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Alternative parameters for the continuation power flow method
Electric Power Systems Research 66 (2003) 105 /113 www.elsevier.com/locate/epsr
Alternative parameters for the continuation power flow method Dilson A. Alves a,*, Luiz C.P. da Silva b, Carlos A. Castro b, Vivaldo F. da Costa b a
Electrical Engineering Department, Electrical Engineering Faculty, Universidade Estadual Paulista (UNESP), C.P. 31-CEP 15385-000 Ilha Solteira, SP, Brazil b School of Electrical and Computer Engineering, State University of Campinas-UNICAMP, C.P. 6101-CEP 13081-970, Campinas, SP, Brazil Received 2 May 2002; received in revised form 18 November 2002; accepted 16 December 2002
Abstract The conventional Newton’s method has been considered inadequate to obtain the maximum loading point (MLP) of power systems. It is due to the Jacobian matrix singularity at this point. However, the MLP can be efficiently computed through parameterization techniques of continuation methods. This paper presents and tests new parameterization schemes, namely the total power losses (real and reactive), the power at the slack bus (real or reactive), the reactive power at generation buses, the reactive power at shunts (capacitor or reactor), the transmission lines power losses (real and reactive), and transmission lines power (real and reactive). Besides their clear physical meaning, which makes easier the development and application of continuation methods for power systems analysis, the main advantage of some of the proposed parameters is that its not necessary to change the parameter in the vicinity of the MLP. Studies on the new parameterization schemes performed on the IEEE 118 buses system show that the illconditioning problems at and near the MLP are eliminated. So, the characteristics of the conventional Newton’s method are not only preserved but also improved. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Voltage stability; Continuation methods; Power flow; Maximum loading; Losses
1. Introduction Voltage stability has become a critical issue because the continuous load increase along with economical and environmental constraints has led systems to operate close to their limits, with reduced stability margins. Thus, an accurate knowledge of how far the current system’s operating point is from the voltage stability limit is crucial to operators, which often need to assess if the system has a secure and feasible operation point following a given disturbance, such as a transmission line outage or sudden change in system loading. Voltage stability static analyses of power systems can be assessed through obtaining critical buses voltage profiles as a function of their loading (PV, QV, and SV curves). These voltage profiles provide considerable insight into the system’s operating conditions for different loading levels, and have been used by the
* Corresponding author. Tel.: /55-18-3743-1150; fax: /55-183743-1163. E-mail address: [email protected] (D.A. Alves).
electric power industry for assessing voltage stability margins and the areas prone to voltage collapse [1]. Among other applications, these curves are used to determine transfer limitations between system areas, margins setting, and comparing transmission plans. Automated power flow procedures using the conventional Newton method were adopted by many electric power industries for carrying out these analyses. The power flow equations are essential for such analyses, since they represent a bound on a region of possible operation. When the power flow equations present no solution under a specified loading condition, it is usually concluded that generation and network are not physically capable of supplying the load requirement, demanding modifications on the generation dispatch and/ or on the transmission system in order to securely supply that load requirement [2]. In other words, for systems which loads behave as constant real and reactive power at steady state, the gradual load increment will lead to a saddle-node bifurcation [3] which corresponds to the MLP. A set of MLP’s defines the boundary between the regions of stable and unstable operation [4].
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00024-5
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The computation of such points is important for the knowledge of voltage stability margins, as well as to the modal analysis, which is used to identify system areas with voltage stability problems and to determine the most adequate reactive power support strategy. Modal analysis is most effectively used for system reinforcement studies or improved control measures when made at the MLP or near it [5]. However, the use of conventional power flow methods to obtain the curves is restricted to their upper portions (corresponding to stable operation). At MLP, the Jacobian (J) matrix of the power flow equations becomes singular and, consequently, the conventional methods will show numerical difficulties. They occur even if double-precision computation and anti-divergence algorithms are used [6,7]. Besides, although the use of these methods associated to a good step size control allow the computation of operation points rather close to the MLP [8], there will always be the need to ponder whether non-convergence situations are due to numerical problems or to the system physical limitation, task often not so obvious. Continuation methods are powerful and useful tools for obtaining solution curves for general non-linear algebraic equations by automatically changing the value of a parameter [9]. Unlike conventional power flow programs, it can compute the power flow solution at or near the MLP where traditional programs often fail to converge or take longer time for a solution. Among the many methods described in the literature, the most widely used ones consist of four basic elements: a predictor step, a corrector step, a step size control, and a parameterization procedure [9]. The parameterization is necessary to remove the Jacobian matrix singularity at the MLP, which causes numerical problems either in the predictor or in the corrector step [9,10]. In [11], a secant predictor and an arc-length parameterization are proposed. In [3], a tangent vector predictor combined with a perpendicular intersection corrector and a step size control technique is proposed so as to avoid the need of parameterization. Although, the augmented Jacobian can be non-singular throughout the voltage profiles tracing process, those methods usually has a geometrical rather than a physical meaning, therefore, we are often more interested in another important predictor corrector method. In this paper are proposed a new method obtained with simple modifications in the conventional Newton method for determining MLP of a power system. In order to avoid problems caused by ill-conditioning, a new parameterized equations based either in the well-known real or reactive power losses equations, or real or reactive power at the slack bus, or reactive power at PV bus, or reactive power at shunts (capacitor or reactor), or transmission lines power losses (real and reactive), or transmission lines power (real and reactive) is added. Besides their clear physical meaning, which makes easier the development
and application of continuation methods for power systems analysis, the main advantage of some of the proposed parameters is that its not necessary to change the parameter in the vicinity of the maximum loading point (MLP). In addition, an approach for automatic choice of the continuation parameters in cases when it is necessary is also presented. Besides the fact that it locally removes the singularity, it should be emphasized that the inherent characteristics of the conventional method are enhanced and the region of convergence around the singular solution is enlarged. Several tests were also carried out to compare the performance of the new proposed parameterization schemes for the continuation power flow method, by the use of secant and tangent predictors. The application of these new methodologies to the IEEE 118 buses system shows that complete voltage profiles, as well as the MLP, can be obtained with the desired precision without any numerical difficulties.
2. Continuation power flow The purpose of the continuation power flow consists in to find a continuum of load flow solutions for a given load change scenario. Its objective is to trace bus voltage profiles starting from a known initial solution (the base case) and using a predictor/corrector scheme to find subsequent solutions up to the MLP. From this process, it can be obtained the voltage stability margin, and additional information related to system bus voltages behavior as the loading level increases. In general, the load flow equations can be written as: G(u; V; l)0
(1)
where V is the vector of bus voltage magnitudes, u is the vector of bus voltage angles, l is the loading factor or level and G is the basic set of power flow equations. Eq. (1) can be rewritten as, lPsp P(u; V)0 sp
and
lQsp Q(u; V)0
(2)
where P /Pgen/Pload for load (PQ) and generation (PV) buses and Qsp /Qgen/Qload for PQ buses, and l / 1 corresponds to the base case generation and loading condition. Eq. (2) supposes that the system loading is proportional to the base case and considers constant power factor during the voltage profiles tracing process. Loads was represented as constant real and reactive power, since it gives the most secure operational condition to the power system [1,12]. Traditionally, Eq. (2) is solved by using conventional Newton’s method specifying a gradual increase in l. In this case, l is considered as an independent variable (pre-specified) in Newton’s iterative process. In the continuation power flow methods l can be considered as a dependent variable. Now, Eq. (2) is composed by n/2nPQ/nPV
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equations (nPQ and nPV correspond to the number of PQ and PV buses, respectively) in n/1 unknown variables. The difference among all the continuation methods presented at the literature relates to the way this new variable is included in the power flow problem, and to the procedure adopted to avoid the Jacobian matrix singularity. The addition of parameterized equations to the power flow problem has been used as a standard tool for tracing voltage profiles. 2.1. Predictor and parameterization schemes Once the base case solution for Eq. (1) has been found (V0, u0, l0 /1) by a conventional load flow method, a continuation method may be used to compute further solutions until MLP is reached. First, a predictor step is carried out so as to finding an estimate for the next solution point. Among the several different predictors found in the literature the tangent [10] and the secant [11] methods, illustrated in Fig. 1, are the most popular. In the tangent method, the estimate of the next solution can be found by taking an appropriately sized step in a direction tangential to the solution path at the current solution. The secant method uses the current and the previous solutions in order to estimate the next one. A trivial predictor is the modified zero-order polynomial [11], which uses the current solution and a fixed increment in the parameter as an estimate for the next solution. The main advantage of this method is that it is relatively inexpensive and presents no problems related to the singularity of the Jacobian matrices. On the other hand, the tangent predictor is usually more accurate than the secant predictor. Both will be used in this paper. The tangent vector (t) is obtained by taking the derivatives of both sides of Eq. (1), which gives,
[Gu
GV Gl ][du
dV
107
dl]T [J Gl ]t [0]
where Gu , GV , and Gl are matrices that correspond to the partial derivatives of G with respect to u , V , and l, respectively. Gu and GV compose the conventional load flow Jacobian matrix (J). It should be noted that the column added to J (/Gl ) corresponds to the new variable l. Since the number of unknown variables is higher than the number of equations, one more equation is needed. This is satisfied by setting one of the components (referred to as the continuation parameter) of the t vector to /1 or /1. Eq. (3) now becomes, 2 3 du Gu GV Gl 4 5 0 (4) dV Ja t ek 91 dl where ek is a row vector with null elements except the k th, which is equal to 1, being the sign (9/) determined by its slope. Variable k is chosen in some way that the modified Jacobian matrix (Ja) be non-singular at the MLP. The variable l can be used as the continuation parameter until close to the MLP, where J becomes illconditioned. In order to adds the information lost with the rank reduction of matrix J at the MLP, Eq. (1) is augmented by one equation that removes the singularity of J. This problem is solved by using local parameterization techniques [10], which consists in changing the continuation parameter close to the MLP. In the tangent vector method, the variable chosen to be the new parameter is the one that has the largest normalized entry in the tangent vector, and l becomes a dependent variable. Once the tangent vector t is computed, the estimate for the next solution is given by, [ue
Ve
le ]T [uj
Vj
T lj ] s[du
dV
dl]T
(5)
where ‘e’ stands for estimate, that is, vector t is used to obtain an estimate for u , V , and l starting from the current solution ‘j’. s is a scalar that defines the predictor step size, whose value must be such that the estimate is within the radius of convergence of the corrector. By using Vk or uk as parameters, the new Jacobian matrix may become singular either in the lower or upper part of the PV curve. Therefore, parameter changes may be necessary during the computation process. In this paper, it is shown that many parameterization schemes can be used to remove the Jacobian matrix singularity at the MLP. It follows the basic ideas of the new parameterizations proposed in this paper. Consider now that Eq. (1) is augmented by the following equation: W (V; u; l; m)mW 0 F (V; u; l)0
Fig. 1. Comparison between the continuation methods with tangent and secant predictors, l as parameter.
(3)
(6)
where F is a equation corresponding either to the wellknown equations of the active or reactive power losses, or active or reactive power at the slack bus, or reactive
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power at PV bus, or reactive power at shunts (capacitor or reactor), or transmission lines power losses (real and reactive), or transmission lines power (real and reactive), and W0 is the respective function value at base case. A new parameter m is added to the problem. Now, since one equation is added, l can be treated as a dependent variable and m is regarded as a parameter. For m/1, the converged solution should provide l/1. With Eq. (6), it is possible to specify the desired amount of real power losses, and the solution of Eq. (5) provides the operating point, including the loading factor (l), for which those losses occur. By expanding Eq. (1) augmented by Eq. (6) in Taylor’s series up to the first order terms one get, J Gl Dx 0 Dx Jm (7) DkW Fx Fl Dl Dl where x /[u , V ], Fx stands for derivatives of F with respect to x and DkW /W0Dm is a scalar that define the change in W0, DkW is actually the size of the predictor step, and Jm the modified Jacobian. After solving Eq. (7) for the tangent vector, the estimate for the next solution is given by: [xe
le ]T [x
l]T s[Dx
(8)
If it is not possible to reach the MLP by directly increasing the loading factor l due to Jacobian matrix singularity, one can specified the desired amount of any function (mW0) which is not belong to the conventional power flow equations set, being l regarded as a dependent variable. Therefore, instead of specifying the loading factor and getting the operating point, with Eq. (7), e.g. it is possible to specify the desired amount of real power losses, and the solution of Eq. (7) provides the operating point, including the loading factor, for which those losses occur. In this work, functions F correspond to one of the parameters previously mentioned. 2.2. Corrector step and step-size control Starting from the estimate (xe, le), the actual next solution point is computed by solving G(V; u; l)0
and
W (V; u; l)kpe F (V; u; l)0:
ss0 =ktk2
(10)
where t2 is the Euclidean norm of tangent vector [Dx Dl]T and s0 is a predefined scalar. As the system becomes stressed, the magnitude of the tangent vector increases and s decreases. The efficiency of the process depends on a good choice of s0. In general, its value is system dependent. Finding an optimal value for s0 is beyond the scope of this work.
3. Test results
Dl]T ;
e kW kW skW :
becomes necessary. The efficiency in curve tracing is closely related to the step length control strategy. A simple step size control method is based on the number of iterations of the corrector step. If the number of iterations is small, the system is still under light load condition and the step size for the next iteration can be larger. If the number of iterations is large, the system is stressed and the step size for the next iteration must be smaller. Another step size control method is based on the normalized tangent vector [13]. The step size is
(9)
The Jacobian matrix of Eq. (9) has the same structure of the one used for obtaining the tangent vector in Eq. (7). For light load conditions a change in load will result in a small change in the operating point. Therefore, the step size (s) could be larger. On the other hand, a small change in load would result in large variation of the operating point in case the system is stressed (heavily loaded). In this case, an adequate step size control
The goal is to compare the characteristics of the proposed methods with the purpose of pointing out its features, considering the influence of reactive power (Q ) and transformer tap limits. All tests were carried out for IEEE 118 bus systems. The mismatch convergence threshold was 105 p.u., and the values adopted for s0 and DkW were 0.5 and 1, respectively. The first point of each curve was obtained by using a conventional power flow method. Upper and lower tap limits of respectively 1.05 and 0.95 were adopted. The adjustment of tap in the on load-tap changing (OLTC) transformers consisted of including the tap position as a dependent variable, whereas the controlled bus voltage magnitude was considered as an independent variable. The method of accounting for Q -limit at PV buses and tap limits is the same as in conventional load flow methods. At every iteration, the reactive generation at all PV buses are compared to their respective limits. In case of violation, a PV bus is switched to type PQ. This bus can be switched back to PV in future iterations. Tap limit violations are also checked. The loads were modeled as constant power and parameter l was used to simulate the real and reactive load increment, considering a constant power factor. The increase in load was followed by a generation increase using l. 3.1. Comparison of several parameterization Fig. 2 shows the effect of the choice of some of the alternative continuation parameters in the application of the continuation method for the IEEE 118 buses system. The goal is to compare the performance (number of
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Fig. 2. (a) and (b) voltage magnitude as a function of l and kQpv, respectively. Number of iterations for the new parameterization schemes: (c) with a single step size; (d) with a doubled step size.
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iterations) of the several parameterization during the changes of one state to another. In that way, the corresponding parameters of each state were obtained by using the bus voltage (V52) as parameter; the fixed step size of 0.01 p.u. was used. This approach corresponds to a predictor step using a modified zero-order polynomial procedure [9,11], which uses the current solution and a fixed increment in the parameter (l, Vk , uk , or m) as an approximation point for the next solution. Once obtained all the states, the respective parameters were computed and then used as estimate by respective methods. In that way, it is guaranteed that these values will take the system from the same initial state to the same final state. This procedure was adopted because for each parameterization the estimate values obtained by the tangent vector are different, which would give the comparison of different points of the curve. As one can see from the results, all versions presented practically the same performance. In the vicinity of the MLP, the conventional method presents numerical difficulties, while the proposed methods not only succeed in finding a solution, but they also allowed the obtaining of points beyond the MLP. Fig. 2(a) show how the singularity is moved beyond MLP (to the lower part of the PV curve) when the parameterizing functions are added to the power flow equations. With this methodology, the step size need not be reduced significantly as can be confirmed by Fig. 2(d), where the same problem is solved with doubled step size, and the number of iterations for each point even so remain reduced, despite the PV buses and OLTC encounter limits. It is important to note that the proposed parameterizing techniques result in an excellent convergence performance in the vicinity of MLP. This occurs due to the non-linearity reduction of power flow equations consequently of the new parameter in use as is shown for example by Fig. 2(b), where a voltage bus (V52) is depicted in function of the reactive power at a PV bus 116 (kQpv), the new parameter. In the MLP region, the modified equation system presents a nearly linear behavior, justifying the fewer iteration number of Newton method. In other words, it can be said that while the conventional method has convergence problems due to the high degree of system nonlinearity in the MLP vicinity, the proposed parameterized methods obtain the MLP with easiness due to elimination, or decrease, of this non-linearity in the same region. 3.2. Tangent predictor Fig. 3 illustrates the effect of the choice of the continuation parameter in the application of the continuation method with tangent predictor, for the IEEE 118 system. Fig. 3(a /f) show the respective parameter
and critical bus voltage (V52), as function of l. These curves also show the predicted points, different for each adopted parameter, along with the number of iterations necessary by each corrector step. Starting from a known initial point an estimate to the next point is obtained by solving Eq. (8) and Eq. (10), respectively. Then, a corrector step is performed by solving Eq. (9). This procedure is repeated until the vicinity of the point where the modified Jacobian matrix (Jm) becomes singular, point B in the figures. Besides, it can be observed in these figures, that in case, when l is used as a parameter, the singularity (that corresponds to MLP) occurs at l/1.8664. In the vicinity of the MLP, the conventional method presents numerical difficulties, while all the methods succeed in finding a solution. This is an advantage of using the new parameters rather than l . A reduction in the number of iterations in the corrector step is observed in the vicinity of MLP. This happens due to the step size control method, based on the normalized tangent vector [13]. As one can see, with these approaches, the performance will be good for all the curve points. Table 1 shows another aspect to be observed, the value obtained for MLP will depend on function W and step size (s0) adopted. Then, if one requires to calculate MLP more accurately, it is necessary to go back to previous point, before changing the sign of l, and then reduce the step size, as it is done in other proposed methods [10,11]. Its important to emphasized that the Jacobian matrices obtained by using real or reactive power losses and real power at the slack bus as parameters become singular after MLP has been reached. However, sometimes this happens very close to MLP. That proximity only implies the need of performing a good step size control, which is already done for the case of l. Therefore, the great advantage of using these parameters is that it is not really necessary to carry out a parameter change a little after MLP. On the other hand, a change in the continuation parameter may be necessary during the PV curve tracing process when using the other parameters such as the reactive power at PV bus, the reactive power at shunts (capacitor or reactor), the transmission lines power losses (real and reactive), and the transmission lines power (real and reactive). By using transmission line power losses (real and reactive) as parameters, the new Jacobian matrix may become singular either in the lower or upper part of the PV curve. This is shown in Fig. 3(c), where the singularity occurs close to l/0.5. As the curvature of solutions path is not known a priori, it is very difficult, or even impossible, to identify which transmission line power loss equation is more appropriate to be used as parameter to obtain all the PV curve points, and, in particularly, the MLP. Therefore, an approach to define the parameter changes during the
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Fig. 3. Comparison on the continuation methods by using the tangent predictor and parameterization by functions W .
computation process is needed. So, depending on the PV bus, or shunt, or transmission line chosen to define the parameter either a good step size control or a change in the parameter will be necessary if one wants to trace the complete diagram. Nevertheless, it is important to note that parameterizing results in an excellent convergence performance in the vicinity of MLP.
3.3. Choosing the continuation parameter The choice of the new parameter is based in the following idea. As the load increases, transmission line series reactive losses also increase, while the reactive support from shunts (including line charging) decreases. Usually, most of the voltages remain very close to 1.0
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Table 1 MLP values by using different continuation parameters Parameter
Parameter value
MLP
V52 (p.u.)
kQpv kQsh kPeLT kPsb kPa kPaLT
3.5684 2.3396 0.0130 12.9865 6.1324 0.1138
1.8663 1.8658 1.8657 1.8610 1.8631 1.8663
0.735 0.726 0.725 0.784 0.774 0.747
p.u. even with increased loading. On the other hand, the progressive exhaustion of generators reactive reserves reduces the reactive supply to a bus or area, and reduces the effectiveness of voltage control, with subsequent increase of the losses and loss rate with respect to voltage drop. The continual decrease in voltage causes an increase of the angular difference across transmission line in order to maintain the active power transmission. Small changes in voltage magnitudes will not affect the shunt reactive powers significantly, whereas even a small change in voltage across the lines associated with large angles can cause substantial change in the series reactive loss. In that condition, both the transmission line loss (real or reactive) and the transmission line reactive power parameter choices, i.e. the choice of the transmission line itself, can be based either on transmission line losses rate, or transmission line belonging to the neighborhood of the bus that presents the smallest voltage. The approach used in this paper for the automatic choice of these continuation parameters is based on the transmission line belonging to the neighborhood of the bus that presents the highest rate of voltage variation and that presents the highest rate of increase of losses for the transmission line losses (real or reactive) parameters, and that presents the highest rate of increase of reactive power for transmission line reactive power parameter. When using transmission line active power as the continuation parameter, the line with the highest angular opening is chosen. For shunts, the bus with the largest voltage variation is selected as the continuation parameter. In relation to the choice of reactive power at a PV bus as a parameter, initially, any of the reactive powers at PV buses that are within bounds could be used. However, as the system approaches the MLP several buses will have reached their respective reactive limits. Therefore, it will be necessary to monitor the PV buses (it is already done during the iterative process), and to choose the one that is within its bounds. The increase in the number of iterations could be used as a criterion of the parameter change, kQpv in this case. However, this criterion should be evaluated carefully due to situations as the first point predicted in Fig. 3(a). Due to the curvature characteristics of the function in use, an apparently large number of iterations Eq. (8) is neces-
sary for the corrector. This happens due to the large step size in parameter kQpv, but not due to its proximity to singularity. A more appropriate option would be to observe that there will be the need of parameter change when the number of iterations increases even with step size reduction [14]. Fig. 4 illustrates the performances (number of iterations needed by the corrector step) of the tangent and secant predictor by using three proposed parameters, real and reactive transmission lines power losses, and reactive power at shunts. It can be noted that both procedures present the same performance for practically all the PV curve points.
4. Conclusion This paper presented new parameterizing schemes that allow the complete tracing of the PV curves based on simple modifications of Newton’s method, while keeping their characteristic advantages. It was shown that the proposed modifications not only preserve the characteristics of the conventional method but also improves its convergence at the MLP region. It was also shown that continuation and conventional methods can be switched during the tracing process in order to efficiently determine all the PV curve points with few iterations. Comparisons on the characteristics of the new techniques have shown that it is possible to determine the MLP with the desired precision, without any numerical difficulty. It has been made possible by removing the Jacobian matrix singularity at the MLP, which is done by adding one of the following equation on the power flow problem: power losses (real or reactive), power at the slack bus (real or reactive), reactive power at PV bus, transmission line power losses (real or reactive), transmission line power (real or reactive), and reactive power at shunts. It was shown that with the use of real power at the slack bus or real power losses equation have as main advantage the possibility to obtain the MLP, once the modified Jacobian matrix presents singularity beyond the MLP. Therefore, in these cases, a change in the parameter before MLP will not be necessary, but at most a step reduction in its vicinity. For the remaining parameters it is necessary to make the change during the tracing process. In these cases, an approach is also presented for the automatic choice in order to efficiently determine all the PV curve points with few iterations. In spite of the need for changing these parameters, the procedure is at most similar to those found in the literature [8,10]. The use of such equations also has as advantage the introduction of parameters with clear physical meaning, and of easy implementation.
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Fig. 4. Performance comparison between tangent and secant predictors on the continuation method: (a) transmission line real losses; (b) transmission line reactive losses; (c) reactive power at shunts.
As it is demonstrated by the results of this work, all the proposed methods present an excellent performance at the vicinity of the MLP and the same trajectory solution tracing.
[7]
Acknowledgements
[8]
The two first authors are grateful to the financial support provided by CAPES and FUNDUNESP (Brazilian Research Funding Agencies).
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Alternative parameters for the continuation power flow method Dilson A. Alves a,*, Luiz C.P. da Silva b, Carlos A. Castro b, Vivaldo F. da Costa b a
Electrical Engineering Department, Electrical Engineering Faculty, Universidade Estadual Paulista (UNESP), C.P. 31-CEP 15385-000 Ilha Solteira, SP, Brazil b School of Electrical and Computer Engineering, State University of Campinas-UNICAMP, C.P. 6101-CEP 13081-970, Campinas, SP, Brazil Received 2 May 2002; received in revised form 18 November 2002; accepted 16 December 2002
Abstract The conventional Newton’s method has been considered inadequate to obtain the maximum loading point (MLP) of power systems. It is due to the Jacobian matrix singularity at this point. However, the MLP can be efficiently computed through parameterization techniques of continuation methods. This paper presents and tests new parameterization schemes, namely the total power losses (real and reactive), the power at the slack bus (real or reactive), the reactive power at generation buses, the reactive power at shunts (capacitor or reactor), the transmission lines power losses (real and reactive), and transmission lines power (real and reactive). Besides their clear physical meaning, which makes easier the development and application of continuation methods for power systems analysis, the main advantage of some of the proposed parameters is that its not necessary to change the parameter in the vicinity of the MLP. Studies on the new parameterization schemes performed on the IEEE 118 buses system show that the illconditioning problems at and near the MLP are eliminated. So, the characteristics of the conventional Newton’s method are not only preserved but also improved. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Voltage stability; Continuation methods; Power flow; Maximum loading; Losses
1. Introduction Voltage stability has become a critical issue because the continuous load increase along with economical and environmental constraints has led systems to operate close to their limits, with reduced stability margins. Thus, an accurate knowledge of how far the current system’s operating point is from the voltage stability limit is crucial to operators, which often need to assess if the system has a secure and feasible operation point following a given disturbance, such as a transmission line outage or sudden change in system loading. Voltage stability static analyses of power systems can be assessed through obtaining critical buses voltage profiles as a function of their loading (PV, QV, and SV curves). These voltage profiles provide considerable insight into the system’s operating conditions for different loading levels, and have been used by the
* Corresponding author. Tel.: /55-18-3743-1150; fax: /55-183743-1163. E-mail address: [email protected] (D.A. Alves).
electric power industry for assessing voltage stability margins and the areas prone to voltage collapse [1]. Among other applications, these curves are used to determine transfer limitations between system areas, margins setting, and comparing transmission plans. Automated power flow procedures using the conventional Newton method were adopted by many electric power industries for carrying out these analyses. The power flow equations are essential for such analyses, since they represent a bound on a region of possible operation. When the power flow equations present no solution under a specified loading condition, it is usually concluded that generation and network are not physically capable of supplying the load requirement, demanding modifications on the generation dispatch and/ or on the transmission system in order to securely supply that load requirement [2]. In other words, for systems which loads behave as constant real and reactive power at steady state, the gradual load increment will lead to a saddle-node bifurcation [3] which corresponds to the MLP. A set of MLP’s defines the boundary between the regions of stable and unstable operation [4].
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00024-5
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The computation of such points is important for the knowledge of voltage stability margins, as well as to the modal analysis, which is used to identify system areas with voltage stability problems and to determine the most adequate reactive power support strategy. Modal analysis is most effectively used for system reinforcement studies or improved control measures when made at the MLP or near it [5]. However, the use of conventional power flow methods to obtain the curves is restricted to their upper portions (corresponding to stable operation). At MLP, the Jacobian (J) matrix of the power flow equations becomes singular and, consequently, the conventional methods will show numerical difficulties. They occur even if double-precision computation and anti-divergence algorithms are used [6,7]. Besides, although the use of these methods associated to a good step size control allow the computation of operation points rather close to the MLP [8], there will always be the need to ponder whether non-convergence situations are due to numerical problems or to the system physical limitation, task often not so obvious. Continuation methods are powerful and useful tools for obtaining solution curves for general non-linear algebraic equations by automatically changing the value of a parameter [9]. Unlike conventional power flow programs, it can compute the power flow solution at or near the MLP where traditional programs often fail to converge or take longer time for a solution. Among the many methods described in the literature, the most widely used ones consist of four basic elements: a predictor step, a corrector step, a step size control, and a parameterization procedure [9]. The parameterization is necessary to remove the Jacobian matrix singularity at the MLP, which causes numerical problems either in the predictor or in the corrector step [9,10]. In [11], a secant predictor and an arc-length parameterization are proposed. In [3], a tangent vector predictor combined with a perpendicular intersection corrector and a step size control technique is proposed so as to avoid the need of parameterization. Although, the augmented Jacobian can be non-singular throughout the voltage profiles tracing process, those methods usually has a geometrical rather than a physical meaning, therefore, we are often more interested in another important predictor corrector method. In this paper are proposed a new method obtained with simple modifications in the conventional Newton method for determining MLP of a power system. In order to avoid problems caused by ill-conditioning, a new parameterized equations based either in the well-known real or reactive power losses equations, or real or reactive power at the slack bus, or reactive power at PV bus, or reactive power at shunts (capacitor or reactor), or transmission lines power losses (real and reactive), or transmission lines power (real and reactive) is added. Besides their clear physical meaning, which makes easier the development
and application of continuation methods for power systems analysis, the main advantage of some of the proposed parameters is that its not necessary to change the parameter in the vicinity of the maximum loading point (MLP). In addition, an approach for automatic choice of the continuation parameters in cases when it is necessary is also presented. Besides the fact that it locally removes the singularity, it should be emphasized that the inherent characteristics of the conventional method are enhanced and the region of convergence around the singular solution is enlarged. Several tests were also carried out to compare the performance of the new proposed parameterization schemes for the continuation power flow method, by the use of secant and tangent predictors. The application of these new methodologies to the IEEE 118 buses system shows that complete voltage profiles, as well as the MLP, can be obtained with the desired precision without any numerical difficulties.
2. Continuation power flow The purpose of the continuation power flow consists in to find a continuum of load flow solutions for a given load change scenario. Its objective is to trace bus voltage profiles starting from a known initial solution (the base case) and using a predictor/corrector scheme to find subsequent solutions up to the MLP. From this process, it can be obtained the voltage stability margin, and additional information related to system bus voltages behavior as the loading level increases. In general, the load flow equations can be written as: G(u; V; l)0
(1)
where V is the vector of bus voltage magnitudes, u is the vector of bus voltage angles, l is the loading factor or level and G is the basic set of power flow equations. Eq. (1) can be rewritten as, lPsp P(u; V)0 sp
and
lQsp Q(u; V)0
(2)
where P /Pgen/Pload for load (PQ) and generation (PV) buses and Qsp /Qgen/Qload for PQ buses, and l / 1 corresponds to the base case generation and loading condition. Eq. (2) supposes that the system loading is proportional to the base case and considers constant power factor during the voltage profiles tracing process. Loads was represented as constant real and reactive power, since it gives the most secure operational condition to the power system [1,12]. Traditionally, Eq. (2) is solved by using conventional Newton’s method specifying a gradual increase in l. In this case, l is considered as an independent variable (pre-specified) in Newton’s iterative process. In the continuation power flow methods l can be considered as a dependent variable. Now, Eq. (2) is composed by n/2nPQ/nPV
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equations (nPQ and nPV correspond to the number of PQ and PV buses, respectively) in n/1 unknown variables. The difference among all the continuation methods presented at the literature relates to the way this new variable is included in the power flow problem, and to the procedure adopted to avoid the Jacobian matrix singularity. The addition of parameterized equations to the power flow problem has been used as a standard tool for tracing voltage profiles. 2.1. Predictor and parameterization schemes Once the base case solution for Eq. (1) has been found (V0, u0, l0 /1) by a conventional load flow method, a continuation method may be used to compute further solutions until MLP is reached. First, a predictor step is carried out so as to finding an estimate for the next solution point. Among the several different predictors found in the literature the tangent [10] and the secant [11] methods, illustrated in Fig. 1, are the most popular. In the tangent method, the estimate of the next solution can be found by taking an appropriately sized step in a direction tangential to the solution path at the current solution. The secant method uses the current and the previous solutions in order to estimate the next one. A trivial predictor is the modified zero-order polynomial [11], which uses the current solution and a fixed increment in the parameter as an estimate for the next solution. The main advantage of this method is that it is relatively inexpensive and presents no problems related to the singularity of the Jacobian matrices. On the other hand, the tangent predictor is usually more accurate than the secant predictor. Both will be used in this paper. The tangent vector (t) is obtained by taking the derivatives of both sides of Eq. (1), which gives,
[Gu
GV Gl ][du
dV
107
dl]T [J Gl ]t [0]
where Gu , GV , and Gl are matrices that correspond to the partial derivatives of G with respect to u , V , and l, respectively. Gu and GV compose the conventional load flow Jacobian matrix (J). It should be noted that the column added to J (/Gl ) corresponds to the new variable l. Since the number of unknown variables is higher than the number of equations, one more equation is needed. This is satisfied by setting one of the components (referred to as the continuation parameter) of the t vector to /1 or /1. Eq. (3) now becomes, 2 3 du Gu GV Gl 4 5 0 (4) dV Ja t ek 91 dl where ek is a row vector with null elements except the k th, which is equal to 1, being the sign (9/) determined by its slope. Variable k is chosen in some way that the modified Jacobian matrix (Ja) be non-singular at the MLP. The variable l can be used as the continuation parameter until close to the MLP, where J becomes illconditioned. In order to adds the information lost with the rank reduction of matrix J at the MLP, Eq. (1) is augmented by one equation that removes the singularity of J. This problem is solved by using local parameterization techniques [10], which consists in changing the continuation parameter close to the MLP. In the tangent vector method, the variable chosen to be the new parameter is the one that has the largest normalized entry in the tangent vector, and l becomes a dependent variable. Once the tangent vector t is computed, the estimate for the next solution is given by, [ue
Ve
le ]T [uj
Vj
T lj ] s[du
dV
dl]T
(5)
where ‘e’ stands for estimate, that is, vector t is used to obtain an estimate for u , V , and l starting from the current solution ‘j’. s is a scalar that defines the predictor step size, whose value must be such that the estimate is within the radius of convergence of the corrector. By using Vk or uk as parameters, the new Jacobian matrix may become singular either in the lower or upper part of the PV curve. Therefore, parameter changes may be necessary during the computation process. In this paper, it is shown that many parameterization schemes can be used to remove the Jacobian matrix singularity at the MLP. It follows the basic ideas of the new parameterizations proposed in this paper. Consider now that Eq. (1) is augmented by the following equation: W (V; u; l; m)mW 0 F (V; u; l)0
Fig. 1. Comparison between the continuation methods with tangent and secant predictors, l as parameter.
(3)
(6)
where F is a equation corresponding either to the wellknown equations of the active or reactive power losses, or active or reactive power at the slack bus, or reactive
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power at PV bus, or reactive power at shunts (capacitor or reactor), or transmission lines power losses (real and reactive), or transmission lines power (real and reactive), and W0 is the respective function value at base case. A new parameter m is added to the problem. Now, since one equation is added, l can be treated as a dependent variable and m is regarded as a parameter. For m/1, the converged solution should provide l/1. With Eq. (6), it is possible to specify the desired amount of real power losses, and the solution of Eq. (5) provides the operating point, including the loading factor (l), for which those losses occur. By expanding Eq. (1) augmented by Eq. (6) in Taylor’s series up to the first order terms one get, J Gl Dx 0 Dx Jm (7) DkW Fx Fl Dl Dl where x /[u , V ], Fx stands for derivatives of F with respect to x and DkW /W0Dm is a scalar that define the change in W0, DkW is actually the size of the predictor step, and Jm the modified Jacobian. After solving Eq. (7) for the tangent vector, the estimate for the next solution is given by: [xe
le ]T [x
l]T s[Dx
(8)
If it is not possible to reach the MLP by directly increasing the loading factor l due to Jacobian matrix singularity, one can specified the desired amount of any function (mW0) which is not belong to the conventional power flow equations set, being l regarded as a dependent variable. Therefore, instead of specifying the loading factor and getting the operating point, with Eq. (7), e.g. it is possible to specify the desired amount of real power losses, and the solution of Eq. (7) provides the operating point, including the loading factor, for which those losses occur. In this work, functions F correspond to one of the parameters previously mentioned. 2.2. Corrector step and step-size control Starting from the estimate (xe, le), the actual next solution point is computed by solving G(V; u; l)0
and
W (V; u; l)kpe F (V; u; l)0:
ss0 =ktk2
(10)
where t2 is the Euclidean norm of tangent vector [Dx Dl]T and s0 is a predefined scalar. As the system becomes stressed, the magnitude of the tangent vector increases and s decreases. The efficiency of the process depends on a good choice of s0. In general, its value is system dependent. Finding an optimal value for s0 is beyond the scope of this work.
3. Test results
Dl]T ;
e kW kW skW :
becomes necessary. The efficiency in curve tracing is closely related to the step length control strategy. A simple step size control method is based on the number of iterations of the corrector step. If the number of iterations is small, the system is still under light load condition and the step size for the next iteration can be larger. If the number of iterations is large, the system is stressed and the step size for the next iteration must be smaller. Another step size control method is based on the normalized tangent vector [13]. The step size is
(9)
The Jacobian matrix of Eq. (9) has the same structure of the one used for obtaining the tangent vector in Eq. (7). For light load conditions a change in load will result in a small change in the operating point. Therefore, the step size (s) could be larger. On the other hand, a small change in load would result in large variation of the operating point in case the system is stressed (heavily loaded). In this case, an adequate step size control
The goal is to compare the characteristics of the proposed methods with the purpose of pointing out its features, considering the influence of reactive power (Q ) and transformer tap limits. All tests were carried out for IEEE 118 bus systems. The mismatch convergence threshold was 105 p.u., and the values adopted for s0 and DkW were 0.5 and 1, respectively. The first point of each curve was obtained by using a conventional power flow method. Upper and lower tap limits of respectively 1.05 and 0.95 were adopted. The adjustment of tap in the on load-tap changing (OLTC) transformers consisted of including the tap position as a dependent variable, whereas the controlled bus voltage magnitude was considered as an independent variable. The method of accounting for Q -limit at PV buses and tap limits is the same as in conventional load flow methods. At every iteration, the reactive generation at all PV buses are compared to their respective limits. In case of violation, a PV bus is switched to type PQ. This bus can be switched back to PV in future iterations. Tap limit violations are also checked. The loads were modeled as constant power and parameter l was used to simulate the real and reactive load increment, considering a constant power factor. The increase in load was followed by a generation increase using l. 3.1. Comparison of several parameterization Fig. 2 shows the effect of the choice of some of the alternative continuation parameters in the application of the continuation method for the IEEE 118 buses system. The goal is to compare the performance (number of
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Fig. 2. (a) and (b) voltage magnitude as a function of l and kQpv, respectively. Number of iterations for the new parameterization schemes: (c) with a single step size; (d) with a doubled step size.
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iterations) of the several parameterization during the changes of one state to another. In that way, the corresponding parameters of each state were obtained by using the bus voltage (V52) as parameter; the fixed step size of 0.01 p.u. was used. This approach corresponds to a predictor step using a modified zero-order polynomial procedure [9,11], which uses the current solution and a fixed increment in the parameter (l, Vk , uk , or m) as an approximation point for the next solution. Once obtained all the states, the respective parameters were computed and then used as estimate by respective methods. In that way, it is guaranteed that these values will take the system from the same initial state to the same final state. This procedure was adopted because for each parameterization the estimate values obtained by the tangent vector are different, which would give the comparison of different points of the curve. As one can see from the results, all versions presented practically the same performance. In the vicinity of the MLP, the conventional method presents numerical difficulties, while the proposed methods not only succeed in finding a solution, but they also allowed the obtaining of points beyond the MLP. Fig. 2(a) show how the singularity is moved beyond MLP (to the lower part of the PV curve) when the parameterizing functions are added to the power flow equations. With this methodology, the step size need not be reduced significantly as can be confirmed by Fig. 2(d), where the same problem is solved with doubled step size, and the number of iterations for each point even so remain reduced, despite the PV buses and OLTC encounter limits. It is important to note that the proposed parameterizing techniques result in an excellent convergence performance in the vicinity of MLP. This occurs due to the non-linearity reduction of power flow equations consequently of the new parameter in use as is shown for example by Fig. 2(b), where a voltage bus (V52) is depicted in function of the reactive power at a PV bus 116 (kQpv), the new parameter. In the MLP region, the modified equation system presents a nearly linear behavior, justifying the fewer iteration number of Newton method. In other words, it can be said that while the conventional method has convergence problems due to the high degree of system nonlinearity in the MLP vicinity, the proposed parameterized methods obtain the MLP with easiness due to elimination, or decrease, of this non-linearity in the same region. 3.2. Tangent predictor Fig. 3 illustrates the effect of the choice of the continuation parameter in the application of the continuation method with tangent predictor, for the IEEE 118 system. Fig. 3(a /f) show the respective parameter
and critical bus voltage (V52), as function of l. These curves also show the predicted points, different for each adopted parameter, along with the number of iterations necessary by each corrector step. Starting from a known initial point an estimate to the next point is obtained by solving Eq. (8) and Eq. (10), respectively. Then, a corrector step is performed by solving Eq. (9). This procedure is repeated until the vicinity of the point where the modified Jacobian matrix (Jm) becomes singular, point B in the figures. Besides, it can be observed in these figures, that in case, when l is used as a parameter, the singularity (that corresponds to MLP) occurs at l/1.8664. In the vicinity of the MLP, the conventional method presents numerical difficulties, while all the methods succeed in finding a solution. This is an advantage of using the new parameters rather than l . A reduction in the number of iterations in the corrector step is observed in the vicinity of MLP. This happens due to the step size control method, based on the normalized tangent vector [13]. As one can see, with these approaches, the performance will be good for all the curve points. Table 1 shows another aspect to be observed, the value obtained for MLP will depend on function W and step size (s0) adopted. Then, if one requires to calculate MLP more accurately, it is necessary to go back to previous point, before changing the sign of l, and then reduce the step size, as it is done in other proposed methods [10,11]. Its important to emphasized that the Jacobian matrices obtained by using real or reactive power losses and real power at the slack bus as parameters become singular after MLP has been reached. However, sometimes this happens very close to MLP. That proximity only implies the need of performing a good step size control, which is already done for the case of l. Therefore, the great advantage of using these parameters is that it is not really necessary to carry out a parameter change a little after MLP. On the other hand, a change in the continuation parameter may be necessary during the PV curve tracing process when using the other parameters such as the reactive power at PV bus, the reactive power at shunts (capacitor or reactor), the transmission lines power losses (real and reactive), and the transmission lines power (real and reactive). By using transmission line power losses (real and reactive) as parameters, the new Jacobian matrix may become singular either in the lower or upper part of the PV curve. This is shown in Fig. 3(c), where the singularity occurs close to l/0.5. As the curvature of solutions path is not known a priori, it is very difficult, or even impossible, to identify which transmission line power loss equation is more appropriate to be used as parameter to obtain all the PV curve points, and, in particularly, the MLP. Therefore, an approach to define the parameter changes during the
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Fig. 3. Comparison on the continuation methods by using the tangent predictor and parameterization by functions W .
computation process is needed. So, depending on the PV bus, or shunt, or transmission line chosen to define the parameter either a good step size control or a change in the parameter will be necessary if one wants to trace the complete diagram. Nevertheless, it is important to note that parameterizing results in an excellent convergence performance in the vicinity of MLP.
3.3. Choosing the continuation parameter The choice of the new parameter is based in the following idea. As the load increases, transmission line series reactive losses also increase, while the reactive support from shunts (including line charging) decreases. Usually, most of the voltages remain very close to 1.0
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Table 1 MLP values by using different continuation parameters Parameter
Parameter value
MLP
V52 (p.u.)
kQpv kQsh kPeLT kPsb kPa kPaLT
3.5684 2.3396 0.0130 12.9865 6.1324 0.1138
1.8663 1.8658 1.8657 1.8610 1.8631 1.8663
0.735 0.726 0.725 0.784 0.774 0.747
p.u. even with increased loading. On the other hand, the progressive exhaustion of generators reactive reserves reduces the reactive supply to a bus or area, and reduces the effectiveness of voltage control, with subsequent increase of the losses and loss rate with respect to voltage drop. The continual decrease in voltage causes an increase of the angular difference across transmission line in order to maintain the active power transmission. Small changes in voltage magnitudes will not affect the shunt reactive powers significantly, whereas even a small change in voltage across the lines associated with large angles can cause substantial change in the series reactive loss. In that condition, both the transmission line loss (real or reactive) and the transmission line reactive power parameter choices, i.e. the choice of the transmission line itself, can be based either on transmission line losses rate, or transmission line belonging to the neighborhood of the bus that presents the smallest voltage. The approach used in this paper for the automatic choice of these continuation parameters is based on the transmission line belonging to the neighborhood of the bus that presents the highest rate of voltage variation and that presents the highest rate of increase of losses for the transmission line losses (real or reactive) parameters, and that presents the highest rate of increase of reactive power for transmission line reactive power parameter. When using transmission line active power as the continuation parameter, the line with the highest angular opening is chosen. For shunts, the bus with the largest voltage variation is selected as the continuation parameter. In relation to the choice of reactive power at a PV bus as a parameter, initially, any of the reactive powers at PV buses that are within bounds could be used. However, as the system approaches the MLP several buses will have reached their respective reactive limits. Therefore, it will be necessary to monitor the PV buses (it is already done during the iterative process), and to choose the one that is within its bounds. The increase in the number of iterations could be used as a criterion of the parameter change, kQpv in this case. However, this criterion should be evaluated carefully due to situations as the first point predicted in Fig. 3(a). Due to the curvature characteristics of the function in use, an apparently large number of iterations Eq. (8) is neces-
sary for the corrector. This happens due to the large step size in parameter kQpv, but not due to its proximity to singularity. A more appropriate option would be to observe that there will be the need of parameter change when the number of iterations increases even with step size reduction [14]. Fig. 4 illustrates the performances (number of iterations needed by the corrector step) of the tangent and secant predictor by using three proposed parameters, real and reactive transmission lines power losses, and reactive power at shunts. It can be noted that both procedures present the same performance for practically all the PV curve points.
4. Conclusion This paper presented new parameterizing schemes that allow the complete tracing of the PV curves based on simple modifications of Newton’s method, while keeping their characteristic advantages. It was shown that the proposed modifications not only preserve the characteristics of the conventional method but also improves its convergence at the MLP region. It was also shown that continuation and conventional methods can be switched during the tracing process in order to efficiently determine all the PV curve points with few iterations. Comparisons on the characteristics of the new techniques have shown that it is possible to determine the MLP with the desired precision, without any numerical difficulty. It has been made possible by removing the Jacobian matrix singularity at the MLP, which is done by adding one of the following equation on the power flow problem: power losses (real or reactive), power at the slack bus (real or reactive), reactive power at PV bus, transmission line power losses (real or reactive), transmission line power (real or reactive), and reactive power at shunts. It was shown that with the use of real power at the slack bus or real power losses equation have as main advantage the possibility to obtain the MLP, once the modified Jacobian matrix presents singularity beyond the MLP. Therefore, in these cases, a change in the parameter before MLP will not be necessary, but at most a step reduction in its vicinity. For the remaining parameters it is necessary to make the change during the tracing process. In these cases, an approach is also presented for the automatic choice in order to efficiently determine all the PV curve points with few iterations. In spite of the need for changing these parameters, the procedure is at most similar to those found in the literature [8,10]. The use of such equations also has as advantage the introduction of parameters with clear physical meaning, and of easy implementation.
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Fig. 4. Performance comparison between tangent and secant predictors on the continuation method: (a) transmission line real losses; (b) transmission line reactive losses; (c) reactive power at shunts.
As it is demonstrated by the results of this work, all the proposed methods present an excellent performance at the vicinity of the MLP and the same trajectory solution tracing.
[7]
Acknowledgements
[8]
The two first authors are grateful to the financial support provided by CAPES and FUNDUNESP (Brazilian Research Funding Agencies).
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