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Evaluation of the maximum loadability point of power systems considering the effect of static load models
Energy Conversion and Management 50 (2009) 3202–3210
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Evaluation of the maximum loadability point of power systems considering the effect of static load models Nima Amjady *, Mohammad Hossein Velayati Department of Electrical Engineering, Semnan University, Semnan, Iran
a r t i c l e
i n f o
Article history: Received 29 December 2008 Received in revised form 18 August 2009 Accepted 29 August 2009 Available online 20 September 2009 Keywords: Voltage stability Maximum loadability point (MLP) Load flow Jacobian Static analysis tool
a b s t r a c t Power system stability is an important problem for power system operation. Determination of different stability margins can result in the optimum utilization of power system with minimum risk. Voltage stability is an important subset of power system stability. To correctly analyze the voltage stability of a power system, suitable dynamic models are usually required. However, static analysis tools can give us useful information about long term voltage stability. Especially, maximum loadability point (MLP) of a power system can be effectively estimated by modal analysis of load flow Jacobians. MLP is one of the important boundaries of voltage stability feasible region that loading beyond which is of little practical meaning. In this paper, MLP boundary of power system is analyzed by means of static analysis tools and its differences with the other boundaries of voltage stability, like saddle node bifurcation, are discussed. Effect of reactive power limits of generators and different static load models on the MLP border is also evaluated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [1]. Voltage stability is becoming one of the dominant factors in determining power system operating limits and greatly depending on load characteristics and specific control devices. Load models, overexcitation limits of generators and auto-regulator of transformers will affect voltage stability significantly [2]. For convenience in analysis and for gaining useful insight into the nature of voltage stability problems, it is useful to characterize voltage stability in terms of the large-disturbance and small-disturbance voltage stability [1]. Small-disturbance voltage stability considers the power system’s ability to control voltages after small disturbances, e.g. changes in load. Large-disturbance voltage stability refers to the system’s ability to maintain steady voltages following large disturbances such as system faults, loss of generation, or circuit contingencies [3]. Voltage instability is a complex phenomenon and may occur over a wide time scale depending on the mechanism by which the instability occurs [4]. Therefore, voltage stability may be either a short-term or long-term phenomenon. In shortterm voltage stability, dynamics of fast acting load components such as induction motors, electronically controlled loads, and HVDC converters are considered. The long term voltage stability * Corresponding author. Tel.: +98 021 88797174; fax: +98 021 88880098. E-mail address: [email protected] (N. Amjady). 0196-8904/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.08.026
is characterized by scenarios such as load recovery by the action of on load tap changer or through loads self restoration, delayed corrective control actions such as shunt compensation switching or load shedding. The long term dynamics such as response of power plant controls, boiler dynamics and automatic generation control also affect long term voltage stability [5]. To correctly analyze the voltage stability of a power system, suitable dynamic models are usually required based on nonlinear differential and algebraic equations. However, in many cases, static analysis tools can be used to estimate stability margins, identify factors influencing stability, and screen a wide range of system conditions and a large number of scenarios for long term voltage stability [1]. In the static analysis, it is assumed that the voltage instability is caused by active power or reactive power unbalance [6]. In recent years many research works have been devoted to determine voltage stability margin, which is the margin between the operating point of the power system and the boundaries of voltage stability feasible region [6–13]. One of the important boundaries of this feasible region is the maximum loadability point or MLP. MLP consists of the load flow feasibility boundary at which the load flow Jacobian is singular [10]. Loading beyond the MLP is of little practical meaning. In the literature, several static analysis tools such as decoupled method [7], modal analysis of load flow Jacobians [8,14], multiple load flow solutions [9], continuation and adaptive continuation methods [15–17], particle swarm optimization (PSO) [18] and energy function technique [19–21] have been proposed to determine the MLP boundary of power system.
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In spite of the performed research works in the area, there is a lack of deep analysis of MLP, which is an important boundary of voltage stability feasible region. The main contribution of the paper is providing such an analysis. In this paper, the MLP boundary of voltage stability feasible region is analyzed by the combination of continuation power flow (CPF) method and modal analysis of load flow Jacobians. A detailed point to point analysis of all load flow Jacobian matrices (reduced and unreduced Jacobians) along the continuation curve based on the criteria of determinant and minimum eigenvalue is presented in this paper, not seen in the previous research works on MLP (such as [6,14,17,18]). Moreover, the effect of reactive power limits of generators and different static load models on the MLP border is also evaluated, while, up to the authors’ knowledge, all other research works on MLP only consider constant power load model. Differences between MLP and the other boundaries of voltage stability feasible region, such as SNB (which is usually taken as the voltage collapse point [13]) are also analyzed in the paper. We found important results about the effect of different load models on the MLP boundary and differences between MLP and the other voltage stability feasible region borders. To the best of our knowledge, such a comprehensive analysis of the MLP boundary has been never presented in the previous research works in the area. 2. Load modeling Voltage stability strongly depends on load characteristic [4,22,23]. Dependency of active and reactive powers of loads on voltage is an important major factor to determine the voltage stability status of power system [24]. In [25] a qualitative discussion on the influence of load characteristics on voltage stability has been reported. Load models are classified into two categories: static and dynamic [22]. The static models represent active and reactive powers of load in terms of voltage and frequency as follows:
a a V V x x x0 V0 bV bx V x Q ¼ Q0 x0 V0 P ¼ P0
ð1Þ ð2Þ
where P0 and Q0 stand for the real and reactive powers consumed at a reference voltage V0 and reference frequency x0, respectively. The exponents aV, ax, bV and bx represent the dependency of active and reactive powers on voltage and frequency, respectively. In the voltage stability studies, the dependency of load active and reactive powers on frequency is usually ignored [4,22,23]. So, we reach to:
a V V V0 bV V Q ¼ Q0 V0 P ¼ P0
ð3Þ ð4Þ
The exponents aV and bV depend on the type of load that is being represented, e.g. for constant power load model aV = bV = 0, for constant current load model aV = bV = 1 and for constant impedance load model aV = bV = 2. Another alternative to represent static load models is based on polynomial equations as follows:
" # 2 V V P ¼ P0 ap þ bp þ cp V0 V0 " # 2 V V Q ¼ Q 0 aq þ bq þ cq V0 V0
ð5Þ ð6Þ
where ap, bp and cp indicate percentage of constant impedance, constant current and constant power load models in the active power
(ap + bp + cp = 1). Similarly, aq, bq and cq are defined for reactive power and aq + bq + cq = 1. In other words, the load model of (5 and 6), known as ZIP model [22], is made up of the three different load models. In this paper, the effect of different static load models (constant power, constant current and constant impedance) described above, on the MLP boundary of power system is evaluated. Also, these load models have significant effect on the dynamic evaluation of voltage stability such as determination of Hopf bifurcation [22,23]. 3. Static voltage stability 3.1. Theoretical analysis The small-disturbance voltage stability is a dynamic phenomenon by nature, but the use of steady state analysis tools can provide useful information about it. In steady state voltage stability studies based on the load flow equations it is assumed that all dynamics are died out and all controllers have done their duty. Steady state voltage stability studies investigate long term voltage stability [12], while for the evaluation of short-term voltage stability, dynamic analysis is usually required [1]. The advantage of using algebraic equations compared to differential equations of dynamic studies is the ease of modeling and computation speed. This paper focuses on the small disturbance and long term voltage stability. Continuation method is an efficient approach to evaluate the boundaries of the feasible region of small-disturbance voltage stability. To obtain the continuation curve of power system, a subset of its parameters is changed step by step. This stage of continuation method is known as predictor [3,17] where a known operating point is used to compute direction and amount of changes applied to the system parameters. Then the new operating point is computed by the varied parameters. This stage is known as the corrector [3,17]. Continuation methods can find voltage stability boundaries of the power system such as the MLP boundary which can be found by the static continuation [15–18] and Hopf bifurcation which can be determined by the dynamic continuation [12]. In this paper, continuation power flow (CPF) is used to evaluate the MLP boundary of power system. The power flow equations for a power system can simply be written as:
DP DQ
¼J
" JPh Dh ¼ JQ h DjVj
J PV J QV
#
Dh DjVj
ð7Þ
where h and |V| indicate angle and magnitude of voltage. Matrix J is known as full load flow Jacobian. Two other reduced Jacobian matrices can be found from J as follows:
Dh ¼ J 1 RPh DPjDQ¼0 where JRPh ¼ J Ph
ð8Þ JPV J 1 QV J Q h
DV ¼ J 1 RQV DQ jDP¼0
ð9Þ J Q h J1 Ph J PV .
where JRQV ¼ J QV JRPh and JRQV are reduced active and reactive Jacobian matrices of the system [14,26,27]. To compute JRQV at each operating point, we keep P constant and evaluate voltage stability by considering the incremental relationship between Q and V. Therefore, J 1 RQV is the reduced V–Q Jacobian that its ith diagonal element represents V–Q sensitivity in ith bus [3]. System is voltage stable if all the eigenvalues are positive then V–Q sensitivity is positive for every bus and det(JRQV) is also positive. At the critical point of system voltage stability, one of eigenvalues of JRQV reaches to origin and this matrix becomes singular. At this point, inversion of full Jacobian matrix is not possible [26].
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Full Jacobian matrix, reduced active and reduced reactive Jacobian matrices contain useful information on voltage stability condition of power system. At MLP all of these three Jacobian matrices become singular [26]. In other words, at MLP, two reduced Jacobian matrices contain the same modal information as that of the full Jacobian matrix [26]. This conclusion is valid not only for the critical eigenvector but also for all other eigenvectors. The eigenvectors of load flow Jacobians provide information related to the mechanism of loss of voltage stability [27]. Eigenvector of the full matrix has two set of entries. One set is related to the active power for all buses that named Full Active (FA), and the other related to the reactive power for PQ buses only that named Full Reactive (FR). In the MLP, the active part of the full Jacobian matrix eigenvector can be compared with the eigenvector of the JRPh. Moreover, the reactive part of the full matrix eigenvector can be compared with the eigenvector of the JRQV [26]. The critical eigenvector of JRPh (or JRQV), reveals those buses where active power (reactive power) increments are more detrimental to the system voltage stability. But considerable difference between reduced active Jacobian matrix and reduced reactive Jacobian matrix can be observed. In fact, JRPh contains both PV (generator) and PQ (load) buses, while JRQV contains PQ buses only. Therefore, JRPh offers more information of JRQV. It is noted that for a power system with NPV of PV buses and NPQ of PQ buses JRQV is (NPQ NPQ) and JRPh is (NPQ + NPV) (NPQ + NPV). In real systems voltage instability happens due to a combination of active and reactive power demanding patterns of loads and generators. Therefore both reactive problem and the active problem must be considered altogether [26]. 3.2. MLP and load flow divergence A few research works use the repetitious power flow solutions by increasing the load on the system in some direction and solving the load flow equations at each step until the load flow solution diverges. They supposed this point as the MLP of power system [28,29]. However these techniques have following disadvantages [18]: – The point where load flow solution diverges does not illustrate the maximum loadability point. Note that, divergence of load flow equation is a mathematical failure, while MLP is a physical limitation. Therefore, the divergence of the power flow may be due to mathematical problem in the applied numerical method. – It has also been noticed that the point where the calculations diverge may vary depending on the numerical method which was used. So, the MLP of power system can be different from the load flow divergence point. 3.3. MLP and voltage collapse Voltage collapse typically occurs on power systems, which are heavily loaded, faulted and/or has reactive power shortages. It is the process by which the sequence of events accompanying voltage instability leads to a blackout or abnormally low voltages in a significant part of the power system. Static voltage stability has neglected all the dynamics of generator, excitation system and so on and considered that voltage instability is caused by active and reactive power unbalance. Therefore, the system equations for static voltage stability contain load flow equations only, while the other types of voltage stability problems in power systems such as voltage collapse and oscillatory problems can be analyzed through bifurcation theory [30]. In bifurcation analysis, differential–algebraic equations of power system
Fig. 1. Bifurcation and P–V curve of an inconstant power load.
are usually used to determine the stability boundaries. Note that the system Jacobian (reduced Jacobian or unreduced Jacobian) of a dynamic power system model typically differs from the power flow Jacobian. If the system Jacobian is asymptotically stable, all eigenvalues have negative real parts. On the other hand, all eigenvalues of the load flow Jacobians have positive real parts before occurring MLP. Some earlier works considered nose point of the P–V curve as the voltage collapse point of the power system. However, in the literature it has been shown that in the systems with inconstantpower loads, the real voltage collapse point is the SNB (Saddle Node Bifurcation) of the bifurcation curve (or point B00 on P–V curve in Fig. 1) instead of the nose point of P–V curve (point B0 ) [31]. Only when all the loads are constant-power type, nose point just coincides with the SNB. At SNB, two equilibrium coalesce and then disappear, at this point the reduced and unreduced Jacobian matrices (related to differential–algebraic equations of the power system [12,13]) has a zero eigenvalue. So, the SNB, which can be obtained from bifurcation analysis considering the reduced/unreduced Jacobian, is different from the MLP indicating the point where the load flow Jacobian becomes singular. Hence, the discussion of this subsection and previous one indicate that the MLP boundary of power system (considered in this paper) can be different from the previously considered boundaries for voltage stability such as load flow divergence point and voltage collapse (SNB) point. 4. Numerical results We test the proposed method to determine the MLP boundary of the six bus test system and 24 bus test system. The effect of reactive power limits of generators on the MLP boundary is studied based on the six bus test system and the effect of different static load models on the MLP is evaluated in the 24 bus test system. For continuation power flow (CPF), we use the power system analysis toolbox (PSAT) software package [32]. PSAT is a MATLAB toolbox for electric power system analysis and simulation. All operations can be assessed by means of graphical user interfaces (GUIs). Some of main features of PSAT are: power flow; continuation power flow; optimal power flow; small signal stability analysis; time domain simulation; phasor measurement unit (PMU) placement; and complete graphical user interface. Continuation methods allow us to trace the complete voltage profile (P–V curve) by automatically changing the value of loading parameter, without having to worry about the singularity of the load flow Jacobian matrices. Determinant and eigenvalues of the load flow Jacobian matrices are computed in the MATLAB software package environment [33]. MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where
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problems and solutions are expressed in familiar mathematical notation. MATLAB features a family of add-on application-specific solutions called toolboxes. Toolboxes are comprehensive collections of MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others.
Single line diagram of the six bus test system with three generators, three load buses and 11 transmission lines is shown in Fig. 2. Data of this test system can be obtained from [32]. Fig. 3 shows the variation of voltage magnitude (V) and voltage angle (h) versus loading factor (k) for load buses 4, 5 and 6 considering the reactive power limits of generators. In this experiment, consumption of these buses is increased in proportion to their base load with constant power factor. All load buses 4, 5 and 6 are constant-power type. From Fig. 3, it is seen that the nose point of k–V and k–h curves for each bus occurs at the same loading factor.
Voltage magnitud (P.u)
Fig. 2. Single line diagram of the six bus test system.
1 (3)
0.8 0.6
(2)
0.4 (1)
0.2 0 0
2
4
6
8
10
Loading Factor λ (1):Bus4 (2):Bus 5 (3):Bus6 0
Voltageangle (deg)
(b)
(2)
-10 (3)
-20 (1)
-30 0
2
4
Loading factor Upper part of k–V curve Nose point of k–V curve Lower part of k–V curve
4.1. Results for six bus test system
(a)
Table 1 Determinant of load flow Jacobian matrices versus loading factor with considering reactive power limits of generators.
6
8
10
Loading Factor λ (1):Bus4 (2):Bus 5 (3):Bus6 Fig. 3. (a) k–V curve and (b) k–h curve for load buses in six bus test system with reactive power limits of generators considered.
det(J)
det(JRPh)
det(JRQV)
0
1.23
4.73
3.74
6.054
3.34e1
2.35
1.85
9.602
5.41e3
1.73e1
1.77e1
9.599
4.18e3
1.44e1
1.46e1
1.11e3 2.98e4 2.74e5 6.57e7 3.97e4
5.12e2 1.55e2 1.50e3 3.61e5 2.38e2
4.95e2 1.45e2 1.38e3 3.00e5 2.12e2
9.531 9.477 9.452 9.449 MLP 9.403
The obtained results for determinant of load flow Jacobian matrices, when the loading factor is varied, are shown in Table 1. All determinants of Jacobian matrices of this paper are computed with per unit values for power system quantities. As the CPF proceeds along the continuation curve, the determinants of all Jacobian matrices decrease until they reach to zero. It is noted that by increasing the loading factor from 0 to 9.602 (nose point), the CPF proceeds along the upper part of the continuation curves (Fig. 3). After the nose point, the CPF enters to the lower part of the continuation curves (e.g. the loading factor of 9.599) and proceeds in this part until it reaches to the MLP. In other words, the MLP, which approximately occurs at the loading factor of 9.449, is located at the lower part of the continuation curves (at the loading factor of 9.449, the determinant of all load flow Jacobian matrices is approximately zero). This experiment shows that the MLP boundary can be different from the nose point of continuation curves. Moreover, for the constant power loads, the divergence point of load flow occurs at a point in the upper part of continuation curve. So, the MLP boundary (which is located in the lower part of the continuation curve in this experiment) can be different from the load flow divergence point as well. In Fig. 4, determinant of full load flow Jacobian matrix det(J), determinant of reduced active Jacobian matrix det(JRPh) and determinant of reduced reactive Jacobian matrix det(JRQV) versus loading factor (k) is shown. As seen, the determinant of all three Jacobian matrices decreases with changing the loading factor along the continuation curve and at the MLP boundary reaches to zero. Moreover, from Fig. 4, it can be seen that the nose point of each continuation curve is different from the MLP boundary when reactive power limits of generators are considered. Figs. 5 and 6 show the eigenvalue locus of the three load flow Jacobian matrices at the base loading point (k = 0) and a little after the MLP boundary (k = 9.403, which is the last row of Table 1), respectively. By changing the loading factor k along the continuation curve, one eigenvalue of each load flow Jacobian matrix goes from the right hand side of the complex plane to origin and reaches to it at the MLP boundary. After that the eigenvalue enters to the left hand side of the complex plane as seen from Fig. 6. In Table 2, the determinant of the three load flow Jacobian matrices versus loading factor is shown without considering the reactive power limits of generators. In the table, upper part, nose point and lower part of the continuation curve are shown from top to bottom, respectively. At first, it can be seen that without considering the reactive power limits of generators, the MLP boundary occurs at the higher loading factors (11.168 in Table 2 versus 9.449 in Table 1). Besides, the MLP coincides with the nose point of the continuation curve when the reactive power limits are not considered. This is another important result derived in this paper, which shows the effect of reactive limits of the power system
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2
determinant
(a)
1
Nose Point MLP
0
0 -1
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
7
8
9
10
(b)
determinant
Loading Factor λ 4 2 0 0
1
2
3
4
5
6
Loading Factor λ 4
determinant
(c)
2 0 0
1
2
3
4
5
6
Loading Factor λ Fig. 4. (a) det(J)–k, (b) det(JRPh)–k and (c) det(JRQV)–k with reactive power limits of generators considered.
eig of Full Jacobian matrix
1
5
0.5 Im aginary
Imaginary
eig of Full Jacobian matrix
10
0 -5 -10 0
5
10
15
20
25
0 -0.5 -1 -5
30
0
5
10
eig of Reduced Reactive Jacobian matrix
eig of Reduced Active Jacobian matrix
0.5
0.5
0.5
0.5
-1 10
15
20
Real
0 -0.5 -1 0
10
20
30
Real
Im aginary
1
Im aginary
1
-0.5
20
eig of Reduced Active Jacobian matrix
1 Imaginary
Imaginary
eig of Reduced Reactive Jacobian matrix
1
0
15
Real
Real
0 -0.5 -1 -5
0
5
10
15
0 -0.5 -1 -10
Real
0
10
20
Real
Fig. 5. Eigenvalue locus of the three load flow Jacobian matrices for base loading point (k = 0).
Fig. 6. Eigenvalue locus of the three load flow Jacobian matrices a little after MLP boundary (k = 9.403).
on the situation of MLP boundary based on the continuation curve. In Fig. 7, determinant of the three load flow Jacobian matrices versus loading factor (k) is shown. As seen, the determinant of all three Jacobian matrices decreases with increasing the loading factor and at the MLP boundary reaches to zero. Again, from Fig. 7, it can be seen that the nose point of each continuation curve coincides with the MLP boundary when the reactive power limits of generators are not considered.
Table 2 Determinant of load flow Jacobian matrices versus loading factor without considering reactive power limits of generators. Loading factor
det(J)
det(JRPh)
det(JRQV)
Upper part of k–V curve
0.35 4.899 8.7498
1.24 5.51e1 1.75e1
4.73 3.07 1.63
3.74 2.44 1.27
Nose point of k–V curve Lower part of k–V curve
11.168 MLP 11.1678
1.19e4 4.5e4
4.01e3 1.56e2
2.14e3 8.24e3
4.2. Results for 24 bus test system The 24 bus test system, shown in Fig. 8, consists of 11 generators, 24 buses, five transformers, 33 branches and one static condenser. Data of this test system can be obtained from [32].
For a better illustration of the effect of different load models on the MLP boundary, reactive power limits of the power system are not considered in this experiment. Continuation curve for bus 3
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2
determinant
(a)
0 -1
0
2
4
0
2
4
6
8
10
12
6
8
10
12
8
10
12
Loading Factor λ
5
determinant
(b)
0
1
0
-5
Loading Factor λ
(c) determinant
4 2 0 -2
0
2
4
6
Loading Factor λ Fig. 7. (a) det(J)–k, (b) det(JRPh)–k and (c) det(JRQV)–k without considering reactive power limits of generators.
Line15-21(2) Synchrono.. Line15-21(1)
~ G ~ G
Station23/B18 Line18-21(2)
Synchrono..
Station22/B22 Line17-18
Line18-21(1)
Station24/B17 Synchrono..
G ~
Line21-22
General L..
Station21/B21 Line17-22 Synchrono..
Synchrono..
~ G
Line16-17
Line16-19
~ G
General L.. Line20-23(1)
Station18/B20
Synchrono..
~ G
Station19/B19
Station20/B16
Line19-20(2)
Station17/B23
General L..
Line19-20(1)
Line15-16
Station14/B15
Line20-23(2)
General L..
Line13-23 General L..
Line14-16
Station16/B13 Line12-23 Line11-13 Line11-14
General L.. G
Station13/B24
~
Synchrono.. 2-Winding..
G ~
Synchrono..
Station11/B12
General L..
Station12/B11
General L..
Line12-13
Station15/B14 Line15-24
2-Winding.. General L.. 2-Winding..
2-Winding..
Station8/B9
2-Winding.. General L..
Line3-9
Station7/B10
Station9/B3
Line6-10
Line8-10
Line8-9
General L..
Station1/B1
Station5/B8
General Load
Line2-4
~ G Station10/B4
Line1-5
Synchrono..
Line7-8
General L..
Station3/B7
Station2/B2 Line1-2
Line2-6
G ~ General L..
Station6/B6
General L..
General L..
Line1-3
Station4/B5
Line4-9
Line5-10
Synchrono..
General L..
G ~
Synchrono..
Fig. 8. Single line diagram of the 24 buses test system.
Shunt/Filter
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1
Voltage magnitude at bus 3 (P.u)
0.9
Loading factor
0.8 Upper part of k–V curve
0.41469 0.63275 0.84051 0.93567 1.0961 1.1544 1.1915
Nose point Lower part of k–V curve
1.1998 1.1706 1.0959
MLP
0.7 0.6
Nose point
0.5 0.4 0.3
P
eig(JRPh)
eig(JRQV)
0.6663 0.7124 0.7083 0.6838 0.5649 0.4371 0.2019
0.6666 0.7126 0.7095 0.6866 0.5754 0.4567 0.2264
6.8430 6.6481 6.0223 5.5244 4.0138 2.8577 1.2158
80.2153 0.7964 1.4336
0.2806 1.2254 2.4608
1.2661 5.1320 10.9890
Z I
0.2
Table 5 Minimum eigenvalue of load flow Jacobian matrices versus loading factor for constant impedance load model.
0.1 0
eig(J)
0
0.2
0.4
0.6
0.8
Loading factor λ
1
1.2
Loading factor
eig(J)
eig(JRPh)
eig(JRQV)
Upper part of k–V curve
0.41093 0.63132 0.8493 0.95375 1.1436 1.2231 1.2865 1.3268
0.6673 0.7145 0.7129 0.6910 0.5842 0.4740 0.2802 0.0632
0.6676 0.7147 0.7141 0.6935 0.5937 0.4914 0.3060 0.0778
6.8495 6.6651 6.0692 5.5997 4.2042 3.1648 1.7289 0.37391
Nose point Lower part of k–V curve
1.3339 1.2958
0.5739 1.1612
0.8301 1.8913
3.5869 8.4827
1.4
Fig. 9. Continuation curve for load bus 3 of the 24 bus test system with constant power P (blue color), constant current I (red color) and constant impedance Z (black color) load models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of the 24 bus test system with three static load models is shown in Fig. 9. MLP boundary and nose point are indicated on the three continuation curves of Fig. 9. As seen, three different continuation curves with different nose points are obtained for constant power, constant current and constant impedance load models. Moreover, in this case, the loadability margin in terms of MLP has its largest value for constant impedance load and its lowest value for constant power load. MLP for constant current load is between MLP values of constant impedance and constant power load models. Minimum eigenvalue of the load flow Jacobian matrices in different values of the loading factor for constant power, constant current and constant impedance load models is shown in Tables 3–5, respectively. The minimum eigenvalue versus loading factor for three static load models is graphically shown in Figs. 10–12, respectively. Despite the previous experiment using the determinant of the load flow Jacobian matrices to evaluate the MLP boundary, in this experiment the minimum eigenvalue of the load flow Jacobians is used for this purpose to further study the other aspects of the MLP boundary. At the MLP boundary the minimum eigenvalue of all three Jacobian matrices reaches to origin and so the Jacobian matrices become singular. As seen from Tables 3–5, the minimum eigenvalue decreases along the continuation curve, like the determinant of the Jacobian matrices. This matter is graphically shown in Figs. 10–12. However, some other important characteristics can be seen from the minimum eigenvalue behaviors. At first, for constant power load model, the minimum eigenvalue of all load flow Jacobian matrices monotonically decreases by increasing the Table 3 Minimum eigenvalue of load flow Jacobian matrices versus loading factor for constant power load model. Loading factor Upper part of k–V curve
0.0854 0.38865 0.82458 0.99625 1.1317
Nose point Lower part of k–V curve
1.1422 1.1421 1.1406 1.1368
eig(J)
eig(JRPh)
eig(JRQV)
0.7993 0.7579 0.6421 0.5325 0.2105
0.7995 0.7591 0.6481 0.5452 0.2348
6.9177 6.3174 4.8403 3.6975 1.2707
0.0062 0.0741 0.0909 0.1782
0.0075 0.0919 0.1135 0.2210
0.0366 0.4325 0.5306 1.0411
loading factor until it reaches to zero at the nose point (for this load model, MLP coincides the nose point). By entering to the lower part of the continuation curve, the minimum eigenvalue becomes negative (Table 3) indicating unstable equilibrium points in this part of the continuation curve for constant power load model. On the other hand, for constant current and constant impedance load models, the minimum eigenvalue of the full load flow Jacobian matrix J and reduced active Jacobian matrix (JRPh) does not decrease monotonically as seen from Tables 4 and 5 and Figs. 11 and 12. The minimum eigenvalue of the reduced reactive Jacobian matrix (JRQV) monotonically decreases even for the constant current and constant impedance load models. Moreover, for these load models, MLP occurs in the upper part of the continuation curve before the nose point (at nose point the minimum eigenvalue becomes negative). Also, this matter is graphically shown in Fig. 9. From Tables 3–5, it can be seen that by proceeding along the lower part of the continuation curve, the minimum eigenvalue of all load flow Jacobian matrices for all three load models becomes more negative (negative number with more absolute value) indicating more instability depth. This characteristic cannot be seen from the determinant of load flow Jacobian matrices, since the determinants show irregular behaviors in the lower part of the continuation curve, especially for the constant current and constant impedance load models. It is noted that the minimum eigenvalue of all three load flow Jacobian matrices show a nonlinear behavior with respect to the loading factor. For instance, while in a loading factor, ith eigenvalue is the minimum one, in another loading factor, jth eigenvalue (j – i) may become the minimum one. This characteristic is also seen in the augmented and reduced Jacobian matrices [12].
5. Conclusion This paper discusses about MLP, an important boundary of voltage stability feasible region. MLP boundary of power system has
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(a)
Full Jacobianmatrix
Minimum eigenvalue
0.8 0.6 0.4 0.2 0 -0.2 0
0.4
0.2
0.6
0.8
1
1.2
1.4
Loading Factor λ
(b)
Reduced Active Jacobian matrix
(c)
Reduced Reactive Jacobian matrix 8
Minimum eigenvalue
Minimum eigenvalue
0. 0.6 0.4 0.2 0 -0.2 0
0.5
1
6 4 2 0 -2 0
1.5
0.5
Loading Factor λ
1
1.5
Loading Factor λ
Fig. 10. Minimum eigenvalue of the load flow Jacobian matrices for constant power load model.
(a)
Full Jacobian matrix
Minimum eigenvalue
1 0. 5 0 -0.5 -1 -1.5 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Loading Factor λ
(b)
Reduced Active Jacobian matrix
(c)
0 -1 -2 -3 0
Reduced Reactive Jacobian matrix 10
Minimum eigenvalue
Minimum eigenvalue
1
5 0 -5 -10 -15
0.5
1
1.5
Loading Factor λ
0
0.5
1
1.5
Loading Factor λ
Fig. 11. Minimum eigenvalue of the load flow Jacobian matrices for constant current load model.
been analyzed by means of modal analysis of load flow Jacobians and its differences with saddle node bifurcation, nose point of continuation curves and load flow divergence point have been discussed. Effect of reactive power limits of generators and different static load models on the MLP border is also evaluated. It has been shown that MLP coincides with the nose point of continuation curve (which is the usual imagination of the MLP boundary) only with constant power load models and without considering reactive
power limits of the power system. However, when reactive power limits of the power system are considered or voltage dependent load models (constant current and constant impedance load models) are included, MLP boundary becomes different from the nose point of the continuation curve (in these cases, MLP can occur in the lower or upper part of the continuation curve). Also, application of the determinant and minimum eigenvalue of the load flow Jacobian matrices as measures to indicate MLP boundary are also
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(a)
Full Jacobian matrix
Minimum eigenvalue
1 0.5 0 -0.5 -1 -1.5 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Loading Factor λ
(b)
0
-1
-2
Reduced Reactive Jacobian matrix
(c) Minimum eigenvalue
Minimum eigenvalue
Reduced Active Jacobian matrix 1
0.5
1
10 5 0 -5 -10
1.5
Loading Factor λ
0
0.5
1
1.5
Loading Factor λ
Fig. 12. Minimum eigenvalue of the load flow Jacobian matrices for constant impedance load model.
shown and discussed. The research work is under way in order to evaluate sensitivity of the MLP boundary with respect to the controllable parameters of the power system, e.g. the voltage set-point of generators and setting of tap changing transformers. In this way, we can develop an optimal power flow (OPF) considering MLP boundary as an objective function, which enhances voltage stability feasible region and loadability of power system. References [1] Kundur P, Paserba J, Ajjarapu V, Anderson G, Bose A, Canizares C. Definition and classification of power system stability. IEEE Trans Power Syst 2004;19(3):1387–401. [2] Li Huawei, Fan Yu, Wu Tao. Impact of load characteristics and low-voltage load shedding on dynamic voltage stability. IEEE CCECE/CCGEI, Ottawa; May 2006. p. 2249–52. [3] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [4] Morison K, Hamadani H, Wang L. Load modeling for voltage stability studies. In: Proceedings of the IEEE power systems conference and exposition (PSCE); 2006. p. 564–8. [5] Cutsem TV, Vournas C. Voltage stability of electric power systems. Boston/ USA: Kluwer Academic Publishers; 1998. [6] Zhang YP, Huang W, Liu ZQ, Yang JY, Cai XL, Zhang JH. Research on the relationship of the singular point for load flow Jacobian matrix and the critical point of voltage collapse. In: Proceeding of the IEEE power engineering society general meeting, vol. 3; 2005. p. 2939–43. [7] Feng Zhihong et al. The static voltage stability analysis methods for many generators power system – singularity decoupled method. In: Proceedings of CSEE, vol. 12. No. 3; 1992. p. 10–18. [8] Feng Zhihong et al. The static voltage stability eigenvalue analysis method for many generators power system. J Tsinghua Univ 1991;31(4):19–27. [9] Sekine Y, Yokoyama A. A static voltage stability index based on multiple load flow solutions. In: Proceedings of bulk power system voltage phenomenavoltage stability and security; May 1989. p. 65–72. [10] Chen H, Wang Y, Zhou R. Transient and voltage stability enhancement via coordinated excitation and UPFC control. IEE Proc Gener Transm Distrib 2001;148(3):201–8. [11] Amjady N. Dynamic voltage security assessment by a neural network based method. Electr Power Syst Res 2003;66(3):215–26. [12] Amjady N, Ansari MR. Small disturbance voltage stability assessment of power systems by modal analysis and dynamic simulation. Int J Energy Convers Manage 2008;49(10):2629–41. [13] Huang GM, Zhao L, Song X. A new bifurcation analysis for power system dynamic voltage stability studies. In: Proceedings of IEEE power engineering society winter meeting, vol. 2; 2002. p. 882–7. [14] Frag ali El-Sheikhi et al. Voltage stability assessment using modal analysis of power systems including flexible ac transmission system (FACTS). In:
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Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Evaluation of the maximum loadability point of power systems considering the effect of static load models Nima Amjady *, Mohammad Hossein Velayati Department of Electrical Engineering, Semnan University, Semnan, Iran
a r t i c l e
i n f o
Article history: Received 29 December 2008 Received in revised form 18 August 2009 Accepted 29 August 2009 Available online 20 September 2009 Keywords: Voltage stability Maximum loadability point (MLP) Load flow Jacobian Static analysis tool
a b s t r a c t Power system stability is an important problem for power system operation. Determination of different stability margins can result in the optimum utilization of power system with minimum risk. Voltage stability is an important subset of power system stability. To correctly analyze the voltage stability of a power system, suitable dynamic models are usually required. However, static analysis tools can give us useful information about long term voltage stability. Especially, maximum loadability point (MLP) of a power system can be effectively estimated by modal analysis of load flow Jacobians. MLP is one of the important boundaries of voltage stability feasible region that loading beyond which is of little practical meaning. In this paper, MLP boundary of power system is analyzed by means of static analysis tools and its differences with the other boundaries of voltage stability, like saddle node bifurcation, are discussed. Effect of reactive power limits of generators and different static load models on the MLP border is also evaluated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [1]. Voltage stability is becoming one of the dominant factors in determining power system operating limits and greatly depending on load characteristics and specific control devices. Load models, overexcitation limits of generators and auto-regulator of transformers will affect voltage stability significantly [2]. For convenience in analysis and for gaining useful insight into the nature of voltage stability problems, it is useful to characterize voltage stability in terms of the large-disturbance and small-disturbance voltage stability [1]. Small-disturbance voltage stability considers the power system’s ability to control voltages after small disturbances, e.g. changes in load. Large-disturbance voltage stability refers to the system’s ability to maintain steady voltages following large disturbances such as system faults, loss of generation, or circuit contingencies [3]. Voltage instability is a complex phenomenon and may occur over a wide time scale depending on the mechanism by which the instability occurs [4]. Therefore, voltage stability may be either a short-term or long-term phenomenon. In shortterm voltage stability, dynamics of fast acting load components such as induction motors, electronically controlled loads, and HVDC converters are considered. The long term voltage stability * Corresponding author. Tel.: +98 021 88797174; fax: +98 021 88880098. E-mail address: [email protected] (N. Amjady). 0196-8904/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.08.026
is characterized by scenarios such as load recovery by the action of on load tap changer or through loads self restoration, delayed corrective control actions such as shunt compensation switching or load shedding. The long term dynamics such as response of power plant controls, boiler dynamics and automatic generation control also affect long term voltage stability [5]. To correctly analyze the voltage stability of a power system, suitable dynamic models are usually required based on nonlinear differential and algebraic equations. However, in many cases, static analysis tools can be used to estimate stability margins, identify factors influencing stability, and screen a wide range of system conditions and a large number of scenarios for long term voltage stability [1]. In the static analysis, it is assumed that the voltage instability is caused by active power or reactive power unbalance [6]. In recent years many research works have been devoted to determine voltage stability margin, which is the margin between the operating point of the power system and the boundaries of voltage stability feasible region [6–13]. One of the important boundaries of this feasible region is the maximum loadability point or MLP. MLP consists of the load flow feasibility boundary at which the load flow Jacobian is singular [10]. Loading beyond the MLP is of little practical meaning. In the literature, several static analysis tools such as decoupled method [7], modal analysis of load flow Jacobians [8,14], multiple load flow solutions [9], continuation and adaptive continuation methods [15–17], particle swarm optimization (PSO) [18] and energy function technique [19–21] have been proposed to determine the MLP boundary of power system.
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In spite of the performed research works in the area, there is a lack of deep analysis of MLP, which is an important boundary of voltage stability feasible region. The main contribution of the paper is providing such an analysis. In this paper, the MLP boundary of voltage stability feasible region is analyzed by the combination of continuation power flow (CPF) method and modal analysis of load flow Jacobians. A detailed point to point analysis of all load flow Jacobian matrices (reduced and unreduced Jacobians) along the continuation curve based on the criteria of determinant and minimum eigenvalue is presented in this paper, not seen in the previous research works on MLP (such as [6,14,17,18]). Moreover, the effect of reactive power limits of generators and different static load models on the MLP border is also evaluated, while, up to the authors’ knowledge, all other research works on MLP only consider constant power load model. Differences between MLP and the other boundaries of voltage stability feasible region, such as SNB (which is usually taken as the voltage collapse point [13]) are also analyzed in the paper. We found important results about the effect of different load models on the MLP boundary and differences between MLP and the other voltage stability feasible region borders. To the best of our knowledge, such a comprehensive analysis of the MLP boundary has been never presented in the previous research works in the area. 2. Load modeling Voltage stability strongly depends on load characteristic [4,22,23]. Dependency of active and reactive powers of loads on voltage is an important major factor to determine the voltage stability status of power system [24]. In [25] a qualitative discussion on the influence of load characteristics on voltage stability has been reported. Load models are classified into two categories: static and dynamic [22]. The static models represent active and reactive powers of load in terms of voltage and frequency as follows:
a a V V x x x0 V0 bV bx V x Q ¼ Q0 x0 V0 P ¼ P0
ð1Þ ð2Þ
where P0 and Q0 stand for the real and reactive powers consumed at a reference voltage V0 and reference frequency x0, respectively. The exponents aV, ax, bV and bx represent the dependency of active and reactive powers on voltage and frequency, respectively. In the voltage stability studies, the dependency of load active and reactive powers on frequency is usually ignored [4,22,23]. So, we reach to:
a V V V0 bV V Q ¼ Q0 V0 P ¼ P0
ð3Þ ð4Þ
The exponents aV and bV depend on the type of load that is being represented, e.g. for constant power load model aV = bV = 0, for constant current load model aV = bV = 1 and for constant impedance load model aV = bV = 2. Another alternative to represent static load models is based on polynomial equations as follows:
" # 2 V V P ¼ P0 ap þ bp þ cp V0 V0 " # 2 V V Q ¼ Q 0 aq þ bq þ cq V0 V0
ð5Þ ð6Þ
where ap, bp and cp indicate percentage of constant impedance, constant current and constant power load models in the active power
(ap + bp + cp = 1). Similarly, aq, bq and cq are defined for reactive power and aq + bq + cq = 1. In other words, the load model of (5 and 6), known as ZIP model [22], is made up of the three different load models. In this paper, the effect of different static load models (constant power, constant current and constant impedance) described above, on the MLP boundary of power system is evaluated. Also, these load models have significant effect on the dynamic evaluation of voltage stability such as determination of Hopf bifurcation [22,23]. 3. Static voltage stability 3.1. Theoretical analysis The small-disturbance voltage stability is a dynamic phenomenon by nature, but the use of steady state analysis tools can provide useful information about it. In steady state voltage stability studies based on the load flow equations it is assumed that all dynamics are died out and all controllers have done their duty. Steady state voltage stability studies investigate long term voltage stability [12], while for the evaluation of short-term voltage stability, dynamic analysis is usually required [1]. The advantage of using algebraic equations compared to differential equations of dynamic studies is the ease of modeling and computation speed. This paper focuses on the small disturbance and long term voltage stability. Continuation method is an efficient approach to evaluate the boundaries of the feasible region of small-disturbance voltage stability. To obtain the continuation curve of power system, a subset of its parameters is changed step by step. This stage of continuation method is known as predictor [3,17] where a known operating point is used to compute direction and amount of changes applied to the system parameters. Then the new operating point is computed by the varied parameters. This stage is known as the corrector [3,17]. Continuation methods can find voltage stability boundaries of the power system such as the MLP boundary which can be found by the static continuation [15–18] and Hopf bifurcation which can be determined by the dynamic continuation [12]. In this paper, continuation power flow (CPF) is used to evaluate the MLP boundary of power system. The power flow equations for a power system can simply be written as:
DP DQ
¼J
" JPh Dh ¼ JQ h DjVj
J PV J QV
#
Dh DjVj
ð7Þ
where h and |V| indicate angle and magnitude of voltage. Matrix J is known as full load flow Jacobian. Two other reduced Jacobian matrices can be found from J as follows:
Dh ¼ J 1 RPh DPjDQ¼0 where JRPh ¼ J Ph
ð8Þ JPV J 1 QV J Q h
DV ¼ J 1 RQV DQ jDP¼0
ð9Þ J Q h J1 Ph J PV .
where JRQV ¼ J QV JRPh and JRQV are reduced active and reactive Jacobian matrices of the system [14,26,27]. To compute JRQV at each operating point, we keep P constant and evaluate voltage stability by considering the incremental relationship between Q and V. Therefore, J 1 RQV is the reduced V–Q Jacobian that its ith diagonal element represents V–Q sensitivity in ith bus [3]. System is voltage stable if all the eigenvalues are positive then V–Q sensitivity is positive for every bus and det(JRQV) is also positive. At the critical point of system voltage stability, one of eigenvalues of JRQV reaches to origin and this matrix becomes singular. At this point, inversion of full Jacobian matrix is not possible [26].
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Full Jacobian matrix, reduced active and reduced reactive Jacobian matrices contain useful information on voltage stability condition of power system. At MLP all of these three Jacobian matrices become singular [26]. In other words, at MLP, two reduced Jacobian matrices contain the same modal information as that of the full Jacobian matrix [26]. This conclusion is valid not only for the critical eigenvector but also for all other eigenvectors. The eigenvectors of load flow Jacobians provide information related to the mechanism of loss of voltage stability [27]. Eigenvector of the full matrix has two set of entries. One set is related to the active power for all buses that named Full Active (FA), and the other related to the reactive power for PQ buses only that named Full Reactive (FR). In the MLP, the active part of the full Jacobian matrix eigenvector can be compared with the eigenvector of the JRPh. Moreover, the reactive part of the full matrix eigenvector can be compared with the eigenvector of the JRQV [26]. The critical eigenvector of JRPh (or JRQV), reveals those buses where active power (reactive power) increments are more detrimental to the system voltage stability. But considerable difference between reduced active Jacobian matrix and reduced reactive Jacobian matrix can be observed. In fact, JRPh contains both PV (generator) and PQ (load) buses, while JRQV contains PQ buses only. Therefore, JRPh offers more information of JRQV. It is noted that for a power system with NPV of PV buses and NPQ of PQ buses JRQV is (NPQ NPQ) and JRPh is (NPQ + NPV) (NPQ + NPV). In real systems voltage instability happens due to a combination of active and reactive power demanding patterns of loads and generators. Therefore both reactive problem and the active problem must be considered altogether [26]. 3.2. MLP and load flow divergence A few research works use the repetitious power flow solutions by increasing the load on the system in some direction and solving the load flow equations at each step until the load flow solution diverges. They supposed this point as the MLP of power system [28,29]. However these techniques have following disadvantages [18]: – The point where load flow solution diverges does not illustrate the maximum loadability point. Note that, divergence of load flow equation is a mathematical failure, while MLP is a physical limitation. Therefore, the divergence of the power flow may be due to mathematical problem in the applied numerical method. – It has also been noticed that the point where the calculations diverge may vary depending on the numerical method which was used. So, the MLP of power system can be different from the load flow divergence point. 3.3. MLP and voltage collapse Voltage collapse typically occurs on power systems, which are heavily loaded, faulted and/or has reactive power shortages. It is the process by which the sequence of events accompanying voltage instability leads to a blackout or abnormally low voltages in a significant part of the power system. Static voltage stability has neglected all the dynamics of generator, excitation system and so on and considered that voltage instability is caused by active and reactive power unbalance. Therefore, the system equations for static voltage stability contain load flow equations only, while the other types of voltage stability problems in power systems such as voltage collapse and oscillatory problems can be analyzed through bifurcation theory [30]. In bifurcation analysis, differential–algebraic equations of power system
Fig. 1. Bifurcation and P–V curve of an inconstant power load.
are usually used to determine the stability boundaries. Note that the system Jacobian (reduced Jacobian or unreduced Jacobian) of a dynamic power system model typically differs from the power flow Jacobian. If the system Jacobian is asymptotically stable, all eigenvalues have negative real parts. On the other hand, all eigenvalues of the load flow Jacobians have positive real parts before occurring MLP. Some earlier works considered nose point of the P–V curve as the voltage collapse point of the power system. However, in the literature it has been shown that in the systems with inconstantpower loads, the real voltage collapse point is the SNB (Saddle Node Bifurcation) of the bifurcation curve (or point B00 on P–V curve in Fig. 1) instead of the nose point of P–V curve (point B0 ) [31]. Only when all the loads are constant-power type, nose point just coincides with the SNB. At SNB, two equilibrium coalesce and then disappear, at this point the reduced and unreduced Jacobian matrices (related to differential–algebraic equations of the power system [12,13]) has a zero eigenvalue. So, the SNB, which can be obtained from bifurcation analysis considering the reduced/unreduced Jacobian, is different from the MLP indicating the point where the load flow Jacobian becomes singular. Hence, the discussion of this subsection and previous one indicate that the MLP boundary of power system (considered in this paper) can be different from the previously considered boundaries for voltage stability such as load flow divergence point and voltage collapse (SNB) point. 4. Numerical results We test the proposed method to determine the MLP boundary of the six bus test system and 24 bus test system. The effect of reactive power limits of generators on the MLP boundary is studied based on the six bus test system and the effect of different static load models on the MLP is evaluated in the 24 bus test system. For continuation power flow (CPF), we use the power system analysis toolbox (PSAT) software package [32]. PSAT is a MATLAB toolbox for electric power system analysis and simulation. All operations can be assessed by means of graphical user interfaces (GUIs). Some of main features of PSAT are: power flow; continuation power flow; optimal power flow; small signal stability analysis; time domain simulation; phasor measurement unit (PMU) placement; and complete graphical user interface. Continuation methods allow us to trace the complete voltage profile (P–V curve) by automatically changing the value of loading parameter, without having to worry about the singularity of the load flow Jacobian matrices. Determinant and eigenvalues of the load flow Jacobian matrices are computed in the MATLAB software package environment [33]. MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where
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problems and solutions are expressed in familiar mathematical notation. MATLAB features a family of add-on application-specific solutions called toolboxes. Toolboxes are comprehensive collections of MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others.
Single line diagram of the six bus test system with three generators, three load buses and 11 transmission lines is shown in Fig. 2. Data of this test system can be obtained from [32]. Fig. 3 shows the variation of voltage magnitude (V) and voltage angle (h) versus loading factor (k) for load buses 4, 5 and 6 considering the reactive power limits of generators. In this experiment, consumption of these buses is increased in proportion to their base load with constant power factor. All load buses 4, 5 and 6 are constant-power type. From Fig. 3, it is seen that the nose point of k–V and k–h curves for each bus occurs at the same loading factor.
Voltage magnitud (P.u)
Fig. 2. Single line diagram of the six bus test system.
1 (3)
0.8 0.6
(2)
0.4 (1)
0.2 0 0
2
4
6
8
10
Loading Factor λ (1):Bus4 (2):Bus 5 (3):Bus6 0
Voltageangle (deg)
(b)
(2)
-10 (3)
-20 (1)
-30 0
2
4
Loading factor Upper part of k–V curve Nose point of k–V curve Lower part of k–V curve
4.1. Results for six bus test system
(a)
Table 1 Determinant of load flow Jacobian matrices versus loading factor with considering reactive power limits of generators.
6
8
10
Loading Factor λ (1):Bus4 (2):Bus 5 (3):Bus6 Fig. 3. (a) k–V curve and (b) k–h curve for load buses in six bus test system with reactive power limits of generators considered.
det(J)
det(JRPh)
det(JRQV)
0
1.23
4.73
3.74
6.054
3.34e1
2.35
1.85
9.602
5.41e3
1.73e1
1.77e1
9.599
4.18e3
1.44e1
1.46e1
1.11e3 2.98e4 2.74e5 6.57e7 3.97e4
5.12e2 1.55e2 1.50e3 3.61e5 2.38e2
4.95e2 1.45e2 1.38e3 3.00e5 2.12e2
9.531 9.477 9.452 9.449 MLP 9.403
The obtained results for determinant of load flow Jacobian matrices, when the loading factor is varied, are shown in Table 1. All determinants of Jacobian matrices of this paper are computed with per unit values for power system quantities. As the CPF proceeds along the continuation curve, the determinants of all Jacobian matrices decrease until they reach to zero. It is noted that by increasing the loading factor from 0 to 9.602 (nose point), the CPF proceeds along the upper part of the continuation curves (Fig. 3). After the nose point, the CPF enters to the lower part of the continuation curves (e.g. the loading factor of 9.599) and proceeds in this part until it reaches to the MLP. In other words, the MLP, which approximately occurs at the loading factor of 9.449, is located at the lower part of the continuation curves (at the loading factor of 9.449, the determinant of all load flow Jacobian matrices is approximately zero). This experiment shows that the MLP boundary can be different from the nose point of continuation curves. Moreover, for the constant power loads, the divergence point of load flow occurs at a point in the upper part of continuation curve. So, the MLP boundary (which is located in the lower part of the continuation curve in this experiment) can be different from the load flow divergence point as well. In Fig. 4, determinant of full load flow Jacobian matrix det(J), determinant of reduced active Jacobian matrix det(JRPh) and determinant of reduced reactive Jacobian matrix det(JRQV) versus loading factor (k) is shown. As seen, the determinant of all three Jacobian matrices decreases with changing the loading factor along the continuation curve and at the MLP boundary reaches to zero. Moreover, from Fig. 4, it can be seen that the nose point of each continuation curve is different from the MLP boundary when reactive power limits of generators are considered. Figs. 5 and 6 show the eigenvalue locus of the three load flow Jacobian matrices at the base loading point (k = 0) and a little after the MLP boundary (k = 9.403, which is the last row of Table 1), respectively. By changing the loading factor k along the continuation curve, one eigenvalue of each load flow Jacobian matrix goes from the right hand side of the complex plane to origin and reaches to it at the MLP boundary. After that the eigenvalue enters to the left hand side of the complex plane as seen from Fig. 6. In Table 2, the determinant of the three load flow Jacobian matrices versus loading factor is shown without considering the reactive power limits of generators. In the table, upper part, nose point and lower part of the continuation curve are shown from top to bottom, respectively. At first, it can be seen that without considering the reactive power limits of generators, the MLP boundary occurs at the higher loading factors (11.168 in Table 2 versus 9.449 in Table 1). Besides, the MLP coincides with the nose point of the continuation curve when the reactive power limits are not considered. This is another important result derived in this paper, which shows the effect of reactive limits of the power system
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2
determinant
(a)
1
Nose Point MLP
0
0 -1
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
7
8
9
10
(b)
determinant
Loading Factor λ 4 2 0 0
1
2
3
4
5
6
Loading Factor λ 4
determinant
(c)
2 0 0
1
2
3
4
5
6
Loading Factor λ Fig. 4. (a) det(J)–k, (b) det(JRPh)–k and (c) det(JRQV)–k with reactive power limits of generators considered.
eig of Full Jacobian matrix
1
5
0.5 Im aginary
Imaginary
eig of Full Jacobian matrix
10
0 -5 -10 0
5
10
15
20
25
0 -0.5 -1 -5
30
0
5
10
eig of Reduced Reactive Jacobian matrix
eig of Reduced Active Jacobian matrix
0.5
0.5
0.5
0.5
-1 10
15
20
Real
0 -0.5 -1 0
10
20
30
Real
Im aginary
1
Im aginary
1
-0.5
20
eig of Reduced Active Jacobian matrix
1 Imaginary
Imaginary
eig of Reduced Reactive Jacobian matrix
1
0
15
Real
Real
0 -0.5 -1 -5
0
5
10
15
0 -0.5 -1 -10
Real
0
10
20
Real
Fig. 5. Eigenvalue locus of the three load flow Jacobian matrices for base loading point (k = 0).
Fig. 6. Eigenvalue locus of the three load flow Jacobian matrices a little after MLP boundary (k = 9.403).
on the situation of MLP boundary based on the continuation curve. In Fig. 7, determinant of the three load flow Jacobian matrices versus loading factor (k) is shown. As seen, the determinant of all three Jacobian matrices decreases with increasing the loading factor and at the MLP boundary reaches to zero. Again, from Fig. 7, it can be seen that the nose point of each continuation curve coincides with the MLP boundary when the reactive power limits of generators are not considered.
Table 2 Determinant of load flow Jacobian matrices versus loading factor without considering reactive power limits of generators. Loading factor
det(J)
det(JRPh)
det(JRQV)
Upper part of k–V curve
0.35 4.899 8.7498
1.24 5.51e1 1.75e1
4.73 3.07 1.63
3.74 2.44 1.27
Nose point of k–V curve Lower part of k–V curve
11.168 MLP 11.1678
1.19e4 4.5e4
4.01e3 1.56e2
2.14e3 8.24e3
4.2. Results for 24 bus test system The 24 bus test system, shown in Fig. 8, consists of 11 generators, 24 buses, five transformers, 33 branches and one static condenser. Data of this test system can be obtained from [32].
For a better illustration of the effect of different load models on the MLP boundary, reactive power limits of the power system are not considered in this experiment. Continuation curve for bus 3
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2
determinant
(a)
0 -1
0
2
4
0
2
4
6
8
10
12
6
8
10
12
8
10
12
Loading Factor λ
5
determinant
(b)
0
1
0
-5
Loading Factor λ
(c) determinant
4 2 0 -2
0
2
4
6
Loading Factor λ Fig. 7. (a) det(J)–k, (b) det(JRPh)–k and (c) det(JRQV)–k without considering reactive power limits of generators.
Line15-21(2) Synchrono.. Line15-21(1)
~ G ~ G
Station23/B18 Line18-21(2)
Synchrono..
Station22/B22 Line17-18
Line18-21(1)
Station24/B17 Synchrono..
G ~
Line21-22
General L..
Station21/B21 Line17-22 Synchrono..
Synchrono..
~ G
Line16-17
Line16-19
~ G
General L.. Line20-23(1)
Station18/B20
Synchrono..
~ G
Station19/B19
Station20/B16
Line19-20(2)
Station17/B23
General L..
Line19-20(1)
Line15-16
Station14/B15
Line20-23(2)
General L..
Line13-23 General L..
Line14-16
Station16/B13 Line12-23 Line11-13 Line11-14
General L.. G
Station13/B24
~
Synchrono.. 2-Winding..
G ~
Synchrono..
Station11/B12
General L..
Station12/B11
General L..
Line12-13
Station15/B14 Line15-24
2-Winding.. General L.. 2-Winding..
2-Winding..
Station8/B9
2-Winding.. General L..
Line3-9
Station7/B10
Station9/B3
Line6-10
Line8-10
Line8-9
General L..
Station1/B1
Station5/B8
General Load
Line2-4
~ G Station10/B4
Line1-5
Synchrono..
Line7-8
General L..
Station3/B7
Station2/B2 Line1-2
Line2-6
G ~ General L..
Station6/B6
General L..
General L..
Line1-3
Station4/B5
Line4-9
Line5-10
Synchrono..
General L..
G ~
Synchrono..
Fig. 8. Single line diagram of the 24 buses test system.
Shunt/Filter
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1
Voltage magnitude at bus 3 (P.u)
0.9
Loading factor
0.8 Upper part of k–V curve
0.41469 0.63275 0.84051 0.93567 1.0961 1.1544 1.1915
Nose point Lower part of k–V curve
1.1998 1.1706 1.0959
MLP
0.7 0.6
Nose point
0.5 0.4 0.3
P
eig(JRPh)
eig(JRQV)
0.6663 0.7124 0.7083 0.6838 0.5649 0.4371 0.2019
0.6666 0.7126 0.7095 0.6866 0.5754 0.4567 0.2264
6.8430 6.6481 6.0223 5.5244 4.0138 2.8577 1.2158
80.2153 0.7964 1.4336
0.2806 1.2254 2.4608
1.2661 5.1320 10.9890
Z I
0.2
Table 5 Minimum eigenvalue of load flow Jacobian matrices versus loading factor for constant impedance load model.
0.1 0
eig(J)
0
0.2
0.4
0.6
0.8
Loading factor λ
1
1.2
Loading factor
eig(J)
eig(JRPh)
eig(JRQV)
Upper part of k–V curve
0.41093 0.63132 0.8493 0.95375 1.1436 1.2231 1.2865 1.3268
0.6673 0.7145 0.7129 0.6910 0.5842 0.4740 0.2802 0.0632
0.6676 0.7147 0.7141 0.6935 0.5937 0.4914 0.3060 0.0778
6.8495 6.6651 6.0692 5.5997 4.2042 3.1648 1.7289 0.37391
Nose point Lower part of k–V curve
1.3339 1.2958
0.5739 1.1612
0.8301 1.8913
3.5869 8.4827
1.4
Fig. 9. Continuation curve for load bus 3 of the 24 bus test system with constant power P (blue color), constant current I (red color) and constant impedance Z (black color) load models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of the 24 bus test system with three static load models is shown in Fig. 9. MLP boundary and nose point are indicated on the three continuation curves of Fig. 9. As seen, three different continuation curves with different nose points are obtained for constant power, constant current and constant impedance load models. Moreover, in this case, the loadability margin in terms of MLP has its largest value for constant impedance load and its lowest value for constant power load. MLP for constant current load is between MLP values of constant impedance and constant power load models. Minimum eigenvalue of the load flow Jacobian matrices in different values of the loading factor for constant power, constant current and constant impedance load models is shown in Tables 3–5, respectively. The minimum eigenvalue versus loading factor for three static load models is graphically shown in Figs. 10–12, respectively. Despite the previous experiment using the determinant of the load flow Jacobian matrices to evaluate the MLP boundary, in this experiment the minimum eigenvalue of the load flow Jacobians is used for this purpose to further study the other aspects of the MLP boundary. At the MLP boundary the minimum eigenvalue of all three Jacobian matrices reaches to origin and so the Jacobian matrices become singular. As seen from Tables 3–5, the minimum eigenvalue decreases along the continuation curve, like the determinant of the Jacobian matrices. This matter is graphically shown in Figs. 10–12. However, some other important characteristics can be seen from the minimum eigenvalue behaviors. At first, for constant power load model, the minimum eigenvalue of all load flow Jacobian matrices monotonically decreases by increasing the Table 3 Minimum eigenvalue of load flow Jacobian matrices versus loading factor for constant power load model. Loading factor Upper part of k–V curve
0.0854 0.38865 0.82458 0.99625 1.1317
Nose point Lower part of k–V curve
1.1422 1.1421 1.1406 1.1368
eig(J)
eig(JRPh)
eig(JRQV)
0.7993 0.7579 0.6421 0.5325 0.2105
0.7995 0.7591 0.6481 0.5452 0.2348
6.9177 6.3174 4.8403 3.6975 1.2707
0.0062 0.0741 0.0909 0.1782
0.0075 0.0919 0.1135 0.2210
0.0366 0.4325 0.5306 1.0411
loading factor until it reaches to zero at the nose point (for this load model, MLP coincides the nose point). By entering to the lower part of the continuation curve, the minimum eigenvalue becomes negative (Table 3) indicating unstable equilibrium points in this part of the continuation curve for constant power load model. On the other hand, for constant current and constant impedance load models, the minimum eigenvalue of the full load flow Jacobian matrix J and reduced active Jacobian matrix (JRPh) does not decrease monotonically as seen from Tables 4 and 5 and Figs. 11 and 12. The minimum eigenvalue of the reduced reactive Jacobian matrix (JRQV) monotonically decreases even for the constant current and constant impedance load models. Moreover, for these load models, MLP occurs in the upper part of the continuation curve before the nose point (at nose point the minimum eigenvalue becomes negative). Also, this matter is graphically shown in Fig. 9. From Tables 3–5, it can be seen that by proceeding along the lower part of the continuation curve, the minimum eigenvalue of all load flow Jacobian matrices for all three load models becomes more negative (negative number with more absolute value) indicating more instability depth. This characteristic cannot be seen from the determinant of load flow Jacobian matrices, since the determinants show irregular behaviors in the lower part of the continuation curve, especially for the constant current and constant impedance load models. It is noted that the minimum eigenvalue of all three load flow Jacobian matrices show a nonlinear behavior with respect to the loading factor. For instance, while in a loading factor, ith eigenvalue is the minimum one, in another loading factor, jth eigenvalue (j – i) may become the minimum one. This characteristic is also seen in the augmented and reduced Jacobian matrices [12].
5. Conclusion This paper discusses about MLP, an important boundary of voltage stability feasible region. MLP boundary of power system has
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(a)
Full Jacobianmatrix
Minimum eigenvalue
0.8 0.6 0.4 0.2 0 -0.2 0
0.4
0.2
0.6
0.8
1
1.2
1.4
Loading Factor λ
(b)
Reduced Active Jacobian matrix
(c)
Reduced Reactive Jacobian matrix 8
Minimum eigenvalue
Minimum eigenvalue
0. 0.6 0.4 0.2 0 -0.2 0
0.5
1
6 4 2 0 -2 0
1.5
0.5
Loading Factor λ
1
1.5
Loading Factor λ
Fig. 10. Minimum eigenvalue of the load flow Jacobian matrices for constant power load model.
(a)
Full Jacobian matrix
Minimum eigenvalue
1 0. 5 0 -0.5 -1 -1.5 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Loading Factor λ
(b)
Reduced Active Jacobian matrix
(c)
0 -1 -2 -3 0
Reduced Reactive Jacobian matrix 10
Minimum eigenvalue
Minimum eigenvalue
1
5 0 -5 -10 -15
0.5
1
1.5
Loading Factor λ
0
0.5
1
1.5
Loading Factor λ
Fig. 11. Minimum eigenvalue of the load flow Jacobian matrices for constant current load model.
been analyzed by means of modal analysis of load flow Jacobians and its differences with saddle node bifurcation, nose point of continuation curves and load flow divergence point have been discussed. Effect of reactive power limits of generators and different static load models on the MLP border is also evaluated. It has been shown that MLP coincides with the nose point of continuation curve (which is the usual imagination of the MLP boundary) only with constant power load models and without considering reactive
power limits of the power system. However, when reactive power limits of the power system are considered or voltage dependent load models (constant current and constant impedance load models) are included, MLP boundary becomes different from the nose point of the continuation curve (in these cases, MLP can occur in the lower or upper part of the continuation curve). Also, application of the determinant and minimum eigenvalue of the load flow Jacobian matrices as measures to indicate MLP boundary are also
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(a)
Full Jacobian matrix
Minimum eigenvalue
1 0.5 0 -0.5 -1 -1.5 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Loading Factor λ
(b)
0
-1
-2
Reduced Reactive Jacobian matrix
(c) Minimum eigenvalue
Minimum eigenvalue
Reduced Active Jacobian matrix 1
0.5
1
10 5 0 -5 -10
1.5
Loading Factor λ
0
0.5
1
1.5
Loading Factor λ
Fig. 12. Minimum eigenvalue of the load flow Jacobian matrices for constant impedance load model.
shown and discussed. The research work is under way in order to evaluate sensitivity of the MLP boundary with respect to the controllable parameters of the power system, e.g. the voltage set-point of generators and setting of tap changing transformers. In this way, we can develop an optimal power flow (OPF) considering MLP boundary as an objective function, which enhances voltage stability feasible region and loadability of power system. References [1] Kundur P, Paserba J, Ajjarapu V, Anderson G, Bose A, Canizares C. Definition and classification of power system stability. IEEE Trans Power Syst 2004;19(3):1387–401. [2] Li Huawei, Fan Yu, Wu Tao. Impact of load characteristics and low-voltage load shedding on dynamic voltage stability. IEEE CCECE/CCGEI, Ottawa; May 2006. p. 2249–52. [3] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [4] Morison K, Hamadani H, Wang L. Load modeling for voltage stability studies. In: Proceedings of the IEEE power systems conference and exposition (PSCE); 2006. p. 564–8. [5] Cutsem TV, Vournas C. Voltage stability of electric power systems. Boston/ USA: Kluwer Academic Publishers; 1998. [6] Zhang YP, Huang W, Liu ZQ, Yang JY, Cai XL, Zhang JH. Research on the relationship of the singular point for load flow Jacobian matrix and the critical point of voltage collapse. In: Proceeding of the IEEE power engineering society general meeting, vol. 3; 2005. p. 2939–43. [7] Feng Zhihong et al. The static voltage stability analysis methods for many generators power system – singularity decoupled method. In: Proceedings of CSEE, vol. 12. No. 3; 1992. p. 10–18. [8] Feng Zhihong et al. The static voltage stability eigenvalue analysis method for many generators power system. J Tsinghua Univ 1991;31(4):19–27. [9] Sekine Y, Yokoyama A. A static voltage stability index based on multiple load flow solutions. In: Proceedings of bulk power system voltage phenomenavoltage stability and security; May 1989. p. 65–72. [10] Chen H, Wang Y, Zhou R. Transient and voltage stability enhancement via coordinated excitation and UPFC control. IEE Proc Gener Transm Distrib 2001;148(3):201–8. [11] Amjady N. Dynamic voltage security assessment by a neural network based method. Electr Power Syst Res 2003;66(3):215–26. [12] Amjady N, Ansari MR. Small disturbance voltage stability assessment of power systems by modal analysis and dynamic simulation. Int J Energy Convers Manage 2008;49(10):2629–41. [13] Huang GM, Zhao L, Song X. A new bifurcation analysis for power system dynamic voltage stability studies. In: Proceedings of IEEE power engineering society winter meeting, vol. 2; 2002. p. 882–7. [14] Frag ali El-Sheikhi et al. Voltage stability assessment using modal analysis of power systems including flexible ac transmission system (FACTS). In:
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