- Email: [email protected]
Reactive power rescheduling with generator ranking for voltage stability improvement
Energy Conversion and Management 50 (2009) 1129–1135
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Reactive power rescheduling with generator ranking for voltage stability improvement Habibollah Raoufi *, Mohsen Kalantar Center of Excellence for Power System Automation and Operation, Department of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran
a r t i c l e
i n f o
Article history: Received 13 January 2008 Received in revised form 22 October 2008 Accepted 20 November 2008 Available online 6 February 2009 Keywords: Generator ranking Modal analysis Participation factors Reactive power dispatch Voltage stability
a b s t r a c t In a power system, voltage stability margin improvement can be done by regulating generators voltages, transformers tap settings and capacitors/reactors rated reactive powers (susceptances). In this paper, one of these methods, which we name ‘‘reactive power rescheduling with generator ranking”, is considered. In this method, using ‘‘ranking coefficients”, the generators are divided into ‘‘important” and ‘‘less-important” ones and then, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. These ranking coefficients are obtained using ‘‘modal analysis”. In this paper, the method’s performance for two types of ranking coefficients has been analyzed. Also, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method has been simulated. For all simulations, the IEEE 30 bus test system has been used. The simulation results show that in the former method, for either type of ranking coefficients, voltage stability margin is considerably improved and, usually, the system active loss and the system operating cost are increased. Also, in the latter method, voltage stability margin is improved and the system active loss and the system operating cost are decreased. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The transfer of power through a transmission network is accompanied by voltage drops between the generation and consumption points. In normal operating conditions, these drops are in the order of a few percents of the nominal voltage. One of the tasks of power system planners and operators is to check that under heavy stress conditions and/or following credible events, all bus voltages remain within acceptable bounds. In some circumstances, however, in the seconds or minutes following a disturbance, voltages may experience large and progressive falls, which are so pronounced that the system integrity is endangered and power cannot be delivered correctly to customers. This catastrophe is referred to as voltage instability and its calamitous result as voltage collapse. This instability stems from the attempt of load dynamics – especially loads supplied with under load tap changing transformers (ULTC), induction motors and thermostatic loads – to restore power consumption beyond the amount that can be provided by the combined transmission and generation system [1]. Nowadays, there are some voltage stability criteria being implemented. For example, the Western Electricity Coordinating Council (WECC) proposes a minimum voltage stability margin (VSM) requirement of 5% considering simple contingencies, 2.5% for double contingencies, and larger than zero for multiple contingencies. * Corresponding author. Tel.: +98 21 77240492; fax: +98 21 77240490. E-mail address: habibollahraoufi@yahoo.com (H. Raoufi). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.11.013
In a similar way, the ONS (Brazilian System Operator) has also initiated some studies and recommends a minimum VSM requirement of 6% also considering simple contingencies. Both criteria are based on VSM index, which is obtained from PV curve computations and represents the distance from the current operating point to the voltage stability limit [2]. ‘‘Reactive power management” is the general name of methods which try to improve voltage profile/stability by regulating generators voltages, transformers tap settings, reactive sources settings and installing new reactive sources. These methods can be divided into two areas: reactive planning (allocation) and reactive dispatch (re-dispatch, scheduling, rescheduling). Also, the dispatch area can be divided into two areas: off-line reactive dispatch and on-line reactive dispatch. In the reactive planning area, the period of study is the next few months or the next few years and installing the new reactive sources are also considered. In the off-line reactive dispatch area, only installed reactive sources are used and the period of study is the next few days or the next few hours. In the on-line reactive dispatch area, only installed reactive sources are used and the period of study is the next few minutes or the next few seconds [3]. In the off-line reactive dispatch area, voltage profile/stability improvement is done by regulating generators voltages, transformers tap settings and capacitors/reactors rated reactive powers (susceptances). For this purpose, usually two groups of methods are used. In the first group methods – which are referred by names such as ‘‘optimal reactive power dispatch”, ‘‘reactive power optimization”, etc. – a specific optimization problem with specific
1130
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
Nomenclature NB N0 Ni NG NPQ NC NE NT nG Rk, Xk Qci
set of numbers of total buses set of numbers of total buses, excluding slack bus set of numbers of buses adjacent to bus i, including bus i set of numbers of generator buses, including slack bus set of numbers of load buses set of numbers of shunt capacitor/reactor installation buses set of numbers of system branches set of numbers of tap changing transformer branches number of system generators resistance and reactance of branch k (pu) rated reactive power (susceptance) of shunt capacitor/ reactor at bus i (pu)
objective function(s) and specific constraints is defined and then, by solving this problem (using an optimization algorithm), appropriate values for those quantities are obtained [4–9]. In the usual form of these methods, voltage profile/stability improvement is done by minimizing the system active loss [4,5]. In the second group methods – which do not have an accepted name – initiative algorithms are used [10–13]. In this paper, a method from the second group is considered. In this method, which we name ‘‘reactive power rescheduling with generator ranking”, using ‘‘ranking coefficients”, the generators are divided into ‘‘important” and ‘‘less-important” ones and then, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively [12,13]. These ranking coefficients are obtained using ‘‘modal analysis” [14–16]. In [12,13], this method has been used by selecting a specific type of ranking coefficients. But, in this type of ranking coefficients, there is no coefficient related to the slack bus generator. Thus, the reactive power generation of the slack bus generator has not been used effectively. Also, in these references, the effect of ‘‘weighting factor” on the method’s performance has not been studied. In this paper, first, the modal analysis theory is presented in a new and clear form. Then, the ‘‘reactive power rescheduling with generator ranking” method is simulated by selecting two types of ranking coefficients (which one of them presents ranking coefficients for ‘‘all” generators including the slack bus generator). Also, the effect of ‘‘weighting factor” on the method’s performance is studied. In addition, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method is also simulated. For all simulations, the IEEE 30 bus test system is used.
Tk Gij, Bij Vi hi hij Pi, Qi Pgi, Qgi Sk PLoss Cost
generator buses on which reactive power generation at primary operating point is equal to lower or upper limit. In addition, Gunlim is an index for generator buses on which reactive power generation at primary operating point is within lower and upper limits. Finally, f is a function that relates power system main quantities to each other and, in other words, is a symbol of power flow equations. After linearization of power flow equations, depending on equations which contribute in the linearization and quantities which their variations are considered equal to zero, 4 jacobian matrixes (JLarge, JMedium, JSmall and JPF) are obtained. Full linearization of power flow equations leads to
3 3 2 DP s Dhs 6 DP g 7 6 Dhg 7 7 7 6 6 7 7 6 DP 6 Dh 7 7 6 6 L L 7 ¼ J Large 6 7 6 6 DQ L 7 6 DV L 7 7 7 6 6 4 DQ Glim 5 4 DV Glim 5 DQ Gunlim DV Gunlim 2
1 1 0 hs Ps B Pg C B hg C C C B B C B PL C B C ¼ f B hL C B B QL C B VL C C C B B A A @ Q @ V Glim Glim Q Gunlim V Gunlim 0
ð1Þ
where s, g and L are indices for slack bus, all generator buses except slack bus and load buses, respectively. Also, Glim is an index for
ð2Þ
where JLarge is the large jacobian matrix. Also, by removing the equation related to Ps from (1) and assuming Dhs = 0, linearization of (1) leads to
3 3 2 DP g Dhg 7 7 6 6 DP L Dh L 7 7 6 6 7 7 6 6 DQ L DV L 7 7 6 6 7 ¼ J Medium 6 7 6 6 7 6 7 7 7 6 6 5 4 DQ Glim 4 DV Glim 5 2
DQ Gunlim
2. Modal analysis Modal analysis is a method for voltage stability evaluation. In this method, voltage stability analysis is done by computing eigenvalues and right and left eigenvectors of a jacobian matrix (which is obtained from the power flow equations). At an operating point, the relations between main power system quantities (bus voltage magnitude, bus voltage angle, bus active power and bus reactive power) can be expressed by power flow equations as follows:
transformer tap setting of branch k mutual conductance and susceptance between buses i and j (pu) voltage magnitude of bus i (pu) voltage angle of bus i (rad) voltage angle difference between buses i and j (rad) injected active and reactive power at bus i (pu) generated active and reactive power at bus i (pu) apparent power flow in branch k (pu) system active loss (pu) system operating cost (unit of money per hour)
DV Gunlim
3 Dhg 7 6 Dh L 2 3 7 J1 j J2 6 7 6 DV L 7 6 ¼ 4 56 7 6 7 7 J3 j J4 6 4 DV Glim 5 DV Gunlim 2
ð3Þ
where JMedium is the medium jacobian matrix. Also, by removing the equations related to QGunlim from (1) and assuming DVGunlim = 0, linearization of (1) leads to
3 3 2 DP s Dh s 7 7 6 6 6 DP g 7 6 Dh g 7 7 7 6 6 6 DP L 7 ¼ J Small 6 DhL 7 7 7 6 6 7 7 6 6 4 DQ L 5 4 DV L 5 DQ Glim DV Glim 2
ð4Þ
where JSmall is the small jacobian matrix. Finally, by removing the equations related to Ps and QGunlim from (1) and assuming Dhs = 0 and DVGunlim = 0, linearization of (1) leads to
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
3 3 2 DP g Dh g 7 7 6 6 DP L DhL 7 7 6 6 7 7 6 6 6 7 ¼ J PF 6 7 7 7 6 6 5 5 4 4 DQ L DV L 2
DQ Glim
DV Glim
3 2 Dhg 3 7 J5 j J6 6 DhL 7 6 7 6 76 ¼ 4 56 7 7 6 5 J7 j J8 4 DV L DV Glim 2
ð5Þ
where JPF is the power flow jacobian matrix. It is mentionable that both JLarge and JSmall are near singular matrixes (at the normal operating point or at the voltage stability limit). So, in the modal analysis method, usually these matrixes have not been used. In (5), the assumption of DPg = 0 and DPL = 0 leads to
DQ L DV L ¼ ðJ RQV ÞPF DQ Glim DV Glim
ð6Þ
where (JRQV)PF is the reduced reactive jacobian matrix which is obtained from the JPF matrix. This matrix is defined as follows:
ðJ RQV ÞPF ¼ J 8 J7 J 1 5 J6
ð7Þ
This matrix represents the relations between reactive power and voltage magnitude at load buses and some generator buses. Also in (5), the assumption of DQL = 0 and DQGlim = 0 leads to
DP g DP L
¼ ðJ RPh ÞPF
Dhg Dh L
ð8Þ
where (JRPh)PF is the reduced active jacobian matrix which is obtained from the JPF matrix. This matrix is defined as follows:
ðJ RPh ÞPF ¼ J5 J 6 J 1 8 J7
ð9Þ
This matrix represents the relations between active power and voltage angle at all buses (except slack bus). Finally in (3), the assumption of DPg = 0 and DPL = 0 and DQL = 0 leads to
DQ Glim DV Glim ¼ ðJ G ÞMedium DQ Gunlim DV Gunlim
ð10Þ
where (JG)Medium is the generator jacobian matrix which is obtained from the JMedium matrix. This matrix is defined as follows:
ðJ G ÞMedium ¼ J 4 J 3 J 1 1 J2
ð11Þ
This matrix represents the relations between reactive power and voltage magnitude at ‘‘all” generator buses. At the voltage stability limit, the JPF, (JRQV)PF and (JRPh)PF matrixes are singular and the (JG)Medium matrix has an infinite eigenvalue. It is mentionable that modal analysis in [14] is done using the (JRQV)PF matrix, in [15] is done using both (JRQV)PF and (JRPh)PF matrixes and in [16] is done using the (JG)Medium matrix. Bus participation factors are coefficients which are obtained by element by element product of right and left eigenvectors of a specific jacobian matrix (for a specific eigenvalue). Using these coefficients, we can obtain information about the effect of system equipments on voltage stability margin. In [14], reactive participation factors – which we represent them by (RPF)PF – have been defined as element by element product of right and left eigenvectors of the (JRQV)PF matrix (for a specific eigenvalue). These coefficients show that reactive injection (for example by shunt compensation) at which buses leads to considerable increase in voltage stability margin. In [15], active participation factors – which we represent them by (APF)PF – have been defined as element by element product of right and left eigenvectors of the (JRPh)PF matrix (for a specific eigenvalue). These coefficients show that increasing active power
1131
generation (for generator buses) or load shedding (for load buses) at which buses leads to considerable increase in voltage stability margin. In [16], generator reactive participation factors – which we represent them by (RPFG)Medium – have been defined as element by element product of right and left eigenvectors of the (JG)Medium matrix (for a specific eigenvalue). These coefficients show that increasing reactive power generation at which generator buses leads to considerable increase in voltage stability margin. In this paper, modal analysis is used for obtaining some participation factors for generator buses, which will be used for generator ranking from the perspective of their reactive power importance (their effect on voltage stability margin). With respect to (6), it is possible that there is no element related to generator buses in the (JRQV)PF eigenvectors. Thus, the ðRPFÞPF coefficients are not used for generator ranking. In [15], it has been shown that if the system is transferred from the current operating point to the voltage stability limit, the (JRPh)PF matrix is calculated at this limit and the (APF)PF coefficients are computed for the smallest eigenvalue, then, the (APF)PF coefficients can be used for generator ranking (from the perspective of voltage stability). In this case, if a generator have small participation factor, voltage stability improvement is done by increasing its reactive power generation. On the other hand, if a generator have large participation factor, increase/decrease of its reactive power generation leads to negligible change in voltage stability margin. With respect to (8), there is no element related to the slack bus generator in the (JRPh)PF eigenvectors. Thus, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. In [16], it has been shown that if the system is transferred from the current operating point to the voltage stability limit, the (JG)Medium matrix is calculated at this limit and the (RPFG)Medium coefficients are computed for the eigenvalue with the largest absolute, then, the (RPFG)Medium coefficients can be used for generator ranking (from the perspective of voltage stability). In this case, if a generator have large participation factor, voltage stability improvement is done by increasing its reactive power generation. On the other hand, if a generator have small participation factor, increase/decrease of its reactive power generation leads to negligible change in voltage stability margin. With respect to (10), there are elements related to ‘‘all” generators in the (JG)Medium eigenvectors. Thus, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators (including the slack bus generator). 3. Optimal power flow In a power system (which consists of several generators, constant power loads, transmission lines, tap changing transformers and shunt capacitors/reactors), the optimal power flow, which is an optimization problem, is defined as follows [17]:
Min Cost ¼
X
ðai þ bi Pgi þ ci P2gi Þ
ð12Þ
V j ðGij cos hij þ Bij sin hij Þ i 2 N 0
ð13Þ
i2N G
Subject to:
0 ¼ Pi V i
X j2N i
0 ¼ Qi Vi
X
V j ðGij sin hij Bij cos hij Þ i 2 NPQ
ð14Þ
j2N i max Q min ci Q ci Q ci
T k T max T min k k V min V i V max i i min Pgi Pgi Pmax gi max Q min gi Q gi Q gi max Sk Sk k 2 NE
i 2 Nc
ð15Þ
k 2 NT
ð16Þ
i 2 NB
ð17Þ
i 2 NG
ð18Þ
i 2 NG
ð19Þ ð20Þ
1132
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
After solving this optimization problem, an operating point is obtained which we name the ‘‘OPF operating point”. At this operating point, the system active loss is computed as follows:
PLoss ¼
X
Pi ¼
i2N B
X
Rk
k2N E
R2k þ X 2k
ðV 2i þ V 2j 2V i V j cos hij Þ
ð21Þ
where it is assumed that the branch k is located between buses i and j. 4. Usual form of optimal reactive power dispatch In the ‘‘usual form of optimal reactive power dispatch” method, an optimization problem is defined. The objective function of this problem is PLoss. The constraints of this problem consist of (13)– (17), (19) and (20). In addition, it is assumed that active power generation at all generator buses (except slack bus) is the same as the value at the OPF operating point [4,5]. Thus, increase/decrease of PLoss is done by increase/decrease of active power generation at the slack bus generator. Assuming an ascending cost function for the slack bus generator and with respect to constant active power generation (and operating cost) at other generator buses, it can be said that PLoss and Cost have the same behavior, i.e., both are decreased or increased (compared with the OPF operating point). In this method, in addition of (21), PLoss can be expressed [18] using (22):
PLoss ¼
X Rk ½ðPi Pj þ Q i Q j Þ cos hij þ ðQ i P j Pi Q j Þ sin hij V iV j k2N
ð22Þ
E
After solving this optimization problem, an operating point is obtained which we name the ‘‘ORPD operating point”. At this operating point, usually PLoss and Cost has been decreased (compared with the OPF operating point).
ator is less-important, li is positive. So, decreasing objective function is done by decreasing reactive power generation at the ith generator. Thus, these generators have incentive for decreasing their reactive power generation. In this method, it is assumed that active power generation at all generator buses (except slack bus) is the same as the value at the OPF operating point. Thus, increase/decrease of PLoss is done by increase/decrease of active power generation at the slack bus generator. Assuming an ascending cost function for the slack bus generator and with respect to constant active power generation (and operating cost) at other generator buses, it can be said that PLoss and Cost have the same behavior, i.e., both are decreased or increased (compared with the OPF operating point). The algorithm of this method is stated as follows: (1) Assume that k = 0, the OPF operating point is the 0th operating point and l0i ¼ 0 for each i e NG. (2) For each i e NG, calculate the ranking coefficient (RC)i for the kth operating point. Then, calculate the average ranking coefficient RC using (24):
P
RC ¼
i2N G ðRCÞi
(3) For each i 2 N G , calculate
‘‘Ranking coefficients” are positive numbers which rank generators from the perspective of their reactive power importance, i.e., their effect on voltage stability margin. Using these coefficients, some generators are considered ‘‘important” and some generators are considered ‘‘less-important”. In one type of ranking coefficients, the important generator has small ranking coefficient. Thus, each generator that its ranking coefficient is less than average is considered important and other generators are considered less-important. In another type of ranking coefficients, the important generator has large ranking coefficient. Thus, each generator that its ranking coefficient is more than average is considered important and other generators are considered less-important. At the important generators, increase/decrease of reactive power generation leads to considerable increase/decrease in voltage stability margin. On the other hand, at the less-important generators, increase/decrease of reactive power generation leads to negligible change in voltage stability margin. In the ‘‘reactive power rescheduling with generator ranking” method, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. This can be done by adding penalty terms to the optimal power flow objective function as follows:
Min F ¼
X i2NG
ðai þ bi Pgi þ ci P2gi Þ þ
X
ðli Q gi Þ
ð23Þ
i2N G
If the ith generator is important, li is negative. So, decreasing objective function is done by increasing reactive power generation at the ith generator. Thus, these generators have incentive for increasing their reactive power generation. On the other hand, if the ith gener-
lkþ1 using (25): i
lkþ1 ¼ lki þ a½ðRCÞi RC i 2 N G i
ð25Þ
(4) Consider an optimization problem which its objective function is defined by (26) and its constraints consists of (13)–(17), (19) and (20). Also, assume that active power generation at all generator buses (except slack bus) is the same as the value at the 0th operating point. Solve this optimization problem and name the obtained operating point as the ‘‘(k + 1)th operating point”.
Min F ¼
X
ðai þ bi P gi þ ci P2gi Þ þ
i2N G
5. Reactive power rescheduling with generator ranking
ð24Þ
nG
X
ðlikþ1 Q gi Þ
ð26Þ
i2NG
(5) If the VSM index at the (k + 1)th operating point is more than the kth operating point, assume that k = k + 1 and go to 2. Otherwise, go to 6. (6) Select the kth operating point as the ‘‘final operating point”. In the above algorithm, a is the ‘‘weighting factor” and weights ranking coefficients. If the important generator has small ranking coefficient, positive a is used. On the other hand, if the important generator has large ranking coefficient, negative a is used. It is mentionable that the ranking coefficients type and the a value can affect the algorithm’s performance. In [12,13], this method has been used by selecting the (APF)PF coefficients as ranking coefficients (for unknown positive value of a). As stated before, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. So, in each iteration, the li coefficient related to the slack bus generator implicitly has been considered equal to zero. Thus, the slack bus reactive power generation has not been used effectively. Also, the effect of a on the method’s performance has not been studied. In this paper, the ‘‘reactive power rescheduling with generator ranking” method has been simulated using both (APF)PF and (RPFG)Medium coefficients as ranking coefficients. As stated before, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators. Thus, using these coefficients, the reactive power generation of ‘‘all” generators are used effectively. Also, in both cases, the effect of a on the method’s performance has been studied. 6. Simulation results In this paper, the ‘‘reactive power rescheduling with generator ranking” method has been simulated using both (APF)PF and
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
(RPFG)Medium coefficients as ranking coefficients (for various amounts of a). In addition, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method has been simulated. For all simulations, the IEEE 30 bus test system [17,19] has been used. This system consists of 6 generators, 37 transmission lines, 4 tap changing transformers and 9 shunt capacitors. The system generators are located at buses 1, 2, 5, 8, 11 and 13. Also, bus 1 is the slack bus. It is assumed that Qci and Tk change discretely and with 0.005 and 0.02 steps, respectively. In addition, there is no line limits in this system. Also, it is assumed that the VSM index is expressed in percent, the system active loss is expressed in per unit and the system operating cost is expressed in unit of money per hour. All simulations have been done using MATLAB and GAMS (the DICOPT solver). At the OPF operating point, the VSM index, the system active loss and the system operating cost are 114.141, 0.09076919 and 800.6923, respectively. The simulated methods have been described in the following sections.
1133
Fig. 2. The system active loss as a function of a in the second method.
6.1. First method In the first method, the ‘‘usual form of optimal reactive power dispatch” method is used. At the ORPD operating point, the VSM index, the system active loss and the system operating cost are 118.491, 0.09051159 and 800.6065, respectively. So, with the first method, the VSM index is increased by 4.35%, the system active loss is decreased by 0.0002576 and the system operating cost is decreased by 0.0858. Thus, it can be said that in the first method, the VSM index is improved and this improvement is done by decreasing the system active loss and the system operating cost. 6.2. Second method In the second method, the ‘‘reactive power rescheduling with generator ranking” method is simulated using the (APF)PF coefficients as ranking coefficients. As stated before, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. So, in each iteration, the li coefficient related to the slack bus generator is considered equal to zero. In addition, in generator ranking with the (APF)PF coefficients, the important generator has small ranking coefficient. Thus, in this method, positive a is used. Figs. 1–3 show the VSM index, the system active loss and the system operating cost as a function of a, respectively. The horizontal line in all figures represents the value of the corresponding quantity at the OPF operating point. In these figures, the a variations are discrete and in a restricted range, because we cannot change a continuously from zero to positive infinity. In this meth-
Fig. 3. The system operating cost as a function of a in the second method.
od, the VSM index is improved by 2.252% in the worst case (a = 1), by 7.79% in the best case (a = 60) and by 4.987% on average. Also, the system active loss is increased by 0.00338892 in the worst case (a = 50), is decreased by 0.00027687 in the best case (a = 20) and is increased by 0.00109411 on average. Finally, the system operating cost is increased by 1.1283 in the worst case (a = 50), is decreased by 0.0922 in the best case (a = 20) and is increased by 0.3642 on average. It can be seen that for each a, voltage stability margin is improved. But, depending on the a value, this improvement can be done by decreasing/increasing the system active loss and the system operating cost. Also, by increasing the a value, the VSM index, the system active loss and the system operating cost can increase or decrease. In addition, in the case of a – 1 (weighting factor is used), the VSM index is greater than the case of a = 1 (weighting factor is not used). So, using weighting factor leads to considerable increase in the VSM index. 6.3. Third method
Fig. 1. The VSM index as a function of a in the second method.
In the third method, the ‘‘reactive power rescheduling with generator ranking” method is simulated using the (RPFG)Medium coefficients as ranking coefficients. As stated before, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators. So, in each iteration, the li coefficient related to each generator can be equal to a nonzero value. In addition, in generator ranking with the (RPFG)Medium coefficients, the important generator has large ranking coefficient. Thus, in this method, negative a is used.
1134
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
Figs. 4–6 show the VSM index, the system active loss and the system operating cost as a function of a, respectively. The horizontal line in all figures represents the value of the corresponding quantity at the OPF operating point. In these figures, the a variations are discrete and in a restricted range, because we cannot change a continuously from zero to negative infinity. In this method, the VSM index is improved by 8.241% in the worst case (a = 1), by 17.781% in the best case (a = 10) and by 15.969% on average. Also, the system active loss is increased by
0.04250478 in the worst case (a = 90), by 0.00103045 in the best case (a = 1) and by 0.02936408 on average. Finally, the system operating cost is increased by 14.214 in the worst case (a = 90), by 0.3429 in the best case (a = 1) and by 9.8093 on average. It can be seen that for each a, voltage stability margin is improved and this improvement is done by increasing the system active loss and the system operating cost. But, this method has been simulated for 11 values of a. So, it is possible that for a specific value of a, voltage stability margin improvement is done by decreasing the system active loss and the system operating cost. Also, by decreasing the a value, the VSM index, the system active loss and the system operating cost can increase or decrease. In addition, in the case of a – 1 (weighting factor is used), the VSM index is greater than the case of a = 1 (weighting factor is not used). So, using weighting factor leads to considerable increase in the VSM index. 6.4. Comparing three methods
Fig. 4. The VSM index as a function of a in the third method.
Fig. 5. The system active loss as a function of a in the third method.
Fig. 6. The system operating cost as a function of a in the third method.
For comparing these three methods, we have to use average values for the second and third methods. Comparing these three methods from the perspective of the VSM index show that the third method is the most powerful method and after it, the second and first methods are located, respectively. The VSM index in the third method is more than the second method by 10.982% and is more than the first method by 11.619%. Also, the VSM index in the second method is more than the first method by 0.637%. Comparing these three methods from the perspective of the system active loss show that the third method has the most active loss and after it, the second and first methods are located, respectively. The system active loss in the third method is more than the second method by 0.02826997 and is more than the first method by 0.02962168. Also, the system active loss in the second method is more than the first method by 0.00135171. Comparing these three methods from the perspective of the system operating cost show that the third method has the most operating cost and after it, the second and first methods are located, respectively. The system operating cost in the third method is more than the second method by 9.4451 and is more than the first method by 9.8951. Also, the system operating cost in the second method is more than the first method by 0.45. So, it can be said that the first method is the cheapest and the third method is the most expensive. In real power systems which have voltage stability criteria, it is not desirable to ‘‘maximize” voltage stability margin. In fact, it is sufficient to meet voltage stability criteria. Thus, the ‘‘reactive power rescheduling with generator ranking” method should be changed to stop whenever voltage stability criteria are met. Also, this method should be simulated for various amounts of a. For each a, an operating point is obtained. From these operating points, we should select the operating points that meet voltage stability criteria. Finally, from the recent operating points (if exist), one operating point which has the least system operating cost (and system active loss) is selected as the ‘‘final operating point”. If this operating point does not exist, the method cannot meet voltage stability criteria. After applying these changes to the second and third methods, we must use the cheapest method at first and the most expensive method at last. Thus, for meeting voltage stability criteria, we have to use the first method, the second method and the third method, respectively. If a cheap algorithm can meet voltage stability criteria, there is no need to use an expensive algorithm. If the first method can meet voltage stability criteria, there is no need to use the second and third methods. Also, if the first method cannot and the second method can meet voltage stability criteria, there is no need to use the third method. Finally, if the first and second methods cannot meet voltage stability criteria, the third method is used.
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
7. Conclusions In this paper, the results of the simulation of a voltage stability improvement method, which we name ‘‘reactive power rescheduling with generator ranking”, were presented. In this method, using ranking coefficients, the generators were divided into important and less-important ones and then, voltage stability margin was improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. In this paper, using both (APF)PF (which do not present a participation factor for the slack bus generator) and (RPFG)Medium (which present participation factors for ‘‘all” generators including the slack bus generator) coefficients as ranking coefficients, the method was simulated. Also, the ‘‘usual form of optimal reactive power dispatch” method was simulated. The simulation results showed that these three methods can considerably improve the VSM index. In addition, using (RPFG)Medium coefficients as ranking coefficients, the VSM index, the system active loss and the system operating cost had the most values. Also, using (APF)PF coefficients as ranking coefficients, these quantities were less than the previous case but more than the ‘‘usual form of optimal reactive power dispatch” method. In these cases, using weighting factor had a positive effect on improving the VSM index. Finally, using the ‘‘usual form of optimal reactive power dispatch” method, these quantities had the least values. References [1] Van Cutsem T. Voltage instability: phenomena, countermeasures, and analysis methods. Proc IEEE 2000;88(2):208–27. [2] Affonso CM, da Silva LCP, Lima FGM, Soares S. MW and MVar management on supply and demand side for meeting voltage stability margin criteria. IEEE Trans Power Syst 2004;19(3):1538–45.
1135
[3] Sharif SS. Optimal reactive power flow problem. PhD dissertation, University of New Brunswick, Canada, September 1998.. [4] Zhao B, Guo CX, Cao YJ. A multiagent-based particle swarm optimization approach for optimal reactive power dispatch. IEEE Trans Power Syst 2005;20(2):1070–8. [5] Wu QH, Cao YJ, Wen JY. Optimal reactive power dispatch using an adaptive genetic algorithm. Int J Elect Power Energy Syst 1998;20(8):563–9. [6] Bansilal, Thukaram D, Parthasarathy K. Optimal reactive power dispatch algorithm for voltage stability improvement. Int J Elect Power Energy Syst 1996;18(7):461–8. [7] Abido MA, Bakhashwain JM. Optimal VAR dispatch using a multiobjective evolutionary algorithm. Int J Elect Power Energy Syst 2005;27(1):13–20. [8] Yong-jun Zhang, Zhen Ren. Optimal reactive power dispatch considering costs of adjusting the control devices. IEEE Trans Power Syst 2005;20(3):1349–56. [9] Yong-jun Zhang, Zhen Ren. Real-time optimal reactive power dispatch using multi-agent technique. Electric Power Syst Res 2004;69(2-3):259–65. [10] Venkatesh B, Sadasivam G, Khan MA. A new optimal reactive power scheduling method for loss minimization and voltage stability margin maximization using successive multi-objective fuzzy LP technique. IEEE Trans Power Syst 2000;15(2):844–51. [11] Arya LD, Sakravdia DK, Kothari DP. Corrective rescheduling for static voltage stability control. Int J Elect Power Energy Syst 2005;27(1):3–12. [12] Menezes TV, da Silva LCP, da Costa VF. Dynamic VAR sources scheduling for improving voltage stability margin. IEEE Trans Power Syst 2003;18(2):969–71. [13] Menezes TV, da Silva LCP, Affonso CM, da Costa VF. MVAR management on the pre-dispatch problem for improving voltage stability margin. IEE Proc Gen Transm Distrib 2004;151(6):665–72. [14] Gao B, Morison GK, Kundur P. Voltage stability evaluation using modal analysis. IEEE Trans Power Syst 1992;7(4):1529–42. [15] da Silva LCP, Wang Y, da Costa VF, Xu W. Assessment of generator impact on system power transfer capability using modal participation factors. IEE Proc Gen Transm Distrib 2002;149(5):564–70. [16] Huang Z, Bao L, Xu W. Generator ranking using modal analysis. IEE Proc Gen Transm Distrib 2003;150(6):709–16. [17] Lai LL, Ma JT, Yokoyama R, Zhao M. Improved genetic algorithms for optimal power flow under both normal and contingent operation states. Int J Elect Power Energy Syst 1997;19(5):287–92. [18] Bhagwan Das D, Patvardhan C. Reactive power dispatch with a hybrid stochastic search technique. Int J Elect Power Energy Syst 2002;24(9):731–6. [19] Pai MA. Computer techniques in power system analysis. New York: McGrawHill; 1986.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Reactive power rescheduling with generator ranking for voltage stability improvement Habibollah Raoufi *, Mohsen Kalantar Center of Excellence for Power System Automation and Operation, Department of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran
a r t i c l e
i n f o
Article history: Received 13 January 2008 Received in revised form 22 October 2008 Accepted 20 November 2008 Available online 6 February 2009 Keywords: Generator ranking Modal analysis Participation factors Reactive power dispatch Voltage stability
a b s t r a c t In a power system, voltage stability margin improvement can be done by regulating generators voltages, transformers tap settings and capacitors/reactors rated reactive powers (susceptances). In this paper, one of these methods, which we name ‘‘reactive power rescheduling with generator ranking”, is considered. In this method, using ‘‘ranking coefficients”, the generators are divided into ‘‘important” and ‘‘less-important” ones and then, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. These ranking coefficients are obtained using ‘‘modal analysis”. In this paper, the method’s performance for two types of ranking coefficients has been analyzed. Also, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method has been simulated. For all simulations, the IEEE 30 bus test system has been used. The simulation results show that in the former method, for either type of ranking coefficients, voltage stability margin is considerably improved and, usually, the system active loss and the system operating cost are increased. Also, in the latter method, voltage stability margin is improved and the system active loss and the system operating cost are decreased. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The transfer of power through a transmission network is accompanied by voltage drops between the generation and consumption points. In normal operating conditions, these drops are in the order of a few percents of the nominal voltage. One of the tasks of power system planners and operators is to check that under heavy stress conditions and/or following credible events, all bus voltages remain within acceptable bounds. In some circumstances, however, in the seconds or minutes following a disturbance, voltages may experience large and progressive falls, which are so pronounced that the system integrity is endangered and power cannot be delivered correctly to customers. This catastrophe is referred to as voltage instability and its calamitous result as voltage collapse. This instability stems from the attempt of load dynamics – especially loads supplied with under load tap changing transformers (ULTC), induction motors and thermostatic loads – to restore power consumption beyond the amount that can be provided by the combined transmission and generation system [1]. Nowadays, there are some voltage stability criteria being implemented. For example, the Western Electricity Coordinating Council (WECC) proposes a minimum voltage stability margin (VSM) requirement of 5% considering simple contingencies, 2.5% for double contingencies, and larger than zero for multiple contingencies. * Corresponding author. Tel.: +98 21 77240492; fax: +98 21 77240490. E-mail address: habibollahraoufi@yahoo.com (H. Raoufi). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.11.013
In a similar way, the ONS (Brazilian System Operator) has also initiated some studies and recommends a minimum VSM requirement of 6% also considering simple contingencies. Both criteria are based on VSM index, which is obtained from PV curve computations and represents the distance from the current operating point to the voltage stability limit [2]. ‘‘Reactive power management” is the general name of methods which try to improve voltage profile/stability by regulating generators voltages, transformers tap settings, reactive sources settings and installing new reactive sources. These methods can be divided into two areas: reactive planning (allocation) and reactive dispatch (re-dispatch, scheduling, rescheduling). Also, the dispatch area can be divided into two areas: off-line reactive dispatch and on-line reactive dispatch. In the reactive planning area, the period of study is the next few months or the next few years and installing the new reactive sources are also considered. In the off-line reactive dispatch area, only installed reactive sources are used and the period of study is the next few days or the next few hours. In the on-line reactive dispatch area, only installed reactive sources are used and the period of study is the next few minutes or the next few seconds [3]. In the off-line reactive dispatch area, voltage profile/stability improvement is done by regulating generators voltages, transformers tap settings and capacitors/reactors rated reactive powers (susceptances). For this purpose, usually two groups of methods are used. In the first group methods – which are referred by names such as ‘‘optimal reactive power dispatch”, ‘‘reactive power optimization”, etc. – a specific optimization problem with specific
1130
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
Nomenclature NB N0 Ni NG NPQ NC NE NT nG Rk, Xk Qci
set of numbers of total buses set of numbers of total buses, excluding slack bus set of numbers of buses adjacent to bus i, including bus i set of numbers of generator buses, including slack bus set of numbers of load buses set of numbers of shunt capacitor/reactor installation buses set of numbers of system branches set of numbers of tap changing transformer branches number of system generators resistance and reactance of branch k (pu) rated reactive power (susceptance) of shunt capacitor/ reactor at bus i (pu)
objective function(s) and specific constraints is defined and then, by solving this problem (using an optimization algorithm), appropriate values for those quantities are obtained [4–9]. In the usual form of these methods, voltage profile/stability improvement is done by minimizing the system active loss [4,5]. In the second group methods – which do not have an accepted name – initiative algorithms are used [10–13]. In this paper, a method from the second group is considered. In this method, which we name ‘‘reactive power rescheduling with generator ranking”, using ‘‘ranking coefficients”, the generators are divided into ‘‘important” and ‘‘less-important” ones and then, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively [12,13]. These ranking coefficients are obtained using ‘‘modal analysis” [14–16]. In [12,13], this method has been used by selecting a specific type of ranking coefficients. But, in this type of ranking coefficients, there is no coefficient related to the slack bus generator. Thus, the reactive power generation of the slack bus generator has not been used effectively. Also, in these references, the effect of ‘‘weighting factor” on the method’s performance has not been studied. In this paper, first, the modal analysis theory is presented in a new and clear form. Then, the ‘‘reactive power rescheduling with generator ranking” method is simulated by selecting two types of ranking coefficients (which one of them presents ranking coefficients for ‘‘all” generators including the slack bus generator). Also, the effect of ‘‘weighting factor” on the method’s performance is studied. In addition, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method is also simulated. For all simulations, the IEEE 30 bus test system is used.
Tk Gij, Bij Vi hi hij Pi, Qi Pgi, Qgi Sk PLoss Cost
generator buses on which reactive power generation at primary operating point is equal to lower or upper limit. In addition, Gunlim is an index for generator buses on which reactive power generation at primary operating point is within lower and upper limits. Finally, f is a function that relates power system main quantities to each other and, in other words, is a symbol of power flow equations. After linearization of power flow equations, depending on equations which contribute in the linearization and quantities which their variations are considered equal to zero, 4 jacobian matrixes (JLarge, JMedium, JSmall and JPF) are obtained. Full linearization of power flow equations leads to
3 3 2 DP s Dhs 6 DP g 7 6 Dhg 7 7 7 6 6 7 7 6 DP 6 Dh 7 7 6 6 L L 7 ¼ J Large 6 7 6 6 DQ L 7 6 DV L 7 7 7 6 6 4 DQ Glim 5 4 DV Glim 5 DQ Gunlim DV Gunlim 2
1 1 0 hs Ps B Pg C B hg C C C B B C B PL C B C ¼ f B hL C B B QL C B VL C C C B B A A @ Q @ V Glim Glim Q Gunlim V Gunlim 0
ð1Þ
where s, g and L are indices for slack bus, all generator buses except slack bus and load buses, respectively. Also, Glim is an index for
ð2Þ
where JLarge is the large jacobian matrix. Also, by removing the equation related to Ps from (1) and assuming Dhs = 0, linearization of (1) leads to
3 3 2 DP g Dhg 7 7 6 6 DP L Dh L 7 7 6 6 7 7 6 6 DQ L DV L 7 7 6 6 7 ¼ J Medium 6 7 6 6 7 6 7 7 7 6 6 5 4 DQ Glim 4 DV Glim 5 2
DQ Gunlim
2. Modal analysis Modal analysis is a method for voltage stability evaluation. In this method, voltage stability analysis is done by computing eigenvalues and right and left eigenvectors of a jacobian matrix (which is obtained from the power flow equations). At an operating point, the relations between main power system quantities (bus voltage magnitude, bus voltage angle, bus active power and bus reactive power) can be expressed by power flow equations as follows:
transformer tap setting of branch k mutual conductance and susceptance between buses i and j (pu) voltage magnitude of bus i (pu) voltage angle of bus i (rad) voltage angle difference between buses i and j (rad) injected active and reactive power at bus i (pu) generated active and reactive power at bus i (pu) apparent power flow in branch k (pu) system active loss (pu) system operating cost (unit of money per hour)
DV Gunlim
3 Dhg 7 6 Dh L 2 3 7 J1 j J2 6 7 6 DV L 7 6 ¼ 4 56 7 6 7 7 J3 j J4 6 4 DV Glim 5 DV Gunlim 2
ð3Þ
where JMedium is the medium jacobian matrix. Also, by removing the equations related to QGunlim from (1) and assuming DVGunlim = 0, linearization of (1) leads to
3 3 2 DP s Dh s 7 7 6 6 6 DP g 7 6 Dh g 7 7 7 6 6 6 DP L 7 ¼ J Small 6 DhL 7 7 7 6 6 7 7 6 6 4 DQ L 5 4 DV L 5 DQ Glim DV Glim 2
ð4Þ
where JSmall is the small jacobian matrix. Finally, by removing the equations related to Ps and QGunlim from (1) and assuming Dhs = 0 and DVGunlim = 0, linearization of (1) leads to
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
3 3 2 DP g Dh g 7 7 6 6 DP L DhL 7 7 6 6 7 7 6 6 6 7 ¼ J PF 6 7 7 7 6 6 5 5 4 4 DQ L DV L 2
DQ Glim
DV Glim
3 2 Dhg 3 7 J5 j J6 6 DhL 7 6 7 6 76 ¼ 4 56 7 7 6 5 J7 j J8 4 DV L DV Glim 2
ð5Þ
where JPF is the power flow jacobian matrix. It is mentionable that both JLarge and JSmall are near singular matrixes (at the normal operating point or at the voltage stability limit). So, in the modal analysis method, usually these matrixes have not been used. In (5), the assumption of DPg = 0 and DPL = 0 leads to
DQ L DV L ¼ ðJ RQV ÞPF DQ Glim DV Glim
ð6Þ
where (JRQV)PF is the reduced reactive jacobian matrix which is obtained from the JPF matrix. This matrix is defined as follows:
ðJ RQV ÞPF ¼ J 8 J7 J 1 5 J6
ð7Þ
This matrix represents the relations between reactive power and voltage magnitude at load buses and some generator buses. Also in (5), the assumption of DQL = 0 and DQGlim = 0 leads to
DP g DP L
¼ ðJ RPh ÞPF
Dhg Dh L
ð8Þ
where (JRPh)PF is the reduced active jacobian matrix which is obtained from the JPF matrix. This matrix is defined as follows:
ðJ RPh ÞPF ¼ J5 J 6 J 1 8 J7
ð9Þ
This matrix represents the relations between active power and voltage angle at all buses (except slack bus). Finally in (3), the assumption of DPg = 0 and DPL = 0 and DQL = 0 leads to
DQ Glim DV Glim ¼ ðJ G ÞMedium DQ Gunlim DV Gunlim
ð10Þ
where (JG)Medium is the generator jacobian matrix which is obtained from the JMedium matrix. This matrix is defined as follows:
ðJ G ÞMedium ¼ J 4 J 3 J 1 1 J2
ð11Þ
This matrix represents the relations between reactive power and voltage magnitude at ‘‘all” generator buses. At the voltage stability limit, the JPF, (JRQV)PF and (JRPh)PF matrixes are singular and the (JG)Medium matrix has an infinite eigenvalue. It is mentionable that modal analysis in [14] is done using the (JRQV)PF matrix, in [15] is done using both (JRQV)PF and (JRPh)PF matrixes and in [16] is done using the (JG)Medium matrix. Bus participation factors are coefficients which are obtained by element by element product of right and left eigenvectors of a specific jacobian matrix (for a specific eigenvalue). Using these coefficients, we can obtain information about the effect of system equipments on voltage stability margin. In [14], reactive participation factors – which we represent them by (RPF)PF – have been defined as element by element product of right and left eigenvectors of the (JRQV)PF matrix (for a specific eigenvalue). These coefficients show that reactive injection (for example by shunt compensation) at which buses leads to considerable increase in voltage stability margin. In [15], active participation factors – which we represent them by (APF)PF – have been defined as element by element product of right and left eigenvectors of the (JRPh)PF matrix (for a specific eigenvalue). These coefficients show that increasing active power
1131
generation (for generator buses) or load shedding (for load buses) at which buses leads to considerable increase in voltage stability margin. In [16], generator reactive participation factors – which we represent them by (RPFG)Medium – have been defined as element by element product of right and left eigenvectors of the (JG)Medium matrix (for a specific eigenvalue). These coefficients show that increasing reactive power generation at which generator buses leads to considerable increase in voltage stability margin. In this paper, modal analysis is used for obtaining some participation factors for generator buses, which will be used for generator ranking from the perspective of their reactive power importance (their effect on voltage stability margin). With respect to (6), it is possible that there is no element related to generator buses in the (JRQV)PF eigenvectors. Thus, the ðRPFÞPF coefficients are not used for generator ranking. In [15], it has been shown that if the system is transferred from the current operating point to the voltage stability limit, the (JRPh)PF matrix is calculated at this limit and the (APF)PF coefficients are computed for the smallest eigenvalue, then, the (APF)PF coefficients can be used for generator ranking (from the perspective of voltage stability). In this case, if a generator have small participation factor, voltage stability improvement is done by increasing its reactive power generation. On the other hand, if a generator have large participation factor, increase/decrease of its reactive power generation leads to negligible change in voltage stability margin. With respect to (8), there is no element related to the slack bus generator in the (JRPh)PF eigenvectors. Thus, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. In [16], it has been shown that if the system is transferred from the current operating point to the voltage stability limit, the (JG)Medium matrix is calculated at this limit and the (RPFG)Medium coefficients are computed for the eigenvalue with the largest absolute, then, the (RPFG)Medium coefficients can be used for generator ranking (from the perspective of voltage stability). In this case, if a generator have large participation factor, voltage stability improvement is done by increasing its reactive power generation. On the other hand, if a generator have small participation factor, increase/decrease of its reactive power generation leads to negligible change in voltage stability margin. With respect to (10), there are elements related to ‘‘all” generators in the (JG)Medium eigenvectors. Thus, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators (including the slack bus generator). 3. Optimal power flow In a power system (which consists of several generators, constant power loads, transmission lines, tap changing transformers and shunt capacitors/reactors), the optimal power flow, which is an optimization problem, is defined as follows [17]:
Min Cost ¼
X
ðai þ bi Pgi þ ci P2gi Þ
ð12Þ
V j ðGij cos hij þ Bij sin hij Þ i 2 N 0
ð13Þ
i2N G
Subject to:
0 ¼ Pi V i
X j2N i
0 ¼ Qi Vi
X
V j ðGij sin hij Bij cos hij Þ i 2 NPQ
ð14Þ
j2N i max Q min ci Q ci Q ci
T k T max T min k k V min V i V max i i min Pgi Pgi Pmax gi max Q min gi Q gi Q gi max Sk Sk k 2 NE
i 2 Nc
ð15Þ
k 2 NT
ð16Þ
i 2 NB
ð17Þ
i 2 NG
ð18Þ
i 2 NG
ð19Þ ð20Þ
1132
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
After solving this optimization problem, an operating point is obtained which we name the ‘‘OPF operating point”. At this operating point, the system active loss is computed as follows:
PLoss ¼
X
Pi ¼
i2N B
X
Rk
k2N E
R2k þ X 2k
ðV 2i þ V 2j 2V i V j cos hij Þ
ð21Þ
where it is assumed that the branch k is located between buses i and j. 4. Usual form of optimal reactive power dispatch In the ‘‘usual form of optimal reactive power dispatch” method, an optimization problem is defined. The objective function of this problem is PLoss. The constraints of this problem consist of (13)– (17), (19) and (20). In addition, it is assumed that active power generation at all generator buses (except slack bus) is the same as the value at the OPF operating point [4,5]. Thus, increase/decrease of PLoss is done by increase/decrease of active power generation at the slack bus generator. Assuming an ascending cost function for the slack bus generator and with respect to constant active power generation (and operating cost) at other generator buses, it can be said that PLoss and Cost have the same behavior, i.e., both are decreased or increased (compared with the OPF operating point). In this method, in addition of (21), PLoss can be expressed [18] using (22):
PLoss ¼
X Rk ½ðPi Pj þ Q i Q j Þ cos hij þ ðQ i P j Pi Q j Þ sin hij V iV j k2N
ð22Þ
E
After solving this optimization problem, an operating point is obtained which we name the ‘‘ORPD operating point”. At this operating point, usually PLoss and Cost has been decreased (compared with the OPF operating point).
ator is less-important, li is positive. So, decreasing objective function is done by decreasing reactive power generation at the ith generator. Thus, these generators have incentive for decreasing their reactive power generation. In this method, it is assumed that active power generation at all generator buses (except slack bus) is the same as the value at the OPF operating point. Thus, increase/decrease of PLoss is done by increase/decrease of active power generation at the slack bus generator. Assuming an ascending cost function for the slack bus generator and with respect to constant active power generation (and operating cost) at other generator buses, it can be said that PLoss and Cost have the same behavior, i.e., both are decreased or increased (compared with the OPF operating point). The algorithm of this method is stated as follows: (1) Assume that k = 0, the OPF operating point is the 0th operating point and l0i ¼ 0 for each i e NG. (2) For each i e NG, calculate the ranking coefficient (RC)i for the kth operating point. Then, calculate the average ranking coefficient RC using (24):
P
RC ¼
i2N G ðRCÞi
(3) For each i 2 N G , calculate
‘‘Ranking coefficients” are positive numbers which rank generators from the perspective of their reactive power importance, i.e., their effect on voltage stability margin. Using these coefficients, some generators are considered ‘‘important” and some generators are considered ‘‘less-important”. In one type of ranking coefficients, the important generator has small ranking coefficient. Thus, each generator that its ranking coefficient is less than average is considered important and other generators are considered less-important. In another type of ranking coefficients, the important generator has large ranking coefficient. Thus, each generator that its ranking coefficient is more than average is considered important and other generators are considered less-important. At the important generators, increase/decrease of reactive power generation leads to considerable increase/decrease in voltage stability margin. On the other hand, at the less-important generators, increase/decrease of reactive power generation leads to negligible change in voltage stability margin. In the ‘‘reactive power rescheduling with generator ranking” method, voltage stability margin is improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. This can be done by adding penalty terms to the optimal power flow objective function as follows:
Min F ¼
X i2NG
ðai þ bi Pgi þ ci P2gi Þ þ
X
ðli Q gi Þ
ð23Þ
i2N G
If the ith generator is important, li is negative. So, decreasing objective function is done by increasing reactive power generation at the ith generator. Thus, these generators have incentive for increasing their reactive power generation. On the other hand, if the ith gener-
lkþ1 using (25): i
lkþ1 ¼ lki þ a½ðRCÞi RC i 2 N G i
ð25Þ
(4) Consider an optimization problem which its objective function is defined by (26) and its constraints consists of (13)–(17), (19) and (20). Also, assume that active power generation at all generator buses (except slack bus) is the same as the value at the 0th operating point. Solve this optimization problem and name the obtained operating point as the ‘‘(k + 1)th operating point”.
Min F ¼
X
ðai þ bi P gi þ ci P2gi Þ þ
i2N G
5. Reactive power rescheduling with generator ranking
ð24Þ
nG
X
ðlikþ1 Q gi Þ
ð26Þ
i2NG
(5) If the VSM index at the (k + 1)th operating point is more than the kth operating point, assume that k = k + 1 and go to 2. Otherwise, go to 6. (6) Select the kth operating point as the ‘‘final operating point”. In the above algorithm, a is the ‘‘weighting factor” and weights ranking coefficients. If the important generator has small ranking coefficient, positive a is used. On the other hand, if the important generator has large ranking coefficient, negative a is used. It is mentionable that the ranking coefficients type and the a value can affect the algorithm’s performance. In [12,13], this method has been used by selecting the (APF)PF coefficients as ranking coefficients (for unknown positive value of a). As stated before, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. So, in each iteration, the li coefficient related to the slack bus generator implicitly has been considered equal to zero. Thus, the slack bus reactive power generation has not been used effectively. Also, the effect of a on the method’s performance has not been studied. In this paper, the ‘‘reactive power rescheduling with generator ranking” method has been simulated using both (APF)PF and (RPFG)Medium coefficients as ranking coefficients. As stated before, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators. Thus, using these coefficients, the reactive power generation of ‘‘all” generators are used effectively. Also, in both cases, the effect of a on the method’s performance has been studied. 6. Simulation results In this paper, the ‘‘reactive power rescheduling with generator ranking” method has been simulated using both (APF)PF and
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
(RPFG)Medium coefficients as ranking coefficients (for various amounts of a). In addition, for comparison purpose, the ‘‘usual form of optimal reactive power dispatch” method has been simulated. For all simulations, the IEEE 30 bus test system [17,19] has been used. This system consists of 6 generators, 37 transmission lines, 4 tap changing transformers and 9 shunt capacitors. The system generators are located at buses 1, 2, 5, 8, 11 and 13. Also, bus 1 is the slack bus. It is assumed that Qci and Tk change discretely and with 0.005 and 0.02 steps, respectively. In addition, there is no line limits in this system. Also, it is assumed that the VSM index is expressed in percent, the system active loss is expressed in per unit and the system operating cost is expressed in unit of money per hour. All simulations have been done using MATLAB and GAMS (the DICOPT solver). At the OPF operating point, the VSM index, the system active loss and the system operating cost are 114.141, 0.09076919 and 800.6923, respectively. The simulated methods have been described in the following sections.
1133
Fig. 2. The system active loss as a function of a in the second method.
6.1. First method In the first method, the ‘‘usual form of optimal reactive power dispatch” method is used. At the ORPD operating point, the VSM index, the system active loss and the system operating cost are 118.491, 0.09051159 and 800.6065, respectively. So, with the first method, the VSM index is increased by 4.35%, the system active loss is decreased by 0.0002576 and the system operating cost is decreased by 0.0858. Thus, it can be said that in the first method, the VSM index is improved and this improvement is done by decreasing the system active loss and the system operating cost. 6.2. Second method In the second method, the ‘‘reactive power rescheduling with generator ranking” method is simulated using the (APF)PF coefficients as ranking coefficients. As stated before, in generator ranking with the (APF)PF coefficients, there is no coefficient related to the slack bus generator. So, in each iteration, the li coefficient related to the slack bus generator is considered equal to zero. In addition, in generator ranking with the (APF)PF coefficients, the important generator has small ranking coefficient. Thus, in this method, positive a is used. Figs. 1–3 show the VSM index, the system active loss and the system operating cost as a function of a, respectively. The horizontal line in all figures represents the value of the corresponding quantity at the OPF operating point. In these figures, the a variations are discrete and in a restricted range, because we cannot change a continuously from zero to positive infinity. In this meth-
Fig. 3. The system operating cost as a function of a in the second method.
od, the VSM index is improved by 2.252% in the worst case (a = 1), by 7.79% in the best case (a = 60) and by 4.987% on average. Also, the system active loss is increased by 0.00338892 in the worst case (a = 50), is decreased by 0.00027687 in the best case (a = 20) and is increased by 0.00109411 on average. Finally, the system operating cost is increased by 1.1283 in the worst case (a = 50), is decreased by 0.0922 in the best case (a = 20) and is increased by 0.3642 on average. It can be seen that for each a, voltage stability margin is improved. But, depending on the a value, this improvement can be done by decreasing/increasing the system active loss and the system operating cost. Also, by increasing the a value, the VSM index, the system active loss and the system operating cost can increase or decrease. In addition, in the case of a – 1 (weighting factor is used), the VSM index is greater than the case of a = 1 (weighting factor is not used). So, using weighting factor leads to considerable increase in the VSM index. 6.3. Third method
Fig. 1. The VSM index as a function of a in the second method.
In the third method, the ‘‘reactive power rescheduling with generator ranking” method is simulated using the (RPFG)Medium coefficients as ranking coefficients. As stated before, in generator ranking with the (RPFG)Medium coefficients, there are coefficients related to ‘‘all” generators. So, in each iteration, the li coefficient related to each generator can be equal to a nonzero value. In addition, in generator ranking with the (RPFG)Medium coefficients, the important generator has large ranking coefficient. Thus, in this method, negative a is used.
1134
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
Figs. 4–6 show the VSM index, the system active loss and the system operating cost as a function of a, respectively. The horizontal line in all figures represents the value of the corresponding quantity at the OPF operating point. In these figures, the a variations are discrete and in a restricted range, because we cannot change a continuously from zero to negative infinity. In this method, the VSM index is improved by 8.241% in the worst case (a = 1), by 17.781% in the best case (a = 10) and by 15.969% on average. Also, the system active loss is increased by
0.04250478 in the worst case (a = 90), by 0.00103045 in the best case (a = 1) and by 0.02936408 on average. Finally, the system operating cost is increased by 14.214 in the worst case (a = 90), by 0.3429 in the best case (a = 1) and by 9.8093 on average. It can be seen that for each a, voltage stability margin is improved and this improvement is done by increasing the system active loss and the system operating cost. But, this method has been simulated for 11 values of a. So, it is possible that for a specific value of a, voltage stability margin improvement is done by decreasing the system active loss and the system operating cost. Also, by decreasing the a value, the VSM index, the system active loss and the system operating cost can increase or decrease. In addition, in the case of a – 1 (weighting factor is used), the VSM index is greater than the case of a = 1 (weighting factor is not used). So, using weighting factor leads to considerable increase in the VSM index. 6.4. Comparing three methods
Fig. 4. The VSM index as a function of a in the third method.
Fig. 5. The system active loss as a function of a in the third method.
Fig. 6. The system operating cost as a function of a in the third method.
For comparing these three methods, we have to use average values for the second and third methods. Comparing these three methods from the perspective of the VSM index show that the third method is the most powerful method and after it, the second and first methods are located, respectively. The VSM index in the third method is more than the second method by 10.982% and is more than the first method by 11.619%. Also, the VSM index in the second method is more than the first method by 0.637%. Comparing these three methods from the perspective of the system active loss show that the third method has the most active loss and after it, the second and first methods are located, respectively. The system active loss in the third method is more than the second method by 0.02826997 and is more than the first method by 0.02962168. Also, the system active loss in the second method is more than the first method by 0.00135171. Comparing these three methods from the perspective of the system operating cost show that the third method has the most operating cost and after it, the second and first methods are located, respectively. The system operating cost in the third method is more than the second method by 9.4451 and is more than the first method by 9.8951. Also, the system operating cost in the second method is more than the first method by 0.45. So, it can be said that the first method is the cheapest and the third method is the most expensive. In real power systems which have voltage stability criteria, it is not desirable to ‘‘maximize” voltage stability margin. In fact, it is sufficient to meet voltage stability criteria. Thus, the ‘‘reactive power rescheduling with generator ranking” method should be changed to stop whenever voltage stability criteria are met. Also, this method should be simulated for various amounts of a. For each a, an operating point is obtained. From these operating points, we should select the operating points that meet voltage stability criteria. Finally, from the recent operating points (if exist), one operating point which has the least system operating cost (and system active loss) is selected as the ‘‘final operating point”. If this operating point does not exist, the method cannot meet voltage stability criteria. After applying these changes to the second and third methods, we must use the cheapest method at first and the most expensive method at last. Thus, for meeting voltage stability criteria, we have to use the first method, the second method and the third method, respectively. If a cheap algorithm can meet voltage stability criteria, there is no need to use an expensive algorithm. If the first method can meet voltage stability criteria, there is no need to use the second and third methods. Also, if the first method cannot and the second method can meet voltage stability criteria, there is no need to use the third method. Finally, if the first and second methods cannot meet voltage stability criteria, the third method is used.
H. Raoufi, M. Kalantar / Energy Conversion and Management 50 (2009) 1129–1135
7. Conclusions In this paper, the results of the simulation of a voltage stability improvement method, which we name ‘‘reactive power rescheduling with generator ranking”, were presented. In this method, using ranking coefficients, the generators were divided into important and less-important ones and then, voltage stability margin was improved by increasing and decreasing reactive power generation at the important and less-important generators, respectively. In this paper, using both (APF)PF (which do not present a participation factor for the slack bus generator) and (RPFG)Medium (which present participation factors for ‘‘all” generators including the slack bus generator) coefficients as ranking coefficients, the method was simulated. Also, the ‘‘usual form of optimal reactive power dispatch” method was simulated. The simulation results showed that these three methods can considerably improve the VSM index. In addition, using (RPFG)Medium coefficients as ranking coefficients, the VSM index, the system active loss and the system operating cost had the most values. Also, using (APF)PF coefficients as ranking coefficients, these quantities were less than the previous case but more than the ‘‘usual form of optimal reactive power dispatch” method. In these cases, using weighting factor had a positive effect on improving the VSM index. Finally, using the ‘‘usual form of optimal reactive power dispatch” method, these quantities had the least values. References [1] Van Cutsem T. Voltage instability: phenomena, countermeasures, and analysis methods. Proc IEEE 2000;88(2):208–27. [2] Affonso CM, da Silva LCP, Lima FGM, Soares S. MW and MVar management on supply and demand side for meeting voltage stability margin criteria. IEEE Trans Power Syst 2004;19(3):1538–45.
1135
[3] Sharif SS. Optimal reactive power flow problem. PhD dissertation, University of New Brunswick, Canada, September 1998.