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A new modeling of loading margin and its sensitivities using rectangular voltage coordinates in voltage stability analysis
Electrical Power and Energy Systems 32 (2010) 290–298
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A new modeling of loading margin and its sensitivities using rectangular voltage coordinates in voltage stability analysis Vander Menengoy da Costa a,*, Magda Rocha Guedes b,1, Arlei Lucas de Sousa Rosa a,2, Marcelo Cantarino c,3 a
Department of Electrical Engineering, Federal University of Juiz de Fora, Campus Universitário – Bairro Martelos, 36036-330 Juiz de Fora – MG, Brazil Federal Center of Technologic Education of Minas Gerais – CEFET, Rua José Peres, 558 36700-000 Leopoldina – MG, Brazil c Centrais Elétricas Brasileiras S.A – ELETROBRÁS, Av. Rio Branco, 53, Centro, 14° andar, 20090-004 Rio de Janeiro – RJ, Brazil b
a r t i c l e
i n f o
Article history: Received 15 October 2008 Received in revised form 19 June 2009 Accepted 25 September 2009
Keywords: Electrical Power Systems Maximum Loading Point Polar Voltage Coordinates Rectangular Voltage Coordinates Sensitivity Analysis Voltage Stability
a b s t r a c t This paper presents new mathematical models to compute the loading margin, as well as to perform the sensitivity analysis of loading margin with respect to different electric system parameters. The innovative idea consists of evaluating the performance of these methods when the power flow equations are expressed with the voltages in rectangular coordinates. The objective is to establish a comparative process with the conventional models expressed in terms of power flow equations with the voltages in polar coordinates. IEEE test system and a South-Southeastern Brazilian network are used in the simulations. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Voltage stability has been one of the major concerns of power systems operators and planners for the last years. The continuous load increase allied to the lack of investments in transmission and generation has led systems to operate very close to their limits. Voltage stability has been a subject widely investigated [1–4]. Its static analysis can be assessed through continuation power flow and point of collapse methods. Over the last years, the rectangular voltage coordinates have been applied to different electric power system areas and remarkable results have been published in the literature, as follows: The algorithm presented in Ref. [5] incorporates a step size optimization factor into the rectangular power flow problem. This algorithm deals with the power flow calculation as a nonlinear programming technique, where the direction and magnitude of solution are determined in order to minimize a certain objective function. The objective function tends to zero when the power flow converges to a solution and, otherwise, stays at a positive value. * Corresponding author. Tel.: +55 32 21023461; fax: +55 32 21023442. E-mail addresses: [email protected] (Vander Menengoy da Costa), [email protected] (M.R. Guedes), [email protected] (Arlei Lucas de Sousa Rosa), [email protected] (M. Cantarino). 1 Tel.: +55 32 88355232; fax: +55 31 3319 5009. 2 Tel.: +55 32 32325352; fax: +55 32 21023442. 3 Tel.: +55 21 25144817; fax: +55 21 25145767. 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.09.005
A current-version rectangular power flow proposed in Ref. [6] introduces a dependent variable for each generation bus together with an additional voltage constraint equation. Besides, except for generation buses, the Jacobian matrix has the elements of the (2 2) off-diagonal blocks equal to those of the bus admittance matrix expanded into real and imaginary coordinates. The elements of the (2 2) diagonal blocks are updated during each iteration according to the load model. In Refs. [7,8] an augmented current injection rectangular power flow formulation is presented. This formulation has the same convergence characteristics as the conventional polar power flow methodology. Many different flexible AC transmission system devices and controls are incorporated into this augmented methodology. Another current-version power flow is proposed in Ref. [9] where the basic idea is to solve an augmented system of equations in which both bus voltages and current injections appear as state variables, and both power and current mismatches are zeroed. Similarly to [6], the elements of the (2 2) diagonal blocks are updated during each iteration in terms of the load model. Basically, the main difference between these two methodologies is that in [6] the state vector is composed exclusively of bus voltages in rectangular coordinates. The current injection rectangular power flow [6] is extended in [10] for solving an unbalanced distribution three-phase power flow problem. This methodology presents an expressive mathematical
Vander Menengoy da Costa et al. / Electrical Power and Energy Systems 32 (2010) 290–298
robustness and converges with a reduced number of iterations. Besides, in Ref. [11] a PV bus representation to be used in three-phase current injection power flow is presented. This approach requires an augmented system of equations to incorporate the reactive power as a dependent variable. A step size optimization factor to be used for solving unbalanced three-phase distribution current injection is presented in Ref. [12]. In addition, this paper extends the concept of this optimization factor to calculate the multiple unbalanced three-phase power flow solutions. As can be seen, the rectangular voltage coordinates have been widely investigated in several studies related to the power system area and final results have been highly encouraging. Therefore, the main objective of this paper is to develop new continuation power flow and point of collapse methods when the power flow equations are expressed with the voltages in rectangular coordinates. In addition, new sensitivity models of loading margin with respect to different parameters using the same kind of coordinates are also presented. Currently, these methods and models presented in the literature are expressed only in terms of polar voltage coordinates. The notations adopted in the paper are the conventional ones whenever possible. Matrices are shown in bold and vectors are shown in bold underlined. 2. Rectangular power flow equations The basic power flow equations written in terms of rectangular voltage coordinates are given by X V rk ðGkm V rm Bkm V mm Þ þ V mk ðGkm V mm þ Bkm V rm Þ þ P Dk P Gk ¼ 0 m2Uk
X
ð1Þ
291
power flow equations to overcome this drawback. A brief review on continuation power flow is presented next. In [13] a mathematical model for the continuation power flow using either additional load change, or voltage magnitude as continuation parameters is presented. In [14] both active and reactive power losses and generated powers at slack bus are tested as continuation parameters. In [15] a tool for evaluating nonlinear effects on power system states due to branch admittance/impedance variations is presented. In [16] a continuation three-phase power flow, which can be used to analyze voltage stability of unbalanced threephase power systems, is proposed. In [17] a fuzzy continuation power flow is developed with the objective of handling simultaneously uncertainties in load parameters and bus injections parameters. These references use power flow equations written in terms of polar voltage coordinates. 3.1. Predictor step To simulate load and generation changes a loading factor c, which characterize the change of load, is employed. The nonlinear Eqs. (1) and (2) are augmented by an extra variable c resulting Fðx; cÞ ¼ 0. The predictor step is used to provide an approximate point of the next solution. A prediction of the next solution is made by taking an appropriately sized step in the direction tangent to t the solution path given by ½ dV r dV m dc . At each prediction, a new continuation parameter is chosen. The procedure adopted in the paper analyses the variation of each state between the two last correct points. The state with the largest percentage variation will be the next continuation parameter. If the continuation parameter is the additional loading c, the prediction in rectangular voltage coordinates is given by
V mk ðGkm V rm Bkm V mm Þ V rk ðGkm V mm þ Bkm V rm Þ þ Q Dk Q Gk ¼ 0
m2Uk
ð5Þ ð2Þ
where V rk þ jV mk is the real and imaginary voltage component at bus k; Gkm þ jBkm is the k–m element of bus admittance matrix; PGk þ Q Gk is the generated complex power at bus k; P Dk þ jQ Dk is the complex power load at bus k. UK denotes the set of buses directly connected to bus k, including itself. Eqs. (1) and (2), denoted by F P ðxÞ ¼ 0 and F Q ðxÞ ¼ 0, represent the power flow equations where x ¼ ½V r V m t . The order of vector x is 2n. n denotes the number of buses. In particular, for a generic bus k, these notations appear as F Pk ðxÞ and F Q k ðxÞ. The set of (1) and (2) will be denoted by FðxÞ throughout the paper. The rectangular Jacobian matrix is formed by the derivative of FðxÞ ¼ 0 with respect to state variables x. The ordinary rectangular power flow is iteratively solved through the following equation:
where pc is the continuation step size regarding c. The components of vector corresponding to a generic bus k, for a constant power load model, are given by PDk and Q Dk . If the continuation parameter is the voltage Vq, where q refers to the bus with the largest voltage variation between the two last correct points, the prediction in rectangular voltage coordinates is given by
# " DP @FðxÞ DV r ¼ DQ @x DV m
stem from Eq. (4). pV is the continuation step size regarding Vq. The estimated solution, denoted by *, is updated from the tangent vector as follows:
ð3Þ
For PV buses, the reactive power equations are replaced by voltage equations. For a PV bus p the voltage constraint in rectangular coordinates is given by
V 2p ¼ V 2rp þ V 2mp
ð4Þ
ð6Þ h i Vr Vm where e ¼ 0 V qq 0 0 V qq 0 . The components of this vector
2
3 2 3 2 3 Vr Vr dV r 6 7 6 7 6 7 4 V m 5 ¼ 4 V m 5 þ 4 dV m 5 c c dc
ð7Þ
3.2. Corrector step 3. Continuation power flow The voltage profile is obtained through successive power flow solutions by simulating load change. However, the voltage profile cannot be traced completely, by using only the conventional power flow, because the Jacobian matrix becomes singular at the maximum loading point. The continuation method is applied to the
The corrector step corresponds to solve the augmented Newton power flow equation with the predicted solution given by (7) as the initial solution. If the continuation parameter is the additional loading c, this step is to simply run an ordinary rectangular power flow at the estimated point. On the other hand, Eq. (8) is used if the continuation parameter is the voltage Vq.
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ð8Þ Finally, the state variables are updated at a given iteration h + 1 as follows:
2
3hþ1
Vr 6 7 4 Vm 5
c
2
3h
2
3h
Vr DV r 6 7 6 7 ¼ 4 V m 5 þ 4 DV m 5
c
ð9Þ
increment the iteration counter, h = h + 1, and solve (11) where the mismatches are given by
R1 ¼ ½DP1
DP 2
DP n
DQ 1
DQ 2
DQ n t
ð12Þ
t
R2 ¼
@FðxÞ w @x
ð13Þ
R3 ¼ 1 w21 w22 w22n ¼ 1
2n X ðwi Þ2
ð14Þ
i¼1
Dc
Step 5: Update the state variables and go back to Step 4. 4. Point of collapse method
2
4.1. Initial aspects The objective of this method is to iteratively calculate the maximum loading point of electric power systems without tracing the continuation curves. A brief review on this subject is presented next. Ref. [18] describes an extension of the point of collapse developed for ac systems studies to the determination of saddle-node bifurcations in power systems including high voltage direct current transmission. Ref. [19] describes the implementation of both point of collapse and continuation methods for computation of voltage collapse points in large ac/dc systems. Besides, Refs. [20,21] calculate the maximum loadability using interior point nonlinear optimization method. Again, the power flow equations are written in terms of polar voltage coordinates. The maximum loading point is characterized by the Jacobian matrix having a simple and unique zero eigenvalue, with nonzero left eigenvector w and right eigenvector v . This condition can be summarized by the set of nonlinear Eq. (10) [18,19] whose solution yields the saddle-node bifurcation point.
Gðx; c; wÞ ¼
8 > > > > <
Fðx; cÞ ¼ 0 @FðxÞt w @x
¼0 > 2n > P > > : 1 ðw2i Þ ¼ 0
ð10Þ
i¼1
The Newton–Raphson iterative method applied to (10) results in (11). The variables are 2n components of x, the 2n components of w and the load increase parameter c.
2
@Fðx;cÞ @x
6 h 6 @ @FðxÞt i 6 4 @x @x :w 0t
0 @FðxÞt @x
2w
3
2 3 2 3 R1 7 Dx 76 7 6 7 4 Dw 5 ¼ 4 R2 5 0 7 5 Dc R3 0
@Fðx;cÞ @c
ð11Þ
Vr
6V 7 6 m7 6 7 4 w 5
2
Vr
3h
6V 7 6 m7 ¼6 7 4 w 5
c
c
2
3 DV r h 6 DV 7 m7 6 þ6 7 4 Dw 5
ð15Þ
Dc
5. Sensitivity analysis of the loading margin In Ref. [22] linear and quadratic estimates to the variation of the loading margin with respect to any system parameter or control are derived. Polar power flow equations are used. Suppose that the equilibrium of the power system satisfy the equation Fðx; k; pÞ ¼ 0. The sensitivity of the load margin L to the change in parameters p is given by Greene et al. [22]
wt f p DL ¼ Mp ¼ t Dp w fk a
ð16Þ
where w is the left eigenvector corresponding to the zero eigenvalue of Jacobian matrix; f p and fk are the derivatives of Fðx; k; pÞ with respect to parameter p and load powers k, respectively. a is a unit vector which specify a pattern of load increase. For constant power loads f k is a diagonal matrix with ones in the rows corresponding to load buses. The terms wt f k a; wt and f k are calculated once regardless of the parameter chosen, and, therefore, the sensitivity to any additional parameters can be computed quickly. The loading margin sensitivities only depend on quantities evaluated at the nominal bifurcation point. Evaluation of the linear sensitivity is very simple. Once the nominal bifurcation point is computed, the linear sensitivity (16) requires computation of the left eigenvector wt and the vector f p . The quadratic estimate of the change in loading margin is given by Greene et al. [22]
1 DM ¼ M p Dp þ M pp Dp2 2
4.2. Solution algorithm
where
The rectangular point of collapse method can be summarized as follows:
Mpp ¼
Step 1: Calculate the estimated maximum loading point through an ordinary power flow. The methodology adopted is to gradually increase the loads until the power flow fails to converge. The estimated value of x and c will be the last point with normal convergence. Step 2: Assemble the rectangular Jacobian matrix at the estimated point. Step 3: Calculate the estimated left eigenvector of the Jacobian matrix associated with its critical eigenvalue. Step 4: Calculate the updated mismatch vector shown in the right-hand side of (11). If all components of this vector are smaller than a specified tolerance, then print results. Otherwise,
3hþ1
1 ½xt f xp þ 2wt f xp xp þ wt f pp wt f k a p xx
ð17Þ
ð18Þ
The quadratic estimate additionally requires the following quantities evaluated at the nose: (i) matrix f xx formed by the derivative of the product of w with the Jacobian matrix with respect to state variables x; (ii) f xp is the derivative of Jacobian matrix with respect to the parameter p; (iii) xp is the sensitivity of the nose equilibrium with respect to p; (iv) f pp is the second-order derivative of Fðx; k; pÞ with respect to parameter p. The sensitivity formulas evaluated at the voltage collapse yield linear and quadratic estimates of the loading margin as a function of any parameter. The performance of these estimates is tested for three different parameters, namely, line susceptance, active power load and bus shunt susceptance, representative of a range of control actions or system uncertainties.
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5.1. Line susceptance
6.2. Analysis of continuation power flow
These variations can represent the operation of a FACTS device or can reflect uncertainty in the network data. The sensitivity analysis can be useful to define the best place for FACTS installation in order to provide a better loading margin. Let k–m be the branch under analysis whose susceptance is bkm. The vector f pp is null. The matrix f xp is the derivative of Jacobian matrix with respect to bkm. The elements of this matrix will be different from zero only in the positions regarding the buses k and m. The components of the vector f p in rectangular coordinates are given by
The step size adopted for the continuation parameter is 0.01 p.u. The buses 526 (IEEE-300) and 1818 (1768-bus) experienced the largest voltage variation from the base case. Figs. 1 and 2 show the voltage profiles related to these buses provided by rectangular
@F Pk ðxÞ ¼ akm ðV mk V rm V rk V mm Þ @bkm
ð19Þ
@F Q k ðxÞ ¼ akm ðV mm V mk þ V rm V rk Þ a2km ðV 2mk þ V 2rk Þ @bkm
ð20Þ
@F Pm ðxÞ ¼ akm ðV mm V rk V rm V mk Þ @bkm
ð21Þ
@F Q m ðxÞ ¼ akm ðV mm V mk þ V rm V rk Þ ðV 2mm þ V 2rm Þ @bkm
ð22Þ
where akm is the tap transformer. 5.2. Active power load Usually, as the load power is reduced, the loading margin is increased. The sensitivity analysis can provide valuable information concerning the system operation. The load power factor is kept constant during the analysis. Let k be the bus whose active power load is PDk . The vector f pp and the matrix f xp are null. The components of the vector f p in rectangular coordinates are given by
@F Pk ðxÞ ¼1 @PDk @F Q k ðxÞ Q Dk ¼ @Q Dk PDk
Fig. 1. Voltage profile at bus 526: IEEE-300 system.
ð23Þ ð24Þ
5.3. Bus shunt susceptance These variations can represent a reactive power support to avoid voltage collapses. Let k be the bus whose shunt susceptance sh bk is selected for sensitivity analysis. The vector f pp is null. The mash trix f xp is the derivative of Jacobian matrix with respect to bk . Therefore, this matrix in rectangular voltage coordinates is composed only of elements 2V rk and 2V mk corresponding to the positions related to the reactive power injected at bus k with respect to real and imaginary voltage components. The only nonzero component of vector f p is given by
@F Q k ðxÞ sh
@bk
¼ ðV 2rk þ V 2mk Þ
Fig. 2. Voltage profile at bus 1818: 1768-bus system.
ð25Þ
6. Results 6.1. Initial considerations In order to perform this study some simulations were accomplished by using the IEEE 300-bus system and a practical Brazilian network composed of 1768 buses, 2527 branches and 119 generation buses. The tolerance adopted for convergence of the iterative process related to continuation power flow and point of collapse methods is 105 p.u. The constant power load model is adopted. Test systems are stressed by keeping a constant power factor. Reactive power generation limits are taking into account.
Fig. 3. Number of iterations in terms of loading: IEEE-300 system.
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and polar coordinates. The maximum loading point corresponds to additional loadings of 0.0246 p.u. (IEEE-300) and 0.0265 p.u. (1768-bus) with voltages V 526 ¼ 0:7828\ 79:59 p:u: and V 1818 ¼ 0:6432\ 78:99 p:u:. As expected, both coordinates yield the same maximum loading point, and the same voltages. Figs. 3 and 4 show the total number of iterations to complete the voltage profiles in both coordinates. The active and reactive bus loads are proportionally increased from the base case. The load power factors are kept constant. Basically, the number of iterations to yield the complete voltage profile is the same regardless of voltage coordinates. Figs. 5 and 6 show the number of iterations to complete the voltage profile when the series and shunt transmission line admittances are gradually reduced from the base case. The objective is to simulate the performance of continuation power flow methodologies under contingency conditions. The branches gradually removed are 37–49 and 2591–2593, regarding the IEEE-300 and 1768-bus systems, respectively. The complete branch removal corresponds to a factor zero. The IEEE-300 voltage profile is obtained even for a complete branch removal. On the other hand, the 1768bus system converges with a partial removal of up to 90%. Both rectangular and polar continuation power flow methods present a similar performance in terms of number of iterations. Figs. 7 and 8 show the number of iterations when the branches 37–49 and 2591–2593 are completely removed and 90% partially removed, respectively. The maximum number of iterations during
Fig. 6. Number of iterations in terms of a partial branch removal: 1768-bus system.
Fig. 4. Number of iterations in terms of loading: 1768-bus system.
Fig. 7. Performance for a full branch removal: IEEE-300 system.
Fig. 5. Number of iterations in terms of a partial branch removal: IEEE-300 system.
Fig. 8. Performance for a partial branch removal of 90%: 1768-bus system.
the corrector process is adopted as twenty. Therefore, twenty-one iterations mean that convergence of the iterative process is not achieved. In this case, the step size is reduced and the continuation parameter may or may not be changed in order to ensure that the corrector process is successful. Basically, both continuation power flow methods present the same number of corrector steps and iterations during each corrector step.
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295
6.3. Analysis of point of collapse method The maximum loading point provided by the rectangular and polar point of collapse methods corresponds to additional loadings of 0.0246 p.u. (IEEE-300) and 0.0265 p.u. (1768-bus) with voltages V 526 ¼ 0:7880\ 79:16 p:u: and V 1818 ¼ 0:6481\ 78:74 p:u:. A good agreement between these results and those ones obtained through continuation power flow can be observed. Figs. 9 and 10 show the maximum absolute power mismatches, at the end of each iteration, regarding the base case. The iterative process related to rectangular point of collapse method results large power mismatches in initial iterations, and the polar point of collapse method, in general, converges with less iterations. Figs. 11–14 show the number of iterations for convergence of the point of collapse method in both coordinates. Firstly, the active and reactive bus loads are proportionally increased from the base case. The load power factors are kept constant. Secondly, the series and shunt transmission line admittances are gradually reduced. The branches under analysis are the same ones as considered in the previous section. In general, the polar point of collapse method presents a better performance in comparison with the rectangular version.
Fig. 11. Performance as a function of loading: IEEE-300 system.
6.4. Sensitivity analysis The nominal parameters of IEEE-300 are the susceptance bkm = 80.6452 p.u. corresponding to branch 7049–49, the active
Fig. 12. Performance as a function of loading: 1768-bus system.
Fig. 9. Convergence trajectory: IEEE-300 system.
Fig. 13. Performance as a function of branch removal: IEEE-300 system.
Fig. 10. Convergence trajectory: 1768-bus system.
load P Dk ¼ 30:74 MW at bus 9052, and a null shunt susceptance sh bk at bus 20. The nominal parameters of 1768-bus are the susceptance bkm = 18.5530 p.u. corresponding to branch 536–559, the active load P Dk ¼ 36:134 MW at bus 2975, and a null shunt suscepsh tance bk at bus 2993. Figs. 15–20 show the loading margin
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Fig. 14. Performance as a function of branch removal: 1768-bus system.
Fig. 16. Line susceptance variation: 1768-bus system.
Fig. 17. Active power load variation: IEEE-300. Fig. 15. Line susceptance variation: IEEE-300 system.
changing as the parameters are varied. Actual results were obtained through the point of collapse method. The following comments can be extracted from Figs. 15–20. As the variation of parameter decreases, the linear and quadratic sensitivity analysis in rectangular coordinates provide very satisfactory results when compared with those generated by exact methods. Moreover, a good agreement between results yielded by both kinds of coordinates is observed. As the variation of parameter increases, the results provided by the quadratic sensitive analysis in both coordinates are better than the ones obtained through the linear sensitivity analysis. As expected, the second order term of the Taylor series improves the quadratic estimates. As a concluding remark, the performance of both kinds of coordinates is very similar. Therefore, the sensitivity analysis of loading margin with respect to different parameters can be efficiently and easily modeled in rectangular voltage coordinates. 6.5. Computational performance Tables 1 and 2 display the computation times, in seconds, required by polar and rectangular coordinates to calculate the linear and quadratic estimates regarding the 1768-bus system. Table 3 displays the relationships between the computation times required by both coordinates for each one of the tasks related to sensitivity calculation. Time relation is not shown for load power variations because the matrix f xp is null.
Fig. 18. Active power load variation: 1768-bus system.
The linear and quadratic rectangular estimates present computational gains around 90% and 36%, respectively, in comparison with the polar approach. Therefore, the use of rectangular coordinates requires smaller computation times and, as a result, leads to a substantially faster sensitivity analysis of loading margin. Certainly, this concluding remark stems from the particular structure
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Vander Menengoy da Costa et al. / Electrical Power and Energy Systems 32 (2010) 290–298 Table 3 Computation time relationships – 1768 bus. Tasks
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
1.9049 3.6573 1.2217 0.9264 1.4364
1.9090 3.8019 * 0.9173 0.7048
1.9045 3.9291 1.1642 0.9369 1.0348
Total relation
1.3663
1.3549
1.3793
sh
of rectangular power flow equations characterized by a quadratic form independent of transcendental functions.
7. Conclusions Fig. 19. Shunt susceptance variation: IEEE-300.
This paper deals with the calculation of the maximum loading point in electrical power systems through the continuation power flow and point of collapse methods. One of the main goals consists of evaluating both methods in rectangular voltage coordinates. The rectangular coordinates applied to both methods present a very simple mathematical modeling when compared to the traditional polar process. Although the polar point of collapse method has presented a slight advantage over the rectangular version, in general, the performance of rectangular coordinates is highly expressive even for adverse conditions investigated in the paper. Additionally, this paper deals with the evaluation of linear and quadratic rectangular sensitivity analysis of loading margin with respect to different parameters. These estimates can be used to evaluate the effect or the efficiency of the parameters variation on the loading margin of electrical power systems. Both linear and quadratic rectangular sensitivity models have a very simple mathematical structure, present very expressive results and propitiate a remarkable computational gain in comparison with the polar coordinates. Finally, since electrical power systems operate very close to their limits, this paper demonstrates that rectangular methods can be regarded as promising additional tools in voltage stability studies.
Fig. 20. Shunt susceptance variation: 1768-bus system.
Table 1 Computation time – polar coordinates – 1768 bus. Tasks
References
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
1.8420 1.8250 0.2480 3.3090 0.1580
1.9510 1.9580 * 3.3930 0.1170
1.8950 1.9960 0.2340 3.4770 0.1190
Total time
7.4820
7.5210
7.8260
sh
Table 2 Computation time – rectangular coordinates – 1768 bus. Tasks
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
0.9670 0.4990 0.2030 3.5720 0.1100
1.0220 0.5150 * 3.6990 0.1660
0.9950 0.5080 0.2010 3.7110 0.1150
Total time
5.4760
5.5510
5.6740
sh
[1] Taylor C. Power systems voltage stability. EPRI – power system engineering series. McGraw-Hill; 1994. [2] Kundur P. Power system stability and control. EPRI – power system engineering series. McGraw-Hill; 1994. [3] Cutsem TV, Vournas C. Voltage stability of electric power systems. Athens, Greece: University of Liège Belgium, National Technical University; 1998. [4] Kamalasadan S, Thukaram D, Srivastava AK. A new intelligent algorithm for online voltage stability assessment and monitoring. Int J Electr Power Energy Syst 2009;31(2):100–10. [5] Iwamoto S, Tamura Y. A load flow calculation method for ill-conditioned power systems. IEEE Trans Power Syst 1981;PAS-100(4):1736–43. [6] Da Costa VM, Martins N, Pereira JLR. Developments in the Newton–Raphson power flow formulation based on current injections. IEEE Trans Power Syst 1999;14(4):1320–6. [7] Da Costa VM, Pereira JLR, Martins N. An augmented Newton–Raphson power flow formulation based on current injections. Int J Electr Power Energy Syst 2001;23(4):305–12. [8] Variz AM, Da Costa VM, Pereira JLR, Martins N. Improved representation of control adjustments into the Newton–Raphson power flow. Int J Electr Power Energy Syst 2003;25(7):501–13. [9] Exposito AG, Ramos ER. Augmented rectangular load flow model. IEEE Trans Power Syst 2002;17(2):271–6. [10] Garcia PAN, Pereira JLR, Da Costa VM, Martins N. Three phase power flow calculations using the current injection method. IEEE Trans Power Syst 2000;15(2):508–14. [11] Garcia PAN, Pereira JLR, Carneiro Jr S, Vinagre MP, Gomes FV. Improvements in the representation of PV buses on three-phase distribution power flow. IEEE Trans Power Delivery 2004;19(2):894–6.
298
Vander Menengoy da Costa et al. / Electrical Power and Energy Systems 32 (2010) 290–298
[12] Da Costa VM, Oliveira ML, Guedes MR. Developments in the analysis of unbalanced three-phase power flow solutions. Int J Electr Power Energy Syst 2007;29(2):501–13. [13] Ajjarapu V, Christy C. The continuation power flow: a tool for steady state voltage stability analysis. IEEE Trans Power Syst 1992;7(1):416–23. [14] Alves DA, da Silva LCP, Castro CA, da Costa VF. Continuation method parameterized by power losses. IEEE Trans Power Syst 2000;2(1):763–8. [15] Flueck AJ, Gonella R, Dondeti JR. A new power sensitivity method of ranking branch outage contingencies for voltage collapse. IEEE Trans Power Syst 2002;17(2):265–70. [16] Zhang XP, Ju P, Handschin E. Continuation three-phase power flow: a tool for voltage stability analysis of unbalanced three-phase power systems. IEEE Trans Power Syst 2005;20(3):1320–9. [17] Kim SS, Kim MK, Park JK. Consideration of multiple uncertainties for evaluation of available transfer capability using fuzzy continuation power flow. Int J Electr Power Energy Syst 2008;30(10):581–93.
[18] Cañizares CA, Alvarado FL, DeMarco CL, Dobson I, Long WF. Point of collapse methods applied to ac/dc power systems. IEEE Trans Power Syst 1992;7(2):673–83. [19] Cañizares CA, Alvarado FL. Point of collapse and continuation methods for large ac/dc systems. IEEE Trans Power Syst 1993;8(1):1–8. [20] Irisarri GD, Wang X, Tong J, Mokhtari S. Maximum loadability of power systems using interior points non-linear optimization method. IEEE Trans Power Syst 1997;12(1):162–72. [21] Dai Y, Mccalley JD, Vittal V. Simplification, expansion and enhancement of direct interior point algorithm for power system maximum loadability. IEEE Trans Power Syst 2000;15(3):1014–21. [22] Greene S, Dobson I, Alvarado FL. Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters. IEEE Trans Power Syst 1997;12(1):262–72.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A new modeling of loading margin and its sensitivities using rectangular voltage coordinates in voltage stability analysis Vander Menengoy da Costa a,*, Magda Rocha Guedes b,1, Arlei Lucas de Sousa Rosa a,2, Marcelo Cantarino c,3 a
Department of Electrical Engineering, Federal University of Juiz de Fora, Campus Universitário – Bairro Martelos, 36036-330 Juiz de Fora – MG, Brazil Federal Center of Technologic Education of Minas Gerais – CEFET, Rua José Peres, 558 36700-000 Leopoldina – MG, Brazil c Centrais Elétricas Brasileiras S.A – ELETROBRÁS, Av. Rio Branco, 53, Centro, 14° andar, 20090-004 Rio de Janeiro – RJ, Brazil b
a r t i c l e
i n f o
Article history: Received 15 October 2008 Received in revised form 19 June 2009 Accepted 25 September 2009
Keywords: Electrical Power Systems Maximum Loading Point Polar Voltage Coordinates Rectangular Voltage Coordinates Sensitivity Analysis Voltage Stability
a b s t r a c t This paper presents new mathematical models to compute the loading margin, as well as to perform the sensitivity analysis of loading margin with respect to different electric system parameters. The innovative idea consists of evaluating the performance of these methods when the power flow equations are expressed with the voltages in rectangular coordinates. The objective is to establish a comparative process with the conventional models expressed in terms of power flow equations with the voltages in polar coordinates. IEEE test system and a South-Southeastern Brazilian network are used in the simulations. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Voltage stability has been one of the major concerns of power systems operators and planners for the last years. The continuous load increase allied to the lack of investments in transmission and generation has led systems to operate very close to their limits. Voltage stability has been a subject widely investigated [1–4]. Its static analysis can be assessed through continuation power flow and point of collapse methods. Over the last years, the rectangular voltage coordinates have been applied to different electric power system areas and remarkable results have been published in the literature, as follows: The algorithm presented in Ref. [5] incorporates a step size optimization factor into the rectangular power flow problem. This algorithm deals with the power flow calculation as a nonlinear programming technique, where the direction and magnitude of solution are determined in order to minimize a certain objective function. The objective function tends to zero when the power flow converges to a solution and, otherwise, stays at a positive value. * Corresponding author. Tel.: +55 32 21023461; fax: +55 32 21023442. E-mail addresses: [email protected] (Vander Menengoy da Costa), [email protected] (M.R. Guedes), [email protected] (Arlei Lucas de Sousa Rosa), [email protected] (M. Cantarino). 1 Tel.: +55 32 88355232; fax: +55 31 3319 5009. 2 Tel.: +55 32 32325352; fax: +55 32 21023442. 3 Tel.: +55 21 25144817; fax: +55 21 25145767. 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.09.005
A current-version rectangular power flow proposed in Ref. [6] introduces a dependent variable for each generation bus together with an additional voltage constraint equation. Besides, except for generation buses, the Jacobian matrix has the elements of the (2 2) off-diagonal blocks equal to those of the bus admittance matrix expanded into real and imaginary coordinates. The elements of the (2 2) diagonal blocks are updated during each iteration according to the load model. In Refs. [7,8] an augmented current injection rectangular power flow formulation is presented. This formulation has the same convergence characteristics as the conventional polar power flow methodology. Many different flexible AC transmission system devices and controls are incorporated into this augmented methodology. Another current-version power flow is proposed in Ref. [9] where the basic idea is to solve an augmented system of equations in which both bus voltages and current injections appear as state variables, and both power and current mismatches are zeroed. Similarly to [6], the elements of the (2 2) diagonal blocks are updated during each iteration in terms of the load model. Basically, the main difference between these two methodologies is that in [6] the state vector is composed exclusively of bus voltages in rectangular coordinates. The current injection rectangular power flow [6] is extended in [10] for solving an unbalanced distribution three-phase power flow problem. This methodology presents an expressive mathematical
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robustness and converges with a reduced number of iterations. Besides, in Ref. [11] a PV bus representation to be used in three-phase current injection power flow is presented. This approach requires an augmented system of equations to incorporate the reactive power as a dependent variable. A step size optimization factor to be used for solving unbalanced three-phase distribution current injection is presented in Ref. [12]. In addition, this paper extends the concept of this optimization factor to calculate the multiple unbalanced three-phase power flow solutions. As can be seen, the rectangular voltage coordinates have been widely investigated in several studies related to the power system area and final results have been highly encouraging. Therefore, the main objective of this paper is to develop new continuation power flow and point of collapse methods when the power flow equations are expressed with the voltages in rectangular coordinates. In addition, new sensitivity models of loading margin with respect to different parameters using the same kind of coordinates are also presented. Currently, these methods and models presented in the literature are expressed only in terms of polar voltage coordinates. The notations adopted in the paper are the conventional ones whenever possible. Matrices are shown in bold and vectors are shown in bold underlined. 2. Rectangular power flow equations The basic power flow equations written in terms of rectangular voltage coordinates are given by X V rk ðGkm V rm Bkm V mm Þ þ V mk ðGkm V mm þ Bkm V rm Þ þ P Dk P Gk ¼ 0 m2Uk
X
ð1Þ
291
power flow equations to overcome this drawback. A brief review on continuation power flow is presented next. In [13] a mathematical model for the continuation power flow using either additional load change, or voltage magnitude as continuation parameters is presented. In [14] both active and reactive power losses and generated powers at slack bus are tested as continuation parameters. In [15] a tool for evaluating nonlinear effects on power system states due to branch admittance/impedance variations is presented. In [16] a continuation three-phase power flow, which can be used to analyze voltage stability of unbalanced threephase power systems, is proposed. In [17] a fuzzy continuation power flow is developed with the objective of handling simultaneously uncertainties in load parameters and bus injections parameters. These references use power flow equations written in terms of polar voltage coordinates. 3.1. Predictor step To simulate load and generation changes a loading factor c, which characterize the change of load, is employed. The nonlinear Eqs. (1) and (2) are augmented by an extra variable c resulting Fðx; cÞ ¼ 0. The predictor step is used to provide an approximate point of the next solution. A prediction of the next solution is made by taking an appropriately sized step in the direction tangent to t the solution path given by ½ dV r dV m dc . At each prediction, a new continuation parameter is chosen. The procedure adopted in the paper analyses the variation of each state between the two last correct points. The state with the largest percentage variation will be the next continuation parameter. If the continuation parameter is the additional loading c, the prediction in rectangular voltage coordinates is given by
V mk ðGkm V rm Bkm V mm Þ V rk ðGkm V mm þ Bkm V rm Þ þ Q Dk Q Gk ¼ 0
m2Uk
ð5Þ ð2Þ
where V rk þ jV mk is the real and imaginary voltage component at bus k; Gkm þ jBkm is the k–m element of bus admittance matrix; PGk þ Q Gk is the generated complex power at bus k; P Dk þ jQ Dk is the complex power load at bus k. UK denotes the set of buses directly connected to bus k, including itself. Eqs. (1) and (2), denoted by F P ðxÞ ¼ 0 and F Q ðxÞ ¼ 0, represent the power flow equations where x ¼ ½V r V m t . The order of vector x is 2n. n denotes the number of buses. In particular, for a generic bus k, these notations appear as F Pk ðxÞ and F Q k ðxÞ. The set of (1) and (2) will be denoted by FðxÞ throughout the paper. The rectangular Jacobian matrix is formed by the derivative of FðxÞ ¼ 0 with respect to state variables x. The ordinary rectangular power flow is iteratively solved through the following equation:
where pc is the continuation step size regarding c. The components of vector corresponding to a generic bus k, for a constant power load model, are given by PDk and Q Dk . If the continuation parameter is the voltage Vq, where q refers to the bus with the largest voltage variation between the two last correct points, the prediction in rectangular voltage coordinates is given by
# " DP @FðxÞ DV r ¼ DQ @x DV m
stem from Eq. (4). pV is the continuation step size regarding Vq. The estimated solution, denoted by *, is updated from the tangent vector as follows:
ð3Þ
For PV buses, the reactive power equations are replaced by voltage equations. For a PV bus p the voltage constraint in rectangular coordinates is given by
V 2p ¼ V 2rp þ V 2mp
ð4Þ
ð6Þ h i Vr Vm where e ¼ 0 V qq 0 0 V qq 0 . The components of this vector
2
3 2 3 2 3 Vr Vr dV r 6 7 6 7 6 7 4 V m 5 ¼ 4 V m 5 þ 4 dV m 5 c c dc
ð7Þ
3.2. Corrector step 3. Continuation power flow The voltage profile is obtained through successive power flow solutions by simulating load change. However, the voltage profile cannot be traced completely, by using only the conventional power flow, because the Jacobian matrix becomes singular at the maximum loading point. The continuation method is applied to the
The corrector step corresponds to solve the augmented Newton power flow equation with the predicted solution given by (7) as the initial solution. If the continuation parameter is the additional loading c, this step is to simply run an ordinary rectangular power flow at the estimated point. On the other hand, Eq. (8) is used if the continuation parameter is the voltage Vq.
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ð8Þ Finally, the state variables are updated at a given iteration h + 1 as follows:
2
3hþ1
Vr 6 7 4 Vm 5
c
2
3h
2
3h
Vr DV r 6 7 6 7 ¼ 4 V m 5 þ 4 DV m 5
c
ð9Þ
increment the iteration counter, h = h + 1, and solve (11) where the mismatches are given by
R1 ¼ ½DP1
DP 2
DP n
DQ 1
DQ 2
DQ n t
ð12Þ
t
R2 ¼
@FðxÞ w @x
ð13Þ
R3 ¼ 1 w21 w22 w22n ¼ 1
2n X ðwi Þ2
ð14Þ
i¼1
Dc
Step 5: Update the state variables and go back to Step 4. 4. Point of collapse method
2
4.1. Initial aspects The objective of this method is to iteratively calculate the maximum loading point of electric power systems without tracing the continuation curves. A brief review on this subject is presented next. Ref. [18] describes an extension of the point of collapse developed for ac systems studies to the determination of saddle-node bifurcations in power systems including high voltage direct current transmission. Ref. [19] describes the implementation of both point of collapse and continuation methods for computation of voltage collapse points in large ac/dc systems. Besides, Refs. [20,21] calculate the maximum loadability using interior point nonlinear optimization method. Again, the power flow equations are written in terms of polar voltage coordinates. The maximum loading point is characterized by the Jacobian matrix having a simple and unique zero eigenvalue, with nonzero left eigenvector w and right eigenvector v . This condition can be summarized by the set of nonlinear Eq. (10) [18,19] whose solution yields the saddle-node bifurcation point.
Gðx; c; wÞ ¼
8 > > > > <
Fðx; cÞ ¼ 0 @FðxÞt w @x
¼0 > 2n > P > > : 1 ðw2i Þ ¼ 0
ð10Þ
i¼1
The Newton–Raphson iterative method applied to (10) results in (11). The variables are 2n components of x, the 2n components of w and the load increase parameter c.
2
@Fðx;cÞ @x
6 h 6 @ @FðxÞt i 6 4 @x @x :w 0t
0 @FðxÞt @x
2w
3
2 3 2 3 R1 7 Dx 76 7 6 7 4 Dw 5 ¼ 4 R2 5 0 7 5 Dc R3 0
@Fðx;cÞ @c
ð11Þ
Vr
6V 7 6 m7 6 7 4 w 5
2
Vr
3h
6V 7 6 m7 ¼6 7 4 w 5
c
c
2
3 DV r h 6 DV 7 m7 6 þ6 7 4 Dw 5
ð15Þ
Dc
5. Sensitivity analysis of the loading margin In Ref. [22] linear and quadratic estimates to the variation of the loading margin with respect to any system parameter or control are derived. Polar power flow equations are used. Suppose that the equilibrium of the power system satisfy the equation Fðx; k; pÞ ¼ 0. The sensitivity of the load margin L to the change in parameters p is given by Greene et al. [22]
wt f p DL ¼ Mp ¼ t Dp w fk a
ð16Þ
where w is the left eigenvector corresponding to the zero eigenvalue of Jacobian matrix; f p and fk are the derivatives of Fðx; k; pÞ with respect to parameter p and load powers k, respectively. a is a unit vector which specify a pattern of load increase. For constant power loads f k is a diagonal matrix with ones in the rows corresponding to load buses. The terms wt f k a; wt and f k are calculated once regardless of the parameter chosen, and, therefore, the sensitivity to any additional parameters can be computed quickly. The loading margin sensitivities only depend on quantities evaluated at the nominal bifurcation point. Evaluation of the linear sensitivity is very simple. Once the nominal bifurcation point is computed, the linear sensitivity (16) requires computation of the left eigenvector wt and the vector f p . The quadratic estimate of the change in loading margin is given by Greene et al. [22]
1 DM ¼ M p Dp þ M pp Dp2 2
4.2. Solution algorithm
where
The rectangular point of collapse method can be summarized as follows:
Mpp ¼
Step 1: Calculate the estimated maximum loading point through an ordinary power flow. The methodology adopted is to gradually increase the loads until the power flow fails to converge. The estimated value of x and c will be the last point with normal convergence. Step 2: Assemble the rectangular Jacobian matrix at the estimated point. Step 3: Calculate the estimated left eigenvector of the Jacobian matrix associated with its critical eigenvalue. Step 4: Calculate the updated mismatch vector shown in the right-hand side of (11). If all components of this vector are smaller than a specified tolerance, then print results. Otherwise,
3hþ1
1 ½xt f xp þ 2wt f xp xp þ wt f pp wt f k a p xx
ð17Þ
ð18Þ
The quadratic estimate additionally requires the following quantities evaluated at the nose: (i) matrix f xx formed by the derivative of the product of w with the Jacobian matrix with respect to state variables x; (ii) f xp is the derivative of Jacobian matrix with respect to the parameter p; (iii) xp is the sensitivity of the nose equilibrium with respect to p; (iv) f pp is the second-order derivative of Fðx; k; pÞ with respect to parameter p. The sensitivity formulas evaluated at the voltage collapse yield linear and quadratic estimates of the loading margin as a function of any parameter. The performance of these estimates is tested for three different parameters, namely, line susceptance, active power load and bus shunt susceptance, representative of a range of control actions or system uncertainties.
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5.1. Line susceptance
6.2. Analysis of continuation power flow
These variations can represent the operation of a FACTS device or can reflect uncertainty in the network data. The sensitivity analysis can be useful to define the best place for FACTS installation in order to provide a better loading margin. Let k–m be the branch under analysis whose susceptance is bkm. The vector f pp is null. The matrix f xp is the derivative of Jacobian matrix with respect to bkm. The elements of this matrix will be different from zero only in the positions regarding the buses k and m. The components of the vector f p in rectangular coordinates are given by
The step size adopted for the continuation parameter is 0.01 p.u. The buses 526 (IEEE-300) and 1818 (1768-bus) experienced the largest voltage variation from the base case. Figs. 1 and 2 show the voltage profiles related to these buses provided by rectangular
@F Pk ðxÞ ¼ akm ðV mk V rm V rk V mm Þ @bkm
ð19Þ
@F Q k ðxÞ ¼ akm ðV mm V mk þ V rm V rk Þ a2km ðV 2mk þ V 2rk Þ @bkm
ð20Þ
@F Pm ðxÞ ¼ akm ðV mm V rk V rm V mk Þ @bkm
ð21Þ
@F Q m ðxÞ ¼ akm ðV mm V mk þ V rm V rk Þ ðV 2mm þ V 2rm Þ @bkm
ð22Þ
where akm is the tap transformer. 5.2. Active power load Usually, as the load power is reduced, the loading margin is increased. The sensitivity analysis can provide valuable information concerning the system operation. The load power factor is kept constant during the analysis. Let k be the bus whose active power load is PDk . The vector f pp and the matrix f xp are null. The components of the vector f p in rectangular coordinates are given by
@F Pk ðxÞ ¼1 @PDk @F Q k ðxÞ Q Dk ¼ @Q Dk PDk
Fig. 1. Voltage profile at bus 526: IEEE-300 system.
ð23Þ ð24Þ
5.3. Bus shunt susceptance These variations can represent a reactive power support to avoid voltage collapses. Let k be the bus whose shunt susceptance sh bk is selected for sensitivity analysis. The vector f pp is null. The mash trix f xp is the derivative of Jacobian matrix with respect to bk . Therefore, this matrix in rectangular voltage coordinates is composed only of elements 2V rk and 2V mk corresponding to the positions related to the reactive power injected at bus k with respect to real and imaginary voltage components. The only nonzero component of vector f p is given by
@F Q k ðxÞ sh
@bk
¼ ðV 2rk þ V 2mk Þ
Fig. 2. Voltage profile at bus 1818: 1768-bus system.
ð25Þ
6. Results 6.1. Initial considerations In order to perform this study some simulations were accomplished by using the IEEE 300-bus system and a practical Brazilian network composed of 1768 buses, 2527 branches and 119 generation buses. The tolerance adopted for convergence of the iterative process related to continuation power flow and point of collapse methods is 105 p.u. The constant power load model is adopted. Test systems are stressed by keeping a constant power factor. Reactive power generation limits are taking into account.
Fig. 3. Number of iterations in terms of loading: IEEE-300 system.
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and polar coordinates. The maximum loading point corresponds to additional loadings of 0.0246 p.u. (IEEE-300) and 0.0265 p.u. (1768-bus) with voltages V 526 ¼ 0:7828\ 79:59 p:u: and V 1818 ¼ 0:6432\ 78:99 p:u:. As expected, both coordinates yield the same maximum loading point, and the same voltages. Figs. 3 and 4 show the total number of iterations to complete the voltage profiles in both coordinates. The active and reactive bus loads are proportionally increased from the base case. The load power factors are kept constant. Basically, the number of iterations to yield the complete voltage profile is the same regardless of voltage coordinates. Figs. 5 and 6 show the number of iterations to complete the voltage profile when the series and shunt transmission line admittances are gradually reduced from the base case. The objective is to simulate the performance of continuation power flow methodologies under contingency conditions. The branches gradually removed are 37–49 and 2591–2593, regarding the IEEE-300 and 1768-bus systems, respectively. The complete branch removal corresponds to a factor zero. The IEEE-300 voltage profile is obtained even for a complete branch removal. On the other hand, the 1768bus system converges with a partial removal of up to 90%. Both rectangular and polar continuation power flow methods present a similar performance in terms of number of iterations. Figs. 7 and 8 show the number of iterations when the branches 37–49 and 2591–2593 are completely removed and 90% partially removed, respectively. The maximum number of iterations during
Fig. 6. Number of iterations in terms of a partial branch removal: 1768-bus system.
Fig. 4. Number of iterations in terms of loading: 1768-bus system.
Fig. 7. Performance for a full branch removal: IEEE-300 system.
Fig. 5. Number of iterations in terms of a partial branch removal: IEEE-300 system.
Fig. 8. Performance for a partial branch removal of 90%: 1768-bus system.
the corrector process is adopted as twenty. Therefore, twenty-one iterations mean that convergence of the iterative process is not achieved. In this case, the step size is reduced and the continuation parameter may or may not be changed in order to ensure that the corrector process is successful. Basically, both continuation power flow methods present the same number of corrector steps and iterations during each corrector step.
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295
6.3. Analysis of point of collapse method The maximum loading point provided by the rectangular and polar point of collapse methods corresponds to additional loadings of 0.0246 p.u. (IEEE-300) and 0.0265 p.u. (1768-bus) with voltages V 526 ¼ 0:7880\ 79:16 p:u: and V 1818 ¼ 0:6481\ 78:74 p:u:. A good agreement between these results and those ones obtained through continuation power flow can be observed. Figs. 9 and 10 show the maximum absolute power mismatches, at the end of each iteration, regarding the base case. The iterative process related to rectangular point of collapse method results large power mismatches in initial iterations, and the polar point of collapse method, in general, converges with less iterations. Figs. 11–14 show the number of iterations for convergence of the point of collapse method in both coordinates. Firstly, the active and reactive bus loads are proportionally increased from the base case. The load power factors are kept constant. Secondly, the series and shunt transmission line admittances are gradually reduced. The branches under analysis are the same ones as considered in the previous section. In general, the polar point of collapse method presents a better performance in comparison with the rectangular version.
Fig. 11. Performance as a function of loading: IEEE-300 system.
6.4. Sensitivity analysis The nominal parameters of IEEE-300 are the susceptance bkm = 80.6452 p.u. corresponding to branch 7049–49, the active
Fig. 12. Performance as a function of loading: 1768-bus system.
Fig. 9. Convergence trajectory: IEEE-300 system.
Fig. 13. Performance as a function of branch removal: IEEE-300 system.
Fig. 10. Convergence trajectory: 1768-bus system.
load P Dk ¼ 30:74 MW at bus 9052, and a null shunt susceptance sh bk at bus 20. The nominal parameters of 1768-bus are the susceptance bkm = 18.5530 p.u. corresponding to branch 536–559, the active load P Dk ¼ 36:134 MW at bus 2975, and a null shunt suscepsh tance bk at bus 2993. Figs. 15–20 show the loading margin
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Fig. 14. Performance as a function of branch removal: 1768-bus system.
Fig. 16. Line susceptance variation: 1768-bus system.
Fig. 17. Active power load variation: IEEE-300. Fig. 15. Line susceptance variation: IEEE-300 system.
changing as the parameters are varied. Actual results were obtained through the point of collapse method. The following comments can be extracted from Figs. 15–20. As the variation of parameter decreases, the linear and quadratic sensitivity analysis in rectangular coordinates provide very satisfactory results when compared with those generated by exact methods. Moreover, a good agreement between results yielded by both kinds of coordinates is observed. As the variation of parameter increases, the results provided by the quadratic sensitive analysis in both coordinates are better than the ones obtained through the linear sensitivity analysis. As expected, the second order term of the Taylor series improves the quadratic estimates. As a concluding remark, the performance of both kinds of coordinates is very similar. Therefore, the sensitivity analysis of loading margin with respect to different parameters can be efficiently and easily modeled in rectangular voltage coordinates. 6.5. Computational performance Tables 1 and 2 display the computation times, in seconds, required by polar and rectangular coordinates to calculate the linear and quadratic estimates regarding the 1768-bus system. Table 3 displays the relationships between the computation times required by both coordinates for each one of the tasks related to sensitivity calculation. Time relation is not shown for load power variations because the matrix f xp is null.
Fig. 18. Active power load variation: 1768-bus system.
The linear and quadratic rectangular estimates present computational gains around 90% and 36%, respectively, in comparison with the polar approach. Therefore, the use of rectangular coordinates requires smaller computation times and, as a result, leads to a substantially faster sensitivity analysis of loading margin. Certainly, this concluding remark stems from the particular structure
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Vander Menengoy da Costa et al. / Electrical Power and Energy Systems 32 (2010) 290–298 Table 3 Computation time relationships – 1768 bus. Tasks
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
1.9049 3.6573 1.2217 0.9264 1.4364
1.9090 3.8019 * 0.9173 0.7048
1.9045 3.9291 1.1642 0.9369 1.0348
Total relation
1.3663
1.3549
1.3793
sh
of rectangular power flow equations characterized by a quadratic form independent of transcendental functions.
7. Conclusions Fig. 19. Shunt susceptance variation: IEEE-300.
This paper deals with the calculation of the maximum loading point in electrical power systems through the continuation power flow and point of collapse methods. One of the main goals consists of evaluating both methods in rectangular voltage coordinates. The rectangular coordinates applied to both methods present a very simple mathematical modeling when compared to the traditional polar process. Although the polar point of collapse method has presented a slight advantage over the rectangular version, in general, the performance of rectangular coordinates is highly expressive even for adverse conditions investigated in the paper. Additionally, this paper deals with the evaluation of linear and quadratic rectangular sensitivity analysis of loading margin with respect to different parameters. These estimates can be used to evaluate the effect or the efficiency of the parameters variation on the loading margin of electrical power systems. Both linear and quadratic rectangular sensitivity models have a very simple mathematical structure, present very expressive results and propitiate a remarkable computational gain in comparison with the polar coordinates. Finally, since electrical power systems operate very close to their limits, this paper demonstrates that rectangular methods can be regarded as promising additional tools in voltage stability studies.
Fig. 20. Shunt susceptance variation: 1768-bus system.
Table 1 Computation time – polar coordinates – 1768 bus. Tasks
References
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
1.8420 1.8250 0.2480 3.3090 0.1580
1.9510 1.9580 * 3.3930 0.1170
1.8950 1.9960 0.2340 3.4770 0.1190
Total time
7.4820
7.5210
7.8260
sh
Table 2 Computation time – rectangular coordinates – 1768 bus. Tasks
Parameters bkm
P Dk
bk
Mp fxx fxp xp Mpp
0.9670 0.4990 0.2030 3.5720 0.1100
1.0220 0.5150 * 3.6990 0.1660
0.9950 0.5080 0.2010 3.7110 0.1150
Total time
5.4760
5.5510
5.6740
sh
[1] Taylor C. Power systems voltage stability. EPRI – power system engineering series. McGraw-Hill; 1994. [2] Kundur P. Power system stability and control. EPRI – power system engineering series. McGraw-Hill; 1994. [3] Cutsem TV, Vournas C. Voltage stability of electric power systems. Athens, Greece: University of Liège Belgium, National Technical University; 1998. [4] Kamalasadan S, Thukaram D, Srivastava AK. A new intelligent algorithm for online voltage stability assessment and monitoring. Int J Electr Power Energy Syst 2009;31(2):100–10. [5] Iwamoto S, Tamura Y. A load flow calculation method for ill-conditioned power systems. IEEE Trans Power Syst 1981;PAS-100(4):1736–43. [6] Da Costa VM, Martins N, Pereira JLR. Developments in the Newton–Raphson power flow formulation based on current injections. IEEE Trans Power Syst 1999;14(4):1320–6. [7] Da Costa VM, Pereira JLR, Martins N. An augmented Newton–Raphson power flow formulation based on current injections. Int J Electr Power Energy Syst 2001;23(4):305–12. [8] Variz AM, Da Costa VM, Pereira JLR, Martins N. Improved representation of control adjustments into the Newton–Raphson power flow. Int J Electr Power Energy Syst 2003;25(7):501–13. [9] Exposito AG, Ramos ER. Augmented rectangular load flow model. IEEE Trans Power Syst 2002;17(2):271–6. [10] Garcia PAN, Pereira JLR, Da Costa VM, Martins N. Three phase power flow calculations using the current injection method. IEEE Trans Power Syst 2000;15(2):508–14. [11] Garcia PAN, Pereira JLR, Carneiro Jr S, Vinagre MP, Gomes FV. Improvements in the representation of PV buses on three-phase distribution power flow. IEEE Trans Power Delivery 2004;19(2):894–6.
298
Vander Menengoy da Costa et al. / Electrical Power and Energy Systems 32 (2010) 290–298
[12] Da Costa VM, Oliveira ML, Guedes MR. Developments in the analysis of unbalanced three-phase power flow solutions. Int J Electr Power Energy Syst 2007;29(2):501–13. [13] Ajjarapu V, Christy C. The continuation power flow: a tool for steady state voltage stability analysis. IEEE Trans Power Syst 1992;7(1):416–23. [14] Alves DA, da Silva LCP, Castro CA, da Costa VF. Continuation method parameterized by power losses. IEEE Trans Power Syst 2000;2(1):763–8. [15] Flueck AJ, Gonella R, Dondeti JR. A new power sensitivity method of ranking branch outage contingencies for voltage collapse. IEEE Trans Power Syst 2002;17(2):265–70. [16] Zhang XP, Ju P, Handschin E. Continuation three-phase power flow: a tool for voltage stability analysis of unbalanced three-phase power systems. IEEE Trans Power Syst 2005;20(3):1320–9. [17] Kim SS, Kim MK, Park JK. Consideration of multiple uncertainties for evaluation of available transfer capability using fuzzy continuation power flow. Int J Electr Power Energy Syst 2008;30(10):581–93.
[18] Cañizares CA, Alvarado FL, DeMarco CL, Dobson I, Long WF. Point of collapse methods applied to ac/dc power systems. IEEE Trans Power Syst 1992;7(2):673–83. [19] Cañizares CA, Alvarado FL. Point of collapse and continuation methods for large ac/dc systems. IEEE Trans Power Syst 1993;8(1):1–8. [20] Irisarri GD, Wang X, Tong J, Mokhtari S. Maximum loadability of power systems using interior points non-linear optimization method. IEEE Trans Power Syst 1997;12(1):162–72. [21] Dai Y, Mccalley JD, Vittal V. Simplification, expansion and enhancement of direct interior point algorithm for power system maximum loadability. IEEE Trans Power Syst 2000;15(3):1014–21. [22] Greene S, Dobson I, Alvarado FL. Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters. IEEE Trans Power Syst 1997;12(1):262–72.