permitted to cause x1 to converge to the solution X s . The basic damped N-R algorithm is reminiscent of the general N-R algorithm of Section 2.1, and it is outlined below. General Damped Newton-Raphson Algorithm 1. i) Choose an initial guess x0. ii) Set ||Ax11| and A1 to 1.0. iii) Set the iteration count i to one. 2. i) Set llAx'"1!! := ||Ax«|| and A i - 1 := A1. ii) Solve the system of linear equations defined by (2.4) for Ax1. iii) Calculate ||Ax'||. iv) Calculate the 3 for j = 1,2,3 via (3.12) and (3.13). v) Calculate X j for j = 1,2,3 via (3.11). vi) Calculate f ( X j ) for j = 1,2,3 via (3.6). vii) Calculate A' via (3.15). viii) If A' > 1.0 or A1 < 0.0 then set A' = 1.0. 3. i) Calculate x i + 1 via (3.8). ii) Calculate f(x i + 1) via (3.6). 4. Termination occurs if one of the following criteria is met. i) The elements of Ax1 are close enough to zero. ii) The elements of G(x') are close enough to zero. iii) The iteration limit has been met. iv) |f(xi+1)-f(x')| is close enough to zero. 12 5. If termination has not occurred, then the iteration count i is incremented by one and then calculations are continued from (2) above. The algorithm described above has functioned properly on 5 node and 28 node loadflow test cases. In chapter 5, results from this algorithm are used to analyze several power systems. As well, Appendix B provides a comparison with the singularity search method of Section 3.6. Further work could be done to this algorithm to ensure that the chosen X j are "close" enough to x1. Especially at the first iteration where there are no previous calculations of ||Ax1_1|| to determine the Zj. However, the algorithm has worked well, except when the nodal loadings have been exceptionately unrealistic. As a brief illustration of the algorithm's performance, two tables have been included with results from the 28 node test case, which is described in Appendix C. The Table 3.1 results are from the actual unaltered base case. Table 3.1 Algorithm Performance when Analytical Solution Exists Iteration Damped N-R Damped N-R N-R Mismatch X Mismatch Distance Distance 1 11.0170 1.0000 11.0170 2 10.0902 1.0000 10.0902 3 1.6257 1.0000 1.6257 4 0.0834 1.0000 0.0834 5 0.0117 0.8757 0.0117 6 0.0016 1.0000 0.0002 7 0.0000 - 0.0000 Table 3.1 verifies that the damped N-R method is essentially the N-R method for cases which converge to an analytical solution. Table 3.2 documents the mismatch distance for the 28 node case for which Q 2g was altered to —0.97 p.u. from the base case —0.50 p.u.. Table 3.2 on the following page illustrates the damping nature of the damped N-R method. 13 Table 3.2 Algorithm Performance when No Analytical Solution Exists Iteration Damped N-R Mismatch Distance Damped N-R X N-R Mismatch Distance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10.9969 9.1295 1.3696 0.0981 0.0598 0.0205 0.0102(99) 0.0102(76) 0.0102(76) 1.0000 1.0000 1.0000 1.0000 1.0000 0.9826 0.0046 0.0000 10.9969 9.1295 1.3696 0.0981 0.0598 0.0205 0.0102(95) 2.1131 0.3852 0.2240 0.0330 0.0136 0.0178 0.0105 0.1242 3 .5 Observation of Newton-Raphson Behaviour The singularity search method presented in Section 3.6 was designed as a result of monitoring the behaviour of various aspects of N-R loadflow cases. The behaviour observed below was available, in part, because the whole Jacobian matrix was formed for analysis. In production type loadflow programs, optimally ordered Gauss elimination is used to solve (2.4), in a way in which after each row of the Jacobian matrix is formed, the elements to the left of the diagonal are eliminated [6]. The resulting upper triangular matrix is then stored in compressed form for the non- zero elements. The complete Jacobian matrix is never available in this elimination scheme. The characteristics which were monitored include the Jacobian matrix's eigenvalues, eigenvec-tors, singular values, and condition numbers. As well, the inner product of the current iteration's normalized N-R correction vector Ax^ was taken with the previous iteration's normalized N-R correction vector Ax^"1, and the inner product of the current iteration's normalized eigenvector E}^  corresponding to the eigenvalue closest to zero was taken with the previous iteration's EJ^~1. Lastly, a record of the mismatch distance was kept at each iteration , along with a record of each iteration's approximate solution x1. After monitoring a number of loadflows, including both diverging and converging cases, a number of trends were observed, and they are listed below. Graphs and tables are included to help illustrate the behaviour. 14 Observations 1. For all cases, the eigenvalue of the Jacobian matrix closest to zero was always a real number. 2. When cases converged the following was true as the solution was approached (Ax ' N )T(Ax ' N - 1 ) = >l.O (3.16) and (Ei,)T(EJ71)=>1.0 (3.17) for each iteration. Since the inner products of (3.16) and (3.17) approach one, this indicates that successive correction vectors and eigenvectors are consistently pointing in the direction of the solution as the solution is approached. Table 3.3 illustrates the behaviour described above for the 5 node base case of Appendix D. Table 3.3 Inner Product Data Iteration (Ax"N ) T (Ax}7 1) (Ej,)T(E^T1) 1 2 0.3595 0.9999 3 0.9464 0.9999 4 0.9914 0.9999 3. When cases diverged (3.16) and (3.17) were satisfied for the first few iterations. Eventually, however, for all diverging cases monitored, the following was observed (Axi , ) T (AxJ7 1 ) «-1.0 (3.18a) or 1.0- |eps| > (Ax^) T(Axjr 1) > -1.0+ |eps| (3.186) and K E ^ H E ^ I ^ l . O (3.19a) or 1.0- |eps| > (Ej J) T(EJr 1) > -1.0+ |eps| (3.196) Where |eps| is equal to some small value which indicates that the inner products are not close to one. Table 3.4 shows typical instances of the described behaviour. The 5 node base case of Ap-pendix D with load P 2 altered to —3.75 p.u. was used to provide the data. 15 Table 3.4 Inner Product Data Iteration (A4)T(Axn l(Ek)W)l Asmall 1 - - 1.4296 2 0.9908 0.9962 0.8324 3 0.9996 0.9987 0.3985 4 0.9999 0.9994 0.1103 5 -0.9999 0.9989 -0.2679 6 -0.9999 0.9994 0.0071 7 0.9453 0.0222 -2.1368 8 -0.8638 0.1814 -0.8843 9 0.8223 0.9882 -0.6798 10 0.9891 0.9986 -0.3104 11 0.9995 0.9994 -0.0350 12 -0.9986 0.9910 1.0767 13 0.9991 0.9980 0.5539 14 0.9999 0.9991 0.2194 For iterations 1-4 the approximate solution x1 would be approaching the singularity. Iterations 4-6 reveal the oscillatory behaviour of the search, in which the correction vector points back on itself, as suggested by (3.18a). At iteration 6 the approximate solution almost lands on the singularity, which is illustrated by A S m a l l w 0 in Fig. 3.2. Thus, iterations 7-8 demonstrate conditions (3.18b) and (3.19b), where the approximate solution has been forced into a region removed from the original singularity and thus the vectors used to compose the inner products of Table 3.4 do not line up. Iterations 9-11 show the approach to another singularity, and iterations 11-14 illustrate the subsequent oscillatory behaviour about this singularity. 4. At the iteration where conditions (3.18) and (3.19) initially occur for diverging cases, the eigenvalue of the Jacobian matrix closest to zero usually changes sign (iteration 5). On all cases where sign changes occurred, they were, for the initial change, positive to negative. A sign change for the eigenvalue closest to zero may not occur for reasons such as: i) If the approximate solution vector has "jumped" to a region removed from the original singularity, the new location is not related to the previous location, and thus either sign is valid. ii) Cases have been observed where the second closest eigenvalue of the Jacobian matrix to zero has been the one to oscillate. 16 Fig. 3.2 below demonstrates the oscillatory behaviour of the eigenvalue of the Jacobian closest to zero. Note how this eigenvalue oscillates about zero, or, in other words, about the point where the Jacobian matrix is singular. Also note how at iteration 6 the eigenvalue is very nearly zero, which causes Ax 6 to be large enough in magnitude to place x 7 in a different region, so that Ax 7 is unrelated to Ax 6 as evidenced by the iteration 7 results in Table 3.4. The data for Fig. 3.2 was from the same case used for Table 3.4. t o m w i i 1 1 1 1 1 1 1 1 1 1 i 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.011.012.013.014.0 I T E R A T I O N Fig. 3.2 The Eigenvalue of the Jacobian Matrix Closest to Zero Versus Iteration. 17 5. For diverging cases, the condition number of the Jacobian matrix ^Largest. "'Smallest ,where a L a r g e 8 t and Smallest are the largest and smallest singular values, respectively, became large as the singularity was approached. Again note the oscillatory behaviour of the condition number in Fig. 3.3 for the same case used for Table 3.4. o d o -, o co o 0.0 1.0 2.0 3-0 4.0 5-0 6.0 7-0 8.0 9.0 10.011.012.013.014.0 I T E R A T I O N Fig. 3.3 The Condition Number of the Jacobian Matrix Versus Iteration 18 6. The mismatch distance at each x' was also observed and is shown in Fig. 3.4 below. The case used for Table 3.4 was used here as well. The smallest mismatch distance obtained below occurred at iteration 6, which is also the point at which the Jacobian matrix was most singular, as illustrated by Fig. 3.2 and Fig. 3.3. o CO ~1 1 1 1 1 1 T 1 1 1 1 T 1 1 1 0.0 1.0 2.0 3.0 4.0 5-0 6.0 7.0 8.0 9.0 10.011.012.013.0140 I T E R A T I O N Fig. 3.4 The Mismatch Distance Versus Iteration. 19 7. Lastly, the X j approximate solution vector was observed for the various test cases. Shown below is the value of a single component of the general vector X j . The component shown in Fig. 3.5 is 03. Fig. 3.5 illustrates that the approximate solution vector "jumped" to another region at iteration 6. Since the co-ordinate shown below is an angle, the mismatch distance replicates itself every 2n radians along this co-ordinate. As it turned out, the region "jumped" to was approximately 2n radians from the original singularity in the 03 co-ordinate direction. o 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.011.012.013.014 0 I T E R A T I O N Fig. 3.5 03 Versus Iteration. 20 The behaviour illustrated by points 1 through 7 above suggested a means by which the minimum mismatch distance could be located. Recall that (3.16) is satisfied for the first few iterations of a diverging case. However, at iteration i - 1, x 1 - 1 is too close to the singularity, and Ax ' - 1 , which satisfies (3.16) with Ax ' - 2 , causes x1 to be in a region where either (3.18a) or (3.18b) is satisfied. This being the case, it was felt the vector Ax 1 - 1 pointed in the direction of the singularity, but it simply "overshot" the singularity. Thus, if divergence has been noted at iteration i by an indicator, such as the change in sign of the smallest eigenvalue of the Jacobian matrix, or a change of (Ax^) T (Axj7 1 ) from being approximately equal to one, then a search could be conducted from data of iteration i — 1 for the multiplier f that minimizes G(x) where x=£Ax i - 1 +x i - 1 , (3.20) or the search could be made for the f which causes the condition number of the Jacobian matrix to be maximized, or which causes the eigenvalue of the Jacobian matrix closest to zero to most nearly equal zero. 3.6 The Singularity Search Method The algorithm described below arrives at the minimum mismatch distance solution indirectly, in that the approximate solution vector x is sought which causes the Jacobian to be most singular. This solution also minimizes the mismatch distance (3.6). Singularity Search Algorithm 1. i) Choose an initial guess x°. ii) Set the iteration count i to one. 2. i) Solve the system of linear equations defined by (2.4) for Ax1. ii) Evaluate the eigenvalues of G'(x'). iii) If this iteration's eigenvalue closest to zero changes sign relative to the previous itera-tion's then go to iv else go to 3. iv) Perform interactive search for the f in (3.20) that forms x which causes G'(x) to possess an eigenvalue within some user defined value of zero. 21 v) Output the mismatch distance as well to verify that its minimization occurs for the x obtained above. vi) Terminate the algorithm. 3. Calculate the next iteration's approximate solution by (2.5). 4. Termination occurs if one of the following criteria is met: i) The elements of Ax1 are close enough to zero. ii) The elements of G(x') are close enough to zero. iii) The iteration limit has been met. 5. If termination has not occurred, increment the iteration count i and continue the calculations from (2) above. This algorithm is not computationally efficient, and could fail to obtain the minimum mis-match distance if, for example, the second closest eigenvalue to zero was the one which was os-cillating. However, this being the first algorithm actually implemented, it served its purpose. It provided the minimum mismatch distance solution in most cases, which permitted the comparison of this solution to the system under study. Note, though, that a version of this algorithm could be made computationally more efficient if 2.ii above was omitted and 2.iii was altered so that (2.18a) and (2.18b) were used as indicators of divergence, which avoids the use of the QR algorithm for finding eigenvalues. As well, optimally ordered Gauss elimination could be used, since the whole Jacobian matrix is not required. Lastly, 2.iv could be replaced by an optimal search technique for the f of (3.20) which minimizes (3.6), the mismatch distance. 3.7 Conclusion Although the coded versions of the methods presented were not yet of the standards of industry, they provided the minimum mismatch distance solutions needed for further analysis. 22 4. ANALYSIS AT THE MINIMUM MISMATCH DISTANCE SOLUTION The implemented versions of the algorithms from sections 3.4 and 3.6 provide the minimum mis-match distance solution x M. The following sections verify the usefulness of this solution, and document several methods of analyzing loadflows with this solution. 4.1 Relationship Between Minimnm Mismatch Distance Solution and System State Firstly, to establish whether x M was related to the system state, system data (such as a nodal load) was varied step by step. This allowed the progression of analytical solutions X s through to minimum mismatch distance solutions x M to be observed. The above procedure was carried out for a number of 5 and 28 node scenarios. Each scenario involved the variation of specific aspects of the system data. The following fist outlines the specific data varied in each scenario. Systematically Varied Parameters (1) Single Node overloads i) P at a single node ii) Q at a single node iii) P and Q at a single node (2) System overload i) P and/or Q at two or more nodes (3) Branch Changes i) single branch impedance ii) two or more branch impedances For all scenarios observed the transition of X s to x M was smooth. Thus the nodal voltage magnitude and angle trends associated with a varied parameter such as a nodal load were continued from the X s to x M solutions. Some results are tabulated below. In Table 4.1 P 2 of the 5 node scenario of Appendix D was systematically varied, and in Table 4.2 Q 2 was varied. A 28 bus comparison is tabulated in chapter 5. 23 Table 4.1 Solution Changes as a Function o/P 2 Components x s(P 2 = -0.9000) x s(P 2 = -3.6600) x M ( P 2 = -3.6700) x M ( P 2 =-3.7500) |V2| 0.958979 0.873228 0.870223 0.869609 0 2 0.010592 -1.053603 -1.084430 -1.090586 |V3| 0.920364 0.759995 0.751730 0.753812 0 3 -0.047593 -0.745485 -0.767400 -0.769002 0 4 0.009324 -0.944259 -0.974753 -0.976707 0 5 0.071411 -0.990737 -1.021486 -1.027636 I Vi I are in p.u., 9\ are in radians and P 2 is in p.u. Variables |Vi|, 0ly |V4|, and |V5| are not listed since they were held constant. The five bus base case solution vector is contained in column two above. In the third column the large angle changes consistent with real power load changes are apparent. The transition from X s to x M solutions is quite smooth and the fourth and fifth column x M solutions continue the trend with the angle changes being the most significant. Table 4.2 Solution Changes as a Function of Q 2 Components x s(Q 2 = -5.1000) x s(Q 2 = -8.6811) x M(Q 2 = -8.6815) x M ( Q 2 =-9.5000) |V2| 0.813355 0.509142 0.507802 0.505207 0 2 0.048586 -0.071672 -0.073220 0.002468 |V3| 0.849913 0.690389 0.689574 0.698136 0 3 -0.049399 -0.210370 -0.211802 -0.162903 0 4 -0.022942 -0.345133 -0.347812 -0.259013 0& 0.113137 0.007567 0.006121 0.079207 I Vi I are in p.u., 0; are in radians and Q2 is in p.u. As in Table 4.1, the solution vector transition from X s to x M is again smooth in Table 4.2. As well, since Q 2 is being varied the voltage magnitudes are affected significantly, which is evident in both the X s and x M solutions. As was mentioned previously, this type of analysis was performed for all the systematically varied parameters listed. In all cases the x M solutions continued the trends established by the X s solutions. Establishing that the x M solution reflected the state of the system allows the loadflow user to interpret this solution and all other quantities calculated from it (currents, fine flows, and line losses) as if it were the actual solution. As a consequence, decisions can be made to upgrade the system based on all the information normally available as loadflow output. 24 The following section discusses how typical loadflow output based on the x M solution can be used to decide at which node system data should be changed in order to achieve proper convergence. 4.2 Loadflow Output Analysis This and the following sections establish methods which can determine problem nodes in loadflows at which problems exist. Loads at specific nodes were increased until the N-R loadflow algorithm described in chapter 2 would diverge. Under these circumstances the x M solution was required and obtained. Then the problem node, which was known in advance, was located via the methods outlined in the succeeding pages. Once these methods were developed enough to successfully analyze scenarios whose problems were known in advance, they were considered reasonable approaches for analyzing cases where the problem areas were not known a priori. For cases in which load was increased at a single node until an x M solution was reached, low voltage magnitudes were evident at the node and in the area surrounding the node. If Q was the load being varied at a single node, the lowest voltage magnitude always occurred at that node ( see Table 4.2). However, if P was the parameter being varied, the lowest voltage magnitude would not necessarily be at the node involved, but the voltage magnitudes would be low in the surrounding region ( as in Table 4.1). This phenomenon is not exemplified most effectively in Table 4.1, but the 28 node case analyzed in chapter 5 provides a better illustration of this behaviour. As a consequence of this behaviour, a subroutine was added to the loadflow code to list the lowest voltage magnitudes (refer to Appendix E). Voltage angles were analyzed next. As load at a node was increased, it was apparent which voltage angles were most sensitive to load changes. However, when analyzing any given loadflow case, one does not have the benefit of having observed a progression of solutions to the one under study. It was noticed, though, that when P was increased at a node, all the voltage angles relative to the slack bus were large compared to a normally converging case. However, this information did not help to specify the problem area, so it was not used. As a result, the sensitivity analysis method of Section 4.4 was used to determine which nodal loads affected voltage angles (and magnitudes) the most. However, more work could be done in this area to determine, for example, whether angle differences between neighbouring nodes might suggest problem areas. Lastly, various quantities which can be calculated from the voltage magnitudes and angles of 25 x M were investigated. These were the line currents, the line power flows, and the line power losses. Each of these quantities was monitored, and a record was kept of the regions in which they were largest. This information only corroborated with the low voltage magnitude information when the overloaded node had one of the highest overall loads. A more effective means to judge where problem areas were would have been to set limits for each of the line quantities in each of the lines based on physical limitations. Such physical information was not available for the cases studied, but future study in this area could be fruitful. The methods discussed in this section were fairly crude, and, in general, they only identified the problem area for any given case. In the following sections two methods are investigated. The first method provides information specifying the general region of the problem node as well as the problem type (P or Q). The second method is successful at pinpointing the problem node and type (P or Q) for cases in which the problem is at a specific node. 4.3 Nodal Mismatch Analysis It was noticed, after observing a number of normally diverging loadflows at the x M solution, that the largest p k or q k mismatches usually occurred at the node which had its P k or Q k load increased. Recall that the p k or q k mismatches in loadflow studies coincide with the ft(x) of G(x) in (2.2). Thus, since the G(x) vector is calculated at each iteration as a prerequisite to forming (2.4), it is available, and can be searched for the largest <ft(x) components in absolute value. A routine to obtain the largest mismatches was implemented , and the results are tabulated below for the 5 node case in Appendix D with Q 2 altered to —9.50 p.u.. Table 4.3 Mismatch Values at the Minimum Mismatch Distance Solution Mismatch Type Mismatch p 2 -0.08063 q 2 -0.74256 p 3 -0.04363 q 3 -0.08412 p„ -0.14612 p 6 0.03544 Note that the q 2 mismatch is the largest in Table 4.3 above. Thus, as well as pinpointing the area of the problem node, one can determine whether the problem type is P or Q by noting whether the largest mismatches are of the p k or q k type. Further examples of this method are provided in the 28 node analysis of chapter 5. 26 4.4 Sensitivity Analysis The first order sensitivity analysis undertaken in this section involves determining how susceptible each |Vj| and 0; are to changes in Pj or Qj. This approach was investigated to see whether |Vj| and 6i were most sensitive to load changes at the problem node. Using such an analysis is not new in power system studies, but applying it at the minimum mismatch distance solution is. A discussion of the theoretical aspects of the sensitivity analysis as presented in [8] is given below. Following this an algorithm is presented and then results are tabulated. The general equations of chapter 2 require some modification to represent conditions at the minimum mismatch distance solution. First, (2.2) is changed to: G(x M ,y)=M (4.1) x M is the minimum mismatch distance solution composed of the components |V£*| and #M The vector y contains specified variables P k and Qk. M is a vector containing the mismatches at x M . For a particular sensitivity calculation the elements n of y will be variable while the elements t will be fixed. Thus, let: ,=(;) I-, The nonlinear system of equations (4.1) can be linearized at x M by expanding them into a Taylor series and retaining only the first order terms. G(x M + Ax,u + Au,t + At) = G(x M ,u,t) + (Ax| - + An^- + At|-)G(x M ,u,t) (4.3) ox on bt where and such that and At=0 G(x M + Ax,u +Au,t +At) PS G(x M ; u, t) ^ ( x M , n ; t ) A x = - ^ ( x M , n , t ) A u (4.4a) 27 Equation (4.4a) is a linear model from which changes in Ax can be computed from changes in Au. The actual calculations performed involved determining the relative change Ax Au k (4.5) for a single variable change Auk. Since, in loadflows, G(x) of (2.2) is composed of pk and qk components which are defined by (3.4) and (3.5), the following is true: G(* ' y ) * U v M , Q ) J (4.6) As well, the uk in (4.5) corresponds to either the Pk or Qk nodal loads or generation. Thus, for example: 6nk[% > y ) * 6Pk W v M,Q); 6uk[X , 7 ) < * 6Qk U(VM,Q) (4.7) However, as indicated in (3.4) and (3.5) each Pk or Qk is contained in precisely one component of p(VM,P) or q(VM,Q). Thus (4.4) can be combined with (4.5) to yield: /0\ 8G , M , Ax (4.8) The one on the right hand side of (4.8) occurs at the kth component. Equation (4.8) is the formulation required to implement a routine in a loadflow program for obtaining the relative sensitivity of equation (4.5). The four types of relative sensitivity that were calculated have the form: A|VS| A|Vi| A6> A0> AP k ' AQ k ' AP k' AQ k The basic algorithm used to obtain the relative sensitivities is listed below. First Order Relative Sensitivity Algorithm (4.9) i) After termination of one of the modified N-R algorithms which has located x M, save the Jacobian <5G (xM,y) 28 ii) Set iteration count i to one. 2. i) Set of (4.4b) to zero. ii) Set component i of to 1.0. 3. Solve (4.4b) for (4.5). 4. Calculate and save it. 5. If i = N go to 6 else increment 6. i) Either search for the largest Ax Aus and display the corresponding Ax or if required display all the sensitivities. ii) Terminate the algorithm. Recall that the components of (4.5) are actually of the form shown in (4.9) for loadflows. Thus, the load or generation change which affects the voltage magnitudes or angles most can be located. Step 4 in the algorithm was included as a means of assigning a magnitude value to the sensitivities for each A U J load or generation investigated. This step was included after empirical results indicated that the components of (4.5) were much larger for the Au; which corresponded to the problem load or generation. Ax Auj i and go to 2. 29 Table 4.4 presents a sensitivity listing for the 5 node base case with P2 altered to -3.75 p.u. Table 4.4 Sensitivity Results Sensitivity Type A|V2|\AP2 A02\AP2 A|V3|\AP2 A03\AP2 A04\AP2 Ah\AP2 A|V2|\AQ2 A02\AQ2 A|V3|\AQ2 A03\AQ2 A04\AQ2 A05\AQ2 A|V2|\AP3 A02\AP3 A|V3|\AP3 A03\AP3 A04\AP3 A05\AP3 A|V2|\AQ3 A02\AQ3 A|V3|\AQ3 A03\AQ3 A04\AQ3 A05\AQ3 A|V2|\AP4 A02\AP4 A|V3|\AP4 A03\AP4 A04\AP4 A05\AP4 A|V2|\AP5 A02\AP5 A|V3|\AP5 A03\AP5 A04\AP5 A05\AP5 Value -1002.27 -10347.9 -2910.62 -7481.45 -10403.5 -10321.4 -58.7197 -606.582 -170.602 -438.557 -609.830 -605.031 -763.186 -7879.39 -2216.27 -5696.68 -7921.69 -7859.23 -341.998 -3531.08 -993.151 -2552.97 -3550.03 -3522.04 -863.159 -8911.48 -2506.50 -6442.92 -8959.23 -8888.68 -991.193 -10233.5 -2878.45 -7398.75 -10288.5 -10207.2 Notice that the A|Vi|\AP2 or A0i\AP2 elements are the largest, indicating that P is the 30 problem load type and that it is at node 2. Also note that the AP 4 and AP 5 sensitivities are large. Since, nodes 4 and 5 are generator nodes, this indicates that generation changes at these nodes would have dramatic effects on the system. One would expect this, since node 2 is adjacent to nodes 4 and 5 (refer to Appendix D), and is overloaded in terms of active power. Table 4.4 illustrates one other aspect of sensitivities calculated at the x M solution point. They can be very large in magnitude. This is due to x M being located at a position in which the Jacobian matrix is nearly singular. 4.5 Conclusion By implementing routines in loadflow programs which output the lowest voltage magnitudes, the nodal mismatches, and the first order relative sensitivities at the x M solution point, contrived overloads at individual nodes can be isolated. As a consequence, the methods should be capable of isolating problem nodes in generally overloaded systems for which the problem area is not known a priori. 31 5. TEST CASES This chapter presents three test cases which verify the assertions of previous chapters. The three test cases are based on the 28 node system from [9], which is fully described in Appendix C. The first test case is an example of a known active power overload at a single node. The second test case contains a known reactive power overload at a single node. Lastly, the third case is overloaded at a number of nodes, and, thus, the problem area is not known a priori. 5.1 Active Power Overload at a Single Node The test case analyzed in this section has P28 overloaded in the system shown below in Fig. 5.1. SLACK NODE Fig. 5.1 28 Node Scenario 32 Table 5.1 below compares selected components of the solutions for three choices of P28. Selected components were tabulated because the whole vector was too large to present. The observed trends in the table below are representative of the trends which could be observed if the whole vector was present. The first column of Table 5.1 contains the 28 node base case solution components in which P 2 8 = —1.45 p.u., and the components of the second column pertain to the P 2 8 = —1.80 p.u. case. A N-R loadflow program corresponding to the algorithm of chapter 2 obtained both of these solutions. In the third column, P 2 8 = -2.05 p.u., and the solution components are from the minimum mismatch distance solution. The singularity search method of Section 3.6 was used to obtain this solution. Table 5.1 Solution Changes as a Function o/P28 Component x s(P 2 8 = -1.45) x p(P 2 8 = -1.80) x M(P 2 8 = -2.05) |v 2 | 1.00016 0.93169 0.89598 02 -0.90328 -0.99966 -1.05278 |v 4 | 1.00016 0.92463 0.88555 -0.95396 -1.05957 -1.11738 |v 8 | 1.00012 0.99346 0.98977 -0.19304 -0.20488 -0.21211 i V a o l 0.97697 0.91433 0.88078 020 -0.75602 -0.83297 -0.87747 |v 2 2 | 0.97732 0.90286 0.85827 022 -0.72310 -0.81227 -0.86856 020 -0.33542 -0.38285 -0.41123 027 -0.51211 -0.57222 -0.60702 |Vas| 1.00016 0.89986 0.83702 #28 -0.83732 -0.97631 -1.07143 | V i | are in p.u., 0i are in radians and P28 is in p.u. As shown in chapter 4 for the 5 node case, the transition from X s to x M solutions is smooth, and the trends from column one to two above are continued in column three. As suggested in chapter 4, the lowest voltage magnitudes often suggest the problem area. The lowest voltage magnitude occurred at node 28 and the next lowest voltage magnitude was at node 22 for the P 2 8 = -2.05 p.u. case. These were in fact the lowest voltage magnitudes of the overall solution vector, though not all the solution components are listed above. 33 The line quantities (currents, power flows, power losses) did not provide much information as to where the problem area was. This was because P 2 8 = -2.05 p.u. is not one of the largest loadings in the system and thus the line quantities associated with this bus were not the largest overall, as indicated in Table 5.2. Table 5.2 28 Node Case Line Quantities for P 2 8 — —2.05 p.u. Line |I| Power Flow Power Losses P Q P Q 22 to 28 2.4229 2.0788 0.0553 0.0822 -0.3004 02 to 21 3.1276 -2.6019 1.0405 0.0763 -0.3815 04 to 19 4.5027 -3.7598 1.3277 0.1095 -0.5474 05 to 06 3.9824 -3.7377 1.1465 0.1110 -0.5551 All quantities are in p.u. However, the line quantities could still be used to indicate the problem area if individual limits were set for each quantity based on physical limitations. If this was done, the current [I|2 — 28 = 2.42 29 p.u. might be over the acceptable limit for this line suggesting an overload in this region. However, such limits were not available for these test cases. The nodal mismatch data is analyzed next. Table 5.3 lists the nodal mismatches for the P 2 8 = -2.05 p.u. scenario. Table 5.3 Nodal Mismatches Node p Mismatch q Mismatch 2 -0.048088 0.002442 4 -0.080227 -0.000825 8 -0.000130 -0.000007 20 0.042004 -0.017502 22 0.025419 -0.008768 26 -0.005980 27 -0.010413 28 -0.053373 -0.010124 The largest active power mismatch occurred at node 4 with the second largest being at node 28. The largest reactive power mismatch occurred at node 20. By referring to Fig. 5.1, one can see that the largest mismatches are all in the region surrounding node 28. Given that the overloading was performed on P 2 8, it is interesting to note that the largest 34 active power mismatch occurred at node 4. Since the largest loading in the region near to node 28 is at node 4, as indicated in Appendix C, it is reasonable to expect, then, that the largest mismatch could be at node 4. Lastly, the sensitivity results are presented in Table 5.4 below for the P 2 8 = -2.05 p.u. case. Components from the worst three sensitivity vectors are listed. The worst three sensitivity vectors of the form (4.5) were determined as indicated in step 6 i) of the sensitivity algorithm in Section 4.4. Table 5.4 Sensitivity Results Sensitivity Type A U J A|V 2|/A U i A0 2/Aui A|V 4|/A U i A0 4/Au ; A|V 8|/A U i A0 8/Aui A|V 2 0|/A U i A 0 2 o / A U J A|V22|/Au, A0 2 2/AUi A0 2 G/Aui A0 2 7/Aui A|V 2 8|/A U i A0 2 8/AUi The sensitivity results above tie all the previous indicators together. The lowest voltage magnitude was at node 28, the largest mismatches were of the active power type, and column one above indicates that the system is most sensitive to active power changes at node 28. Thus, the contrived P 2 8 overloading has been located by the methods discussed, which further substantiates claims that the methods are helpful in determining problem areas in loadflows which would diverge in typical production type programs. = A P 2 £ Au ; = A P 4 U i = A P 2 -248.017 -241.654 -211.611 -305.075 -297.246 -260.231 -276.311 -269.194 -235.798 -344.472 -335.561 -293.950 -20.8652 -20.3299 -17.8053 -24.0501 -23.4229 -20.5212 -222.849 -217.131 -190.175 -216.167 -210.616 -184.458 -244.071 -237.863 -208.331 -214.851 -209.397 -183.388 -110.287 -107.462 -94.1087 -158.315 -154.253 -135.092 -314.927 -306.959 -268.848 -337.983 -329.502 -288.582 35 5.2 Reactive Power Overload at a Single Node In this section, Q28 of the 28 node scenario is overloaded, and the methods described in chapters 3 and 4 will be used to pinpoint this pre-set problem. As in Section 5.1, components of the X s and x M solution vectors will be compared to verify that the x M solution portrays the same trends as do the X s solutions as overloading occurs. This comparison is presented in Table 5.5 below. Table 5.5 Solution Changes as a Function of Q28 Component x s(Q 2 8 = -0.50) x s(Q 2 8 = -0.96) x M(Q 2 8 = -1.65) | V 3 | 1.00015 0.88701 0.93341 03 -0.88022 -0.98294 -0.92018 |v 4 | 1.00016 0.87865 0.93206 04 -0.95396 -1.07737 -1.00658 |v. | 1.00012 0.99182 0.99957 -0.19304 -0.20104 -0.18960 | v 2 0 | 0.97697 0.87714 0.91404 020 -0.75602 -0.83007 -0.77611 | v 2 2 | 0.97732 0.84688 0.83864 022 -0.72310 -0.78606 -0.72823 #26 -0.33542 -0.36979 -0.32857 027 -0.51211 -0.56598 -0.51617 | v 2 8 | 1.00016 0.79477 0.71436 028 -0.83732 -0.93327 -0.86693 |Vi| are in p.u., 0; are in radians and Q2s is in p.u. The results listed in Table 5.5 are not as "nice" as those in Table 5.1 of the previous section. The solution component trends from the column one components to the column two components were not directly translated into the components of column three. For example, some of the voltage magnitudes went up in value instead of down in value in column three. However, from a power engineer's viewpoint, the solution components of column two and three tell basically the same thing in terms of where the lowest voltage magnitudes are located, and that the voltage magnitudes are too low. In both cases the lowest voltage magnitude was at node 28. 36 Next, Table 5.6 lists the nodal mismatches for the Q 2 8 = -1.65 p.u. scenario. Table 5.6 Nodal Mismatches Node p Mismatch q Mismatch 3 0.072930 -0.152432 4 0.091487 -0.147090 8 0.002023 -0.002197 20 -0.122756 -0.148582 22 0.043858 -0.051138 26 -0.058311 -27 -0.063338 -28 0.084167 -0.121249 As in the previous section, the nodal mismatches confirm that the problem is in the area of node 28. The largest mismatches are the reactive power mismatches at nodes 3, 20, 4, and 28. Since the largest mismatches are of the reactive power type, it is reasonable to assume that that is the problem type. Lastly, the sensitivity results are listed in Table 5.7 below, for the Q 2 8 = —1.65 p.u. case. The'listed vector components are from the worst three sensitivity vectors of the form of (4.5). Table 5.7 Sensitivity Results Sensitivity Type -A|V 3|/A U i A0 3/Aui A|V 4|/A U i A0 4/Aui A|V 8|/A U i A0 8/Aui A|V 2 0|/A U i A0 2 O/Aui A|V 2 2|/A U i A0 2 2/Au; A 0 2 c / A U J A 0 2 7 / A U J A|V 2 8|/A U i A0 2 8/Aui The sensitivity vector with the largest components in absolute value had form Ax/AQ 2 8. This combines with the low voltage magnitude at node 28 and the large reactive power mismatches = A Q 2 8 A U J = A P 2 8 U I = A P 4 -274.990 -201.780 -170.966 -274.053 -201.004 -170.294 -295.095 -216.532 -183.440 -326.506 -239.492 -202.844 -20.6765 -15.1680 -12.8510 -23.4781 -17.2086 -14.5780 -243.115 -178.391 -151.149 -200.060 -146.710 -124.291 -314.664 -230.921 -195.712 -180.513 -132.310 -112.143 -95.5593 -70.0467 -59.3460 -144.639 -106.058 -89.8530 -488.340 -358.431 -303.824 -308.475 -226.090 -191.731 37 to implicate node 28 as the problem node and Q to be the problem type. Thus, the pre-set problem was identified correctly. 5.3 System Overload In this section the 28 node base case of Appendix C had the following load changes: P28 — -1.450 P ,u. P28 = -1.600 p.u. Q28 = -0.500 P u. => Q28 = -0.650 p.u. P 1 0 = -0.750 P u. => P10 = -0.900 p.u. Qio = -0.440 P' ,u. => Qio = -0.590 p.u. P08 = -0.608 P .u. P08 = -0.758 p.u. Q08 = -0.070 P ,u. Q08 = -0.230 p.u. P()5 = -3.740 P .u. Po5 = -3.890 p.u. Q05 -0.340 P .u. Q05 = -0.490 p.u. The above load changes were enough to cause the N-R loadflow of chapter 2 to diverge. Thus the singularity search method was used to obtain the minimum mismatch distance solution. Since the load changes occurred simultaneously at various parts of the system, the problem node was not known in advance. In Table 5.8 below, selected components of the minimum mismatch distance solution vector are listed. Table 5.8 Solution Change from Base Case Component XS Base Case x M Overloaded System |v 2 | 1.00016 0.88693 -0.90328 -1.04879 |v 4 | 1.00016 0.87476 04 -0.95396 -1.11490 |v 6 | 1.00037 0.95956 05 -0.39487 -0.43762 |v 8 | 1.00012 0.94492 08 -0.19304 -0.24624 IV10I 1.00014 0.85271 010 -0.67296 -0.79450 |v 2 0 | 0.97697 0.87337 020 -0.75602 -0.86769 024 0.10956 0.08125 027 -0.51211 -0.60122 |v 2 8 | 1.00016 0.83894 028 -0.83732 -0.99120 |Vj| are in p.u. and 6, are in radians 38 The lowest voltage magnitudes occurred at nodes 28 and 10 for the overloaded case. Since the two lowest voltages were in two different areas of the system and the voltages were quite low in general, no specific comment can be made yet about which area is the problem area. Table 5.9 below lists the nodal mismatches for the overloaded case described above. Table 5.9 Nodal Mismatches Node p Mismatch q Mismatch 2 -0.021205 0.002946 4 -0.037031 0.003803 5 -0.000737 0.000044 8 -0.000048 -0.000000 10 -0.004029 -0.001185 20 0.022665 -0.005432 24 0.000005 27 -0.002493 28 -0.015672 -0.001951 The largest mismatches were of the active power type, and they included nodes 2, 4, 20, and 28, which are all in the same area, with node 4 mismatch being the largest. This information combined with the lowest voltage magnitude at node 28 suggests that these nodes comprise the problem area. Table 5.10 lists the sensitivity results which should help clarify which node is the weakest for this case. The three sensitivity columns present the three worst sensitivity vectors. The worst sensitivity vectors were determined by the procedure outlined in step 6i) of the algorithm in Section 4.4. Each of the columns in Table 5.10 corresponds to an active power change in the area which had large active power mismatches. Thus, active power overloading appears to be the problem type. With node 4 corresponding to the largest mismatch and the worst sensitivity vector, it was considered the problem node in this test case. It should be mentioned, however, that node 4 is not the only node where changes could be made to rectify the divergence of typical loadflow programs. It is, though, the probable weak link in the overloaded 28 node system. 39 Table 5.10 Sensitivity Results Sensitivity Type Au; = AP 4 Auj = AP 2 Ui = AP 2 8 A|V 2|/Aui A0 2/Aui A|V 4|/Aui A0 4/Aui A|V 5|/Aui A0 5/Aui A|V 8|/Aui A0 8/Aui A|V 1 0|/Aui A0 l o/Aui -598.300 -741.517 -668.133 -840.675 -95.3489 -91.3759 -54.6731 -63.3426 -497.336 -422.717 -534.311 -517.796 -50.8975 -381.330 -708.352 -691.614 -521.696 -646.515 -582.662 -733.150 -83.1465 -79.6802 -47.6763 -55.2345 -433.692 -368.615 -465.939 -451.527 -44.3820 -332.524 -617.708 -603.101 -520.522 -645.112 -581.303 -731.443 -82.9525 -79.4945 -47.5651 -55.1058 -432.681 -367.756 -464.853 -450.476 -44.2787 -331.749 -616.175 -601.517 A|V 2 0|/Aui A0 2 O/Aui A0 2 4/Aui A0 2 7/Aui A|V 2 8|/Aui A 0 2 8 / A U J Although one cannot be certain that node 4 is the weak node, the choice of node 4 seems reasonable given that it possesses a large load on a radial line in a generally overloaded system. 5.4 Conclusion The previous sections illustrated the capabilities of the analysis methods developed in earlier chap-ters on the 28 node system of Appendix C. The methods, which require the comparison of the voltage magnitudes, nodal mismatches, and the relative sensitivities with respect to P and Q at the minimum mismatch distance solution, were successful at pinpointing weak or problem nodes. As well, the problem type (P or Q) was determined. Since the three test cases studied would not have provided converged solutions on typical loadflow programs used in industry, little useful information would have been known about these scenarios. This implies that the analysis documented in this chapter could be of practical interest. 40 6. S U G G E S T I O N S F O R F U R T H E R R E S E A R C H There axe three major areas for further research, namely the upgrading of the existing program code, the investigation of alternate analysis techniques, and the development of procedures for correcting data at the weak node. To make the algorithms of the Weak Node Method useful for industry, they must be com-putationally efficient. Thus, if production code is required, an investigation of the computational efficiency of these algorithms should be performed, and any improvements should be incorporated into the production code. Voltage angle differences between the ends of a line, line currents, line power flows, and line power losses were not used to help in the identification of weak nodes in the Weak Node Method. If maximum ratings were assigned to each line quantity, based on physical circumstances, a comparison of the actual line quantity results to the ratings would conceivably assist in locating weak nodes. As well, a better understanding of what data is incorrect at the weak node may be obtainable from studying the line quantities with respect to maximum ratings. Further research could also be done to expand the scope of the Weak Node Method. In par-ticular, the development of routines for suggesting corrections to the system data at the weak node could be attempted. Such a project would involve establishing a criteria for suggesting corrections based on the results of an analysis of a large number of test cases. This criteria would then be incorporated into computer code for testing. 41 7. CONCLUSION The Weak Node Method identifies the problem node for power system scenarios which would cause the divergence of the Newton-Raphson loadflow method. The Weak Node Method was formulated by a synthesis of two techniques. With the first technique, the minimum mismatch distance solution is obtained. The second technique pinpoints the weak node at the minimum mismatch distance solution. These two techniques were successfully tested with a number of test cases. In essence, this thesis describes the process of formulating the Weak Node Method, and the subsequent successful testing of the method. Since the Weak Node Method provides useful information about loadflows which would normally diverge, it would be an asset for loadflow users who must correct loadflow cases which have diverged. 42 LIST OF REFERENCE W.F. Tinney and C.E. Hart, "Power Flow Solution by Newton's Method", IEEE Trans. PAS-86, pp. 1449-1460, 1967. H.W. Dommel, W.F. Tinney and W.L. Powell, "Further Developments in Newton's Method for Power System Applications", IEEE Winter Power Meeting, Paper No. 70 CP 161-PWR, New York, 1970. S. Tripathy, G. Prasad, O. Malik and G. Hope, "Load-Flow Solutions for Ill-Conditioned Power Systems by a Newton-Like Method", IEEE Trans. PAS-101, pp. 3648-3657, 1982. U.M. Ascher, R.M.M. Mattheij and R.D. Russell, " Numerical Solution of Boundary Value Problems for Ordinary Differential Equations", Chapter VIII, Prentice-Hall, 1987 (in Press). B. Stott, "Review of Load-Flow Calculation Methods", IEEE Trans. PAS-62, pp. 916-929, 1974. H.W. Dommel, "Notes on Power System Analysis", Univ. of B.C., 1976. S. Iwamoto and Y. Tamura, "A Load Flow Calculation Method for Ill-Conditioned Power Systems", IEEE Trans. PAS-100, pp. 1736-1743, 1981. W.F. Tinney and H.W. Dommel, "Steady-State Sensitivity Analysis", 4th Power Systems Computation Conference, Report No. 3.1/10, Grenoble (France), 1972. S. Abe, N. Hamada, A. Isono and K. Okuda, "Load Flow Convergence in the Vicinity of a Voltage Stability Limit", IEEE Trans. PAS-97, pp. 1983-1993, 1978. G. Strang, "Linear Algebra and Its Applications", Academic Press, 1980. 43 APPENDLX A. Loadflow Solution with the Newton-Raphson Method The information presented in this appendix was obtained directly from [6], pages 117 to 146. The nonlinear equations to be solved are of the power equation form: N V k * £ Y k m V m - ( P k - j Q k ) = 0 k=l,...,N (A.l) m=l where the voltage variable is complex. For the solution, two real variables must be used in place of one complex variable, because (A.l) is not analytic. As real variables, the magnitude |Vk| and the angle 6k are chosen, with V k = |Vk|e^ (A. 2) Though the polar co-ordinates |Vk| and 0k are the variables, computations are done using rectan-gular co-ordinates V k = e k + j f k , {A. 3) to avoid excessive sine and cosine computations. Equation (A.l) is separated into two real equations. These equations, shown below, represent components of the right hand side of (2.4) when multiplied by a negative sign. N p k(V) = R e ( V k ^ Y k m V m ) - P k = 0 {A A) 111=1 N q k(V) = - I m ( V * ^ Y k m V m ) - Q k = 0 (A.5) m = l The right hand side components of (2.4) will be referred to as A P k and AQ k, respectively. To calculate the components of the Jacobian matrix and AP k, A Q k above, several interme-diate quantities are required. First, the nodal current from the approximate voltage solution is required. N I k = X> k l l lV m (-4.6) m=i This current can be obtained by summation, using Y k m = G k m + j B k m , and V n i = e m + j f m such that N Ik = X^ (ak>» +J b k».) (A .7) iu=i where akin = GkiiiEm - B k m f m (-4-8) bkm = B k m e u i -f G k u if u l (-4-9) 44 The Jacobian matrix will contain elements of the form 66 ' 1 '«|V| ' 66 ' 1 '«|V| where A|V| in (2.9) has been replaced by A|V|/|V| resulting in the formulation |V|(6"p/o"|V|), which is simpler to compute. The off-diagonal elements of the Jacobian matrix have the form: 77T~ = l v m L i . r | = aknitk - b k me k 6pk 5qk U . 1 0 ) I V - I F T T 1 = = a k m e k + b k m f k (A.ll) These terms can be calculated whenever a k m and b k m are computed with m # k. To calculate the diagonal elements of the Jacobian and the AP k or AQ k components, the actual power must be calculated. pactual _ j Q - t u a l = y . ^ (A .12 ) The diagonal elements of the Jacobian matrix are then = -Qkctual - Bkk|Vk|2 (A.K) 66 k |V k|^y = Pr u a l+G k k|V k| 2 (A .14 ) SjjJL pac.ua, _ G k k | V k | 2 (,1.15) IV. \j3k _ — /^actual _ rj I v |2 = Q-,ual - Bkk|Vk|2 {A. 16) ' "'*|Vki The right hand side of (2.4) can then be calculated from AP k = Pk - P^tual [A.17) AQ k = Qk - Qr t u a l (A.18) The formulations above are used in the following Newton-Raphson loadflow algorithm which was the core algorithm used for the thesis work. 45 Newton-Raphson Loadflow Algorithm 1. Set iteration count i = 0, and guess voltages V2°';..., V^0> 2. Set node count k = 1. 3. Increase node count by 1. 4. Compute a k m and b k m from (A.8) and (A.9) for each nonzero entry in row k of the admittance matrix. Use these values, except when m = k, to compute the off-diagonal elements of the Jacobian matrix from (A.10) and (A.11). Sum up all aklu and b k m to obtain the complex current Ik. 5. Compute P£ct,,al and Qkctual from (A. 12) and use the values to compute the diagonal elements from (A.13), (A.14), (A.15) and (A.16), and the right-hand sides from (A.17) and (A.18). 6. Go to 3 if k < N, otherwise continue. 7. Find A0k and AjVk|/|Vk| via a linear equation solving routine. 8. If all AO and A|V|/|V| are smaller than e in magnitude then quit. However, if not then continue. 9. Compute new angles 91U.W = 0 + AO, new magnitudes |V|liew = |V|(1 + A | V|/|V|) and convert to rectangular coordinates e + jf . 10. Increase iteration count by 1 and go to 2. Reference [10] was used to write the subroutine for solving linear equations. To calculate the line current or line power flow from node k to m, the following equations must be solved: Iku, = (Vk - Vm)Yk'" + V kY k 0 (A.19) Pku,+jQkm = I L V k M-20) where Y k l u is a branch admittance, in contrast to Y k m, which is an entry in the nodal admittance matrix. The line losses (LL k m) were calculated as follows: LLkui = Pkm + jQkm + Pmk + jQmk {A.21) 46 APPENDIX B. Damped Newton-Raphson and Singularity Search Method Comparison Table B.l below lists the minimum mismatch distance solution vector for the Damped Newton-Raphson (DNR) of Section 3.4 and the Singularity Search Method (SSM) of Section 3.6. The comparison below illustrates that both methods arrive at essentially the same result. The case studied was the 5 node case with P3 = —4.70 p.u.. Table B.l Damped Newton-Raphson and Singularity Search Method Results Component DNR SSM |V2| 0.90076 0.90078 02 -0.82453 -0.82496 |V3| 0.71044 0.71049 03 -0.81771 -0.81793 04 -0.85482 -0.85523 05 -0.76239 -0.76280 |"Vi | are in p.u. and 0, are in radians Table B.l above was included to illustrate that the DNR and SSM algorithms search for the same minimum mismatch distance vector. However, results can vary depending on the user defined accuracy of x M required. 47 APPENDIX C. Twenty Eight Node Test Case Data The 28 node test case data was obtained from [9], and was used to test all of the implemented routines used in the research for this thesis. A line diagram of the test case is shown in Fig. C.l below SLACK NODE Fig. C.l 28 Node Test Case Listed in Table C.l below are the components of the initial voltage guess. Table C.l Initial Voltage Guess Node (i) |Vi| •l 1 | Node (i) I Vi| «. 1 1.05000 0.00000 15 1.00000 0.00000 2 1.00000 0.00000 16 1.00000 0.00000 3 1.00000 0.00000 17 1.00000 0.00000 4 1.00000 0.00000 18 1.00000 0.00000 5 1.00000 0.00000 19 1.00000 0.00000 6 1.00000 0.00000 20 1.00000 0.00000 7 1.00000 0.00000 21 1.00000 0.00000 8 1.00000 0.00000 22 1.00000 0.00000 9 1.00000 0.00000 23 1.05000 0.00000 10 1.00000 0.00000 24 1.05000 0.00000 11 1.00000 0.00000 25 1.05000 0.00000 12 1.00000 0.00000 26 1.05000 0.00000 13 1.00000 0.00000 27 1.05000 0.00000 14 1.00000 0.00000 28 1.00000 0.00000 IVi| are in p.u. and 6, are in radians 48 The load, generation and shunt capacitance data of the base case are listed in Table C.2. Table C.2 Load, Generation and Shunt Capacitance Node Shunt Susceptance P Q 2 1.0252 -2.650 0.220 3 2.0804 -3.810 -0.260 4 1.6665 -3.840 0.020 5 1.5422 -3.740 -0.340 6 — -1.290 -0.030 7 1.0042 -1.350 -0.580 8 0.2080 -0.608 -0.070 9 0.4476 -1.180 0.180 10 0.7768 -0.750 -0.440 11 0.5868 -1.390 -0.520 23 — 1.910 — 24 — 4.400 — 25 — 0.490 — 26 — 3.900 — 27 — 1.520 — 28 1.2069 -1.450 -0.500 All items are in p.u.; generation is positive P , Q ; load is negative P , Q Table C.3 documents the line impedances. Table C.3 Line Impedances line PMJ x« II line i j II i -> J 2 — 21 0.0078 0.0390 13 — 21 0.0296 0.1480 2 — 21 0.0078 0.0390 13 — 21 0.0296 0.1480 3 — 20 0.0058 0.0290 13 — 22 0.0242 0.1210 4 — 19 0.0054 0.0270 13 — 26 0.0070 0.0350 5 — 6 0.0070 0.0350 14 — 15 0.0068 0.0340 6—16 0.0062 0.0310 14 — 20 0.0038 0.0190 6—17 0.0010 0.0050 14 — 27 0.0112 0.0560 7 — 17 0.0196 0.0980 15 — 16 0.0220 0.1100 8—18 0.0280 0.1400 16 — 1 0.0054 0.0270 9—15 0.0118 0.0590 16 — 17 0.0054 0.0270 10 — 15 0.0294 0.1470 16 — 23 0.0070 0.0350 11 — 12 0.0280 0.1400 17 — 18 0.0072 0.0360 11 — 25 0.0258 0.1290 18 — 24 0.0104 0.0520 12 — 14 0.0220 0.1100 19 — 20 0.0038 0.0190 12 — 16 0.0062 0.0310 20 — 21 0.0076 0.0380 13 — 1 0.0200 0.1000 20 — 22 0.0128 0.0640 13 — 14 0.0060 0.0300 28 — 22 0.0140 0.0700 All quantities are in p.u. 49 The 28 node base case solution vector as obtained by the N-R loadflow program used for the thesis research is shown in Table C.4 below. Table C.4 28 Node Solution Vector Node (i) |v,| ». 1 | Node (i) |Vi| «i 1 1.05000 0.00000 15 0.98004 -0.55006 2 1.00016 -0.90328 16 1.00387 -0.20073 3 1.00015 -0.88022 17 0.99506 -0.23644 4 1.00016 -0.95396 18 1.00178 -0.10410 5 1.00037 -0.39487 19 0.98185 -0.83885 6 0.99424 -0.25433 20 0.97697 -0.75602 7 1.00024 -0.37820 21 0.97880 -0.78752 8 1.00012 -0.19304 22 0.97732 -0.72310 9 1.00012 -0.62873 23 1.05000 -0.14425 10 1.00014 -0..67296 24 1.05000 0.10956 11 1.00008 -0.44573 25 1.05000 -0.39336 12 0.98655 -0.30661 26 1.05000 -0.33542 13 0.98511 -0.45830 27 1.05000 -0.51211 14 0.97927 -0.58331 28 1.00016 -0.83732 | Vi I are in p.u. and 0j are in radians Note that these solution components are different than those in [9] in the fourth decimal place. This was due to rounding introduced when converting the impedances to the admittance matrix values. 50 APPENDLX D. Five Node Test Case Data The 5 node test case was obtained from [6], and was used to test all of the implemented routines used in the research for this thesis. A line diagram of the test case is shown below. The boxed-in values are the branch admittance values in per unit. 5 O H 2 3 - J 7 . 5 \ LOAD LOAD SLACK NODE Fig. D.l 5 Node Test Case Listed in Table D.l below are the components of the initial voltage guess. Table D.l Initial Voltage Guess Node (i) 1 2 3 4 5 1.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Vj| are in p.u. and $i are in radians 51 The load and generation of the base case scenario are specified in Table D.2 below. Table D.2 Load and Generation Node P Q 1 — — 2 -0.9 -0.9 3 -1.6 -0.8 4 0.5 — 5 1.5 — All items are in p.u.; generation is positive P, Q; load is negative P, Q The 5 node base case solution vector as .obtained by the N-R loadflow used for the thesis research is shown below in Table D.3. Table D.3 5 Node Solution Vector Node (i) |V;| 0f 1 1.00000 0.00000 2 0.95898 0.01059 3 0.92036 -0.04759 4 1.00000 0.00932 5 1.00000 0.07141 Vi| in p.u. and 6i are in radians 52 APPENDIX E. Routines Required for the Thesis Research The following routines were used during the research for the thesis. 1. Basic Newton-Raphson Loadflow Routine Output: i) Voltage magnitudes ii) Voltage angles iii) Line currents iv) Line power flows v) Line power losses These outputs were available if proper convergence had occurred or if the minimum mismatch solution had been found. The above routines were derived from [6]. 2. Lowest Voltage Magnitude Routine Output: i) Lowest voltage magnitudes 3. Graphic Output Routine This routine output data to files in a form which graphics routines could read. 4. Data Storage Routine This routine required only non-zero entries of the initial voltage guess, the admittance matrix, and the load or generation data. It assembled this base case data in a file in a manner suitable to the N-R loadflow routine. 5. Data Modification Routine Enabled the user to modify small portions of the overall case data without requiring all the case data to be re-entered. 6. Singularity Search Method Routine This routine operates as per Section 3.6. 7. Damped Newton-Raphson Routine This routine operates as per Section3.4. 8. Stability Search Routine A user defined value such as a nodal load or a branch impedance was successively varied automatically until divergence occurred. This was used to acquire diverging test cases and to also pinpoint at which value of the varied parameter divergence occurred. 9. Mismatch Routines 53 Output: i) Mismatch distance ii) Nodal mismatches iii) Largest mismatches in absolute value 10. Inner Product Routine This routine was used to obtain the inner product of iteration i and i — 1 Newton correction vectors or eigenvectors corresponding to the eigenvalue closest to zero. 11. Sensitivity Routine This routine operates as described in chapter 4. The algorithm can be derived from either [6] or [8]. 12. U.B.C. Matrix Routines i) DSING - provides double precision singular values ii) DREIGN - provides double precision complex or real eigenvectors and eigenvalues 13. Condition number routine Output the Jacobian matrix's condition number as calculated from the largest and smallest singular values. 54 

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