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STATCOM-controller using trajectory sensitivity


Electrical Power and Energy Systems 33 (2011) 531–539

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Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes

Improvement of transient stability of power systems with STATCOM-controller using trajectory sensitivity
Dheeman Chatterjee a, Arindam Ghosh b,?
a b

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur-721 302, India School of Engineering Systems, Queensland University of Technology, Brisbane, Qld 4001, Australia

a r t i c l e

i n f o

a b s t r a c t
This paper discusses the use of trajectory sensitivity analysis (TSA) in determining the transient stability margin of a power system compensated by a shunt FACTS device. The shunt device used is static synchronous compensator (STATCOM). It is shown that TSA can be used for the design of controller for the STATCOM. The preferable locations for the placement of the STATCOM for different fault conditions are also identi?ed. The effects of STATCOM in maintaining different bus voltages in the post-fault condition are studied. The STATCOM is modeled by a voltage source connected to the system through a transformer. The systems used for the study are the WSCC 3-machine 9-bus system and the IEEE 16-machine 68bus system. ? 2010 Elsevier Ltd. All rights reserved.

Article history: Received 7 December 2006 Accepted 6 December 2010 Available online 6 January 2011 Keywords: Trajectory sensitivity analysis Transient stability margin STATCOM

1. Introduction The use of FACTS devices at strategic locations with well-designed controllers can help in improving the operational ef?ciency of power systems. This is very important in view of the increasing competition in electric energy industry caused by the undergoing restructuring and deregulation in different parts of the world. With the increasing demand for electrical power in one hand and environmental and economic constraints on building of new power generation and transmission infrastructure on the other, more ef?cient utilization of the existing system has become important. FACTS controllers can be useful tools to meet that requirement. As systems are pushed to their limits, maintaining stability becomes more dif?cult. This requires improved tools for assessing available stability margins of a system. Trajectory sensitivity analysis (TSA) can be a viable option for the stability assessment in power systems. Transient energy function (TEF) method is a tool used for transient stability assessment. But this becomes increasingly complex when detailed models are considered and FACTS devices along with controllers are included in the system. The computation of controlling unstable equilibrium point (UEP) may pose increased computational problems. The use of TSA as an alternative to avoid this problem has been pointed out in [1]. The use of trajectory sensitivity (TS) in ?nding critical values of parameters and dynamic rescheduling of generation has been discussed in [2]. The tech-

? Corresponding author. Tel.: +61 7 3138 2459; fax: +61 7 3138 1516.
E-mail address: a.ghosh@qut.edu.au (A. Ghosh). 0142-0615/$ - see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.12.005

nique to extend the method for systems with both continuous and discrete equations (hybrid systems) is discussed in [3]. A method for reducing the number of trajectory sensitivity calculations to get the most effective control is described in [4]. The use of TSA technique to investigate the Nordel power grid disturbance of January 1, 1997 is discussed in [5]. Application of TSA in transmission system protection to detect unstable power swings and electrical centers is described in [6]. The use of TSA in effective application of a TCSC controller in a multi-machine power system is discussed in [7]. Mathematical modeling and analysis of static compensator (STATCOM) is presented in [8]. Transient stability and power ?ow models of different FACTS devices are given in [9]. In this paper the effect of STATCOM in improving the transient stability condition of a power system is investigated. The sensitivities of state trajectories are used here to assess system stability margin. Fault in one of the lines is simulated as a contingency, which de?nes the nominal trajectory. Sensitivity is computed with respect to fault clearing time. Design of the STATCOM controller is carried out with the help of TSA. The effects of placement of STATCOM controllers at various locations of a power system on the transient stability are studied. Also the improvements in the post-fault steady state voltages at different load buses on application of the STATCOM are studied. The STATCOM is represented here by a voltage source, which is connected to the system through a coupling transformer. The voltage of the source is in phase with the AC system voltage at the point of connection and the magnitude of the voltage is controllable. The systems under consideration are the WSCC 3-machine 9-bus system and the IEEE 16-machine 68-bus system. Load is modeled as constant impedance.

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2. Multi-machine power system model and trajectory sensitivity 2.1. Model of the multi-machine power system In this paper, the synchronous machines are represented by the ?ux-decay model as shown in Fig. 1a [10]. A simpli?ed static exciter model with one gain and one time constant is considered, as in Fig. 1b. The internal voltage of the generator and its angle, Ei are given by

where Gij and Bij are the network transfer conductance and admittance respectively. These are obtained from the augmented YBUS matrix where the admittance corresponding to the transient reactance of the machines are included along with normal YBUS. PLi and QLi are the real and reactive powers loads respectively at the ith bus. 2.2. Trajectory sensitivity analysis Multi-machine power systems are generally modeled by differential and algebraic equations (DAEs) as described in Section 2.1. Let such a system be represented by

Ei \/i ?

  ! x0d 0 1 ? i V i sin?di ? hi ? ? jEqi ej?di ?p=2? xq i

?1?

_ x ? f ?x; y; k?; 0 ? g?x; y; k?;

x?t 0 ? ? x0 y?t0 ? ? y0

?9?

The generator and exciter dynamics are described by the following equations [10,11]

ddi ? xs Dxri ; dt 2Hi

i ? 1; . . . ; m

?2?

dDxri ? Pmi ? Pei ? K Di Dxri ; i ? 1; . . . ; m dt ! 0 dEqi xd i 0 xdi 0 T doi ? ? 0 Eqi ? 0 ? 1 V i cos?di ? hi ? ? Efdi ; dt xd i xdi i ? 1; . . . ; m T Ai dEfdi ? ?Efdi ? ?V refi ? V i ?K Ai ; dt i ? 1; . . . ; m

where x is the state vector, y is a vector of algebraic variables and k is a vector of system parameters. The sensitivities of state trajectories with respect to system parameters can be found by perturbing k from its nominal value k0. The equations for trajectory sensitivity can be found as [1]

?3?

_ w1 ? 0

h i h i @f @f w1 ? @y w2 ? @k ; h i ?@g? ?@g? ? @x w1 ? @g w2 ? @k ; @y
@f @x

h i

w1 ?t0 ? ? 0 ?10? w2 ?t 0 ? ? 0

?4? ?5?

Pei ? E0qi V i sin?di ? hi ?=x0di ? 0:5?1=xq ? 1=x0di ?V 2 sin 2?di ? hi ?; i i ? 1; ::::; m ?6?
where, d is the angular position of the rotor, xr is the rotor speed, Dxr per unit speed deviation, xs is the synchronous speed, m is the number of machines, H is the inertia constant, KD is the damping coef?cient, Pm is the mechanical power input, Pe is the electrical power output, xd and xq are the d-axis and q-axis synchronous reactance, x0d is the d-axis transient reactance, T 0do is the d-axis open circuit time constant, KA and TA are the gain and time constant of the exciter, V is the terminal voltage of the machine in per unit and h is its angle. The dynamics of the network and the stator windings are neglected and the network is represented by a set of algebraic equations for i = 1, . . ., n.

where w1 = ox/ok and w2 = ox/ok. Solving (9) and (10) simultaneously, we get x, y and the sensitivities w1 and w2. However, the sensitivities can be found in a simpler way by using numerical method. Let us choose only one parameter, i.e., k becomes a scalar and the sensitivities with respect to it are studied. Two values of k are chosen (say k1 and k2). The corresponding state vectors x1 and x2 respectively are then computed. Now the sensitivity is de?ned as

Sens ?

x2 ? x1 Dx ? k2 ? k1 Dk

?11?

P Li ?

n?m X j?1

? ? jV i jjV j j Gij cos?hi ? hj ? ? Bij sin?hi ? hj ?

?7?

If Dk is small, the numerical sensitivity is expected to be very near to the analytically calculated trajectory sensitivity value. In this paper the trajectory sensitivity is calculated numerically. In case of power systems, sensitivity of state variables, e.g., the generator rotor angle (d) and per unit speed deviation (Dxr) can be computed as in (11) with respect to some parameter k. These sensitivities give us information about the effect of change of parameters on individual state variables and hence on the generators to which the particular state variable is associated. However to know the overall system condition, we need to sum up all these effects of parameter change on individual generators. To achieve this goal, the norm of the sensitivities is ?rst calculated. The sensitivity norm for the m-machine system of Section 2.1 is given as

Q Li ?

n?m X j?1

jV i jjV j j?Gij sin?hi ? hj ? ? Bij cos?hi ? hj ??

?8?

v????????????????????????????????????????????????????????????????????? # u m " uX @di @dj 2 @ Dxr 2 i t SN ? ? ? @a @a @a i?1

?12?

Fig. 1. (a) Flux-decay model of generator; and (b) static exciter model.

D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539

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where the jth machine is taken as the reference. Then a new term g (ETA) is introduced [2], which is de?ned as

g?

1 max?SN ?

?13?

STATCOM behaves like a constant current source. To incorporate this feature in the simulation, Vsc is kept constant at the pre-speci?ed value when Isc 6 Imax. But whenever the value of Isc exceeds Imax, Vsc is adjusted to make Isc equal to Imax. 3.2. The control scheme A controller is employed along with the STATCOM which determines the value of Vsc and hence decides the amount and direction of compensation as per Eqs. (14) and (15). The block diagram of the control scheme [12] is shown in Fig. 3. The mid-point voltage (Vm) of the line (to which the STATCOM is connected) is taken as the control variable. It is compared with the reference value of the mid-point voltage V ? . The error is integrated to get the STATCOM ref current reference, Iref. A PI controller is used to realize this. The input to the PI is the error and the output of the PI is Iref. As in the open loop case, a maximum limit (Imax) is set on the value of Iref and this is realized by passing the signal Iref through a limiter. The output of this limiter is used to calculate the value of Vsc using Eq. (16) with Isc replaced by Iref. The calculated value of Vsc is then included in the system simulation. The voltage reference V ? is obref tained by adding a signal DV to a ?xed voltage reference Vref. The signal DV is derived from the rate of change of transmitted power (dP/dt) through the line to which the STATCOM is connected [12]. The dP/dt is measured and fed through a proportional controller to get DV. Another important issue to be addressed is the assignment of Vref. In this study, the pre-fault steady state value of line mid-point voltage (denoted by Vm0) is taken as Vref. However, Vm0 is again dependant on the value of Vsc in the pre-fault steady state condition (denoted by Vsc0). Therefore a suitable value of Vsc0 is decided ?rst. The corresponding value of Vm0 is then computed using load ?ow. This is used as Vref in the control loop. 4. Choice of STATCOM location and controller parameters 4.1. The systems under study The systems studied in this paper are the WSCC 3-machine 9bus system and the IEEE 16-machine 68-bus system shown in Fig. 4 and 5 respectively. A 3-phase fault is simulated in one of the lines of the systems. The simulation is done in three steps. At ?rst, the pre-fault system is run for a small time. Then a symmetrical fault is applied at one end of a line. Simulation of the faulted condition continues till the fault is cleared after a time tcl. Then the post-fault system is simulated for a longer time (say 5–10 s) to observe the nature of the transients. 4.2. Variation of TS with fault clearing time in uncompensated system At ?rst, the uncompensated system (i.e. without any STATCOM) is considered. A fault is simulated in different lines of the two systems. The fault is considered to take place near a load bus and may be of self-clearing type or may have to be cleared by isolating the faulty line. To start with, the fault duration is assumed to be 0.05 s. The trajectory sensitivity and g are computed for all the cases. Then the value of tcl is increased in steps and the TS and g are computed again. These values of g for the uncompensated system are denoted as g0. The values of g0 for faults near different load buses (bus 5, 6 and 8) of the 9-bus system are shown in Table 1. Similarly for the 68-bus system, g0 for faults near load buses 16, 24, 7, 4, 12, 33 and 1 are shown. The fault durations considered are 0.10 s, 0.15 s and 0.25 s for the 9-bus system and 0.05 s, 0.10 s and 0.15 s for the 68-bus system. g0 are shown for both self-clearing fault (SCF) and fault cleared by line isolation. It can be observed from the table that the value of g0 decreases with increase in clear-

As the system moves towards instability, the oscillation in TS will be more resulting in larger values of SN. This will make g small. Ideally g should be zero at the point of instability. Therefore the value of g gives an indication of the distance from instability. In this paper TS with respect to fault clearing time and the corresponding g has been used for assessing the relative stability conditions of the system. 3. Modeling of the STATCOM-controller 3.1. Model of the STATCOM The STATCOM is modeled by a voltage source connected to the power system through a coupling transformer. The voltage of the source is the output of a voltage-sourced converter realizing the STATCOM. As shown in Fig. 2, the STATCOM connection is at the mid-point of a transmission line. The phase angle of the voltage of the voltage-sourced converter is the same as that of the mid-point voltage [12]. This ensures that there is exchange of only reactive power and no real power between the STATCOM and the ac system. The expressions for the current ?owing from the STATCOM to the system and the reactive power injection are given as

Isc \w ?

?V sc ? V m ?\hm jX l ?V sc =V m ? 1? Xl

?14?

Q sc ? V 2 ? m

?15?

where Vsc is the magnitude of the voltage of the voltage-sourced converter and Vm is the magnitude of the voltage at the mid-point of the line, hm is the phase angle of the mid-point voltage and Xl is the leakage reactance of the coupling transformer. Eq. (14) can be rewritten as

Isc ?

?V sc ? V m ? Xl

and w ? ?hm ? p=2?

?16?

The leakage reactance of the coupling transformer is taken to be 0.1 p.u. We can see that Vsc determines the direction of the reactive power ?ow. When the STATCOM is used in open loop, i.e. without any controller, the value of Vsc is taken to be constant throughout the simulation. However, this may result in a very high value of STATCOM current (Isc) especially during fault conditions, which makes the simulation unrealistic. To avoid this a maximum limit is set for the STATCOM current, denoted by Imax. In a practical system, this current limit is decided by the rating of the STATCOM. Once this limit is hit, i.e. the STATCOM current reaches Imax, the

Fig. 2. Representation of STATCOM.

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D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539

Fig. 3. Block diagram of control scheme.

4.3. Effect of STATCOM placement in open loop Next, the effect of placement of the STATCOM in open loop (i.e. without any controller) is studied. All the fault locations shown in Table 1 are considered for the study. The value of tcl is taken as 0.15 s for the 9-bus system and 0.10 s for the 68-bus system. The values of Vm0 of different lines (without STATCOM) of the 9-bus system are found to range between 1.0087 p.u. and 1.0244 p.u. Therefore, to ensure reactive power injection by the STATCOM in the steady state, the value of Vsc0 is taken to be 1.05 p.u. (>1.0244 p.u.). Similarly for the 68-bus system, the values of Vm0 of the different lines considered are between 0.999 p.u. and 1.059 p.u. Therefore, the value of Vsc0 is taken to be 1.06 p.u. (>1.059 p.u.). STATCOM is placed in different locations (except the faulty line) and the TS and g are computed for each of the cases. These g values are normalized with respect to the corresponding g0. Therefore, a normalized g value more than 1.0 indicates improvement in stability and vice versa. Table 2A shows the normalized g for different STATCOM locations in case of (i) fault near bus 6 of the 9-bus system and (ii) fault near bus 7 of the 68-bus system. It can be seen from Table 2A that STATCOM causes improvement in stability in most of the cases (as indicated by normalized g > 1). Naturally, those locations of STATCOM are preferable (from transient stability improvement point of view) for which the value

Fig. 4. Single line diagram of the WSCC 9-bus system.

ing time for all the fault locations. As per the discussion of Section 2.2, this indicates deterioration in stability. It is in accordance with the expectation that the stability condition of a power system would deteriorate as the fault duration is increased. This shows the effectiveness of TS and g as a stability margin indicator.

Fig. 5. Single line diagram of the 16-machine 68-bus system.

D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539 Table 1 Value of g0 for different fault location and fault type. System 3-machine 9-bus system Self-clearing fault Faulty bus Fault clearing time 0.10 s 5 6 8 16-machine 68-bus system Faulty bus 0.1636 0.1687 0.1310 0.15 s 0.1420 0.1423 0.1140 0.25 s 0.1024 0.1092 0.0611 5–4 5–7 6–4 6–9 8–7 8–9 Fault line 0.15 s 0.0397 0.0583 0.0837 0.0693 0.1631 0.0870 0.1235 16–21 24–16 7–6 4–5 12–11 33–34 33–32 1–31 Fault cleared by line isolation Faulty line Fault clearing time 0.10 s 0.1488 0.1301 0.1466 0.1411 0.1254 0.1354 0.15 s 0.1333 0.1285 0.1296 0.1335 0.1106 0.1185 0.25 s

535

0.1148 0.1014 0.1111 0.1300 0.0982 0.0522

Fault clearing time 0.05 s 0.10 s 0.0575 0.0704 0.0947 0.0794 0.1681 0.1439 0.1347

Fault clearing time 0.05 s 0.0618 0.0773 0.0992 0.0879 0.1727 0.1608 0.1507 0.1337 0.10 s 0.0528 0.0705 0.0927 0.0818 0.1682 0.1340 0.1265 0.1307 0.15 s 0.0239 0.0584 0.0814 0.0722 0.1632 0.0984 0.0525 0.1232

16 24 7 4 12 33 1

0.0653 0.0772 0.1014 0.0854 0.1726 0.1794 0.1368

Table 2A Variation of normalized g with STATCOM location. 9-bus system STATCOM in line Fault near bus 6 SCF 5–4 5–7 6–4 6–9 8–7 8–9 1.0084 1.0830 0.9796 1.0134 1.1377 1.2136 Lines 6–4 isolated 1.0216 1.1049 – 0.9846 1.1590 1.2932 4–5 4–14 5–6 5–8 6–11 7–8 8–9 10–11 10–13 13–14 14–15 68-bus system STATCOM in line Fault near bus 7 SCF 1.0084 1.0127 1.0275 1.0201 1.0285 1.0169 1.0169 1.0422 1.0412 1.0327 1.0370 Lines 7–6 isolated 1.0129 1.0173 1.0324 1.0248 1.0334 1.0205 1.0173 1.0464 1.0464 1.0378 1.0410

of g are higher. For example, in case of a fault near bus 6 of the 9bus system, the preferable placement location (PPL) for STATCOM is lines 8–9 for both types of fault. Similarly, for a fault near bus 7 of the 68-bus system, the PPL are lines 10–11 and 10–13 for both types of fault. The PPL for other fault locations are also identi?ed and are shown in Table 2B. We can see that the effectiveness of the STATCOM varies with the fault location, resulting in different PPL. Therefore, it is dif?cult to identify a single location for the FACTS device such that when it is placed in that particular location, it will most effectively counter instability due to fault in any corner of the system. However, if one or a few critical fault locations are known where the fault is most likely to occur and/or where the fault makes the system most vul-

nerable to instability, then one can use this TSA based method to ?nd out the best possible location for the device. In practical systems, the operators and planners do have this sort of information (regarding fault prone lines) and hence this method can be very useful. 4.4. Application of the STATCOM with controller The STATCOM is used to improve transient stability condition of the system as well as to maintain the control variable (line midpoint voltage) at the pre-speci?ed value in the post-fault steady state. However the effects of the STATCOM depends largely on the proper functioning of the controller. Therefore choice of suit-

Table 2B Preferrable STATCOM placement locations for different fault locations and fault types. 9-bus system Faulty bus 5 6 8 PPL (line) 6–9 8–9 6–9 Faulty line 5–4 5–7 6–4 6–9 8–7 8–9 PPL (line) 8–7 8–7 8–9 8–9 6–9 6–9 68-bus system Faulty bus 7 4 12 16 24 33 1 PPL (line) 10–11 14–15 5–6 17–27 16–17 30–32 30–31 Faulty line 7–6 4–5 12–11 16–21 24–16 33–34 33–32 1–31 PPL (line) 10–11, 10–13 14–15 5–6 23–24 16–17 30–32 30–32 30–31

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D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539

able values of controller constants KP and KI is very important. As shown in the previous subsection, TS and g gives us a fair idea about the transient stability margin of a system. Therefore g is used here as a metric for the choice of the controller parameters. Let us start with the case of a fault in lines 16–21 of the 68-bus system, which is cleared by isolating the line. The STATCOM is placed in lines 23–24, which is identi?ed as the PPL for this fault location (Table 2B). Trajectory sensitivity and normalized g are computed for different combinations of KP and KI. A 3-dimensional plot of variation of normalized g with the controller constants is shown in Fig. 6. For the same fault and STATCOM locations, the variation of normalized g with KI for three different values of KP (KP = 0.1, 0.5 and 1.0) are shown in Fig. 7a. It can be observed from the two ?gures that higher values of g can be obtained with the controller constant combinations (KP = 0.1 and 9 6 KI 6 10). Similarly Fig. 7b shows the variation of normalized g with KI for a fault in lines 8–9 of the 9-bus system. The STATCOM is in lines 6–9, the PPL for this case. Here plots are shown for KP = 0.2, 0.5 and 0.8. Higher values of g can be obtained in this case with controller constant combinations (KP = 0.8 and KI = 11) or (KP = 0.5 and KI = 10). Hence the STATCOM will be most effective in transient stability improvement of the system when the controller constants are in these ranges. Table 3 shows the most suitable combination of KP

and KI values found by this study for fault (followed by line isolation) at different lines of the two systems with the STATCOM placed in the corresponding PPL. Next, we consider the cases of self-clearing fault. As before, normalized g values for different KP and KI are found. The variation of normalized g with KI for different values of KP (KP = 2, 1 and 0.2) are shown in Fig. 7c for a fault near bus 33 of the 68-bus system with the STATCOM in lines 30–32. It can be seen that the suitable combinations of constants in this case are (KP = 2, KI = 15). Similarly Fig. 7d shows the variation of normalized g with KI for a fault near bus 6 of the 9-bus system. Here the STATCOM is in lines 8–9. Plots are shown for KP = 0.2, 0.5 and 0.8. The best possible combinations of constants in this case are (KP = 0.8, KI = 13) and (KP = 0.5, KI = 12). The most suitable combination of KP and KI values for self-clearing faults at different buses of the two systems with the STATCOM placed in the corresponding PPL are given in Table 3. 4.5. Choice of controller constants suitable for both type of fault A method to choose the STATCOM controller constants for transient stability improvement has been described in the previous subsection. The constants are chosen for either self-clearing fault or for fault cleared by line isolation. However, the type of the fault is generally not known beforehand. Therefore the controller needs to be designed such that it can serve both the cases. This choice can be made by a compromise of the results for the two types of faults. A new term called Combination Index (C.I.) is invoked for this purpose. Suppose, for a particular combination of KP and KI, gscf and gf are the g values in case of self-clearing fault and fault cleared by line isolation respectively. Then C.I. is de?ned as the arithmetic mean of gscf and gf.

C:I: ?

?gscf ? gf ? 2

Fig. 6. Normalized g with controller constants; fault in lines 16–21, STATCOM in lines 23–24 of 68-bus system.

C.I. is computed for different combinations of KP and KI and used to make the choice of constants. Earlier, that combination of KP and KI was being considered as most suitable for which the value of g was found to be maximum. Now, in this case, that combination of KP and KI is considered suitable for which C.I. is maximum. Let us denote these as combined constants. Plot of variation of C.I. with KI for a particular KP is shown in Fig. 8a along with the corresponding plots of normalized g for both types of faults. The fault is in lines 16–21 and the STATCOM is in lines 23–24 of the 68-bus system. KP is chosen as 0.1. Fig. 8b shows the plots of variation of C.I.

Fig. 7. Variation of normalized g with KI for different KP; (a) fault in lines 16–21, 68-bus system (b) fault in lines 8–9, 9-bus system, (c) SCF in bus 33, 68-bus system, (d) SCF in bus 6, 9-bus system; STATCOM in the corresponding PPL as per Table 2B.

D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539 Table 3 Suitable controller constants and corresponding normalized g for different fault and STATCOM locations. System STATCOM in line Self-clearing fault Fault bus 9-bus system 6–9 8–9 68-bus system 10–11 10–13 23–24 17–27 30–32 8 6 7 7 16 16 33 (KP, KI) 0.8, 12.0 0.8, 13.0 1.8, 1.8, 1.2, 2.0, 2.0, 7.0 11.0 11.5 11.0 15.0 Fault cleared by line isolation

537

g
1.0325 1.1138 1.0243 1.0243 1.0174 1.0157 1.0250

Faulty line 8–7 8–9 6–4 6–9 7–6 7–6 16–21 16–21 33–34

(KP, KI) 0.8, 0.8, 0.8, 0.8, 0.8, 1.8, 0.1, 2.0, 2.0, 3.0 11.0 18.0 18.0 11.0 11.0 10.0 11.0 12.0

g
1.0353 1.0751 1.0895 1.1146 1.0259 1.0270 1.0663 1.0341 1.0343

Fig. 8. Variation of (a) normalized g and C.I. with KI, (b) C.I. with KI for different KP.

with KI for three different values of KP for the same fault and STATCOM locations. From these ?gures, one can clearly identify the combined constants to be KP = 0.1 and KI = 6.00. The same method is carried out for other fault and STATCOM locations also and the combined constants found are shown in Table 4 along with the corresponding normalized g values. The best possible controller constants to serve each type of fault were found separately in the previous subsection (Table 3). A comparison of those with the combined constants, i.e. the controller constants suitable for both type of fault (Table 4) shows the following: (i) In some cases, the individual best possible constants for both types of fault are the same. For example, for a fault in lines

7–6 and STATCOM in lines 10–13 of the 68-bus system, KP = 1.8, KI = 11.0 for both types of fault. Similarly, for fault in lines 16–21 and STATCOM in lines 17–27, KP = 2.0, KI = 11.0 for both types of fault. Naturally the combined constants in these cases are also the same as the individual best possible constants. (ii) The situation is different in some other cases (shown in bold face in Table 4). For a fault in lines 33–34 and STATCOM in lines 30–32 of the 68-bus system, the best possible constants for self-clearing fault are KP = 2.0, KI = 15.0 whereas for fault with line isolation the constants are KP = 2.0, KI = 12.0. The combined constants in this case are KP = 2.0, KI = 12.0, i.e. in the case of self-clearing fault, the combined constants (corresponding g = 1.0229) are different from the individual best possible constant values (corresponding g = 1.0250). Similarly, for a SCF in bus 8 of the 9-bus system with STATCOM in lines 6–9, the combined constants (corresponding g = 1.0325) are found to be different from the individual best possible constant values (corresponding g = 1.0193). (iii) It is clear from the results that in some cases the use of combined constants results in slightly lower value of g and hence a comparatively less improvement of stability in one fault type. However the difference is too small and in exchange the STATCOM controller is being designed to operate under both types of faults, which is more suitable for practical application. 5. Veri?cation of STATCOM performance 5.1. Veri?cation of transient stability condition Sensitivity of state trajectories and g have been computed and used to assess system stability condition. The controller constants are chosen on the basis of this assessment. To verify these results,

Table 4 Controller constants suitable for both types of fault and corresponding normalized g for different fault locations. System STATCOM in line Self-clearing fault Fault bus 9-bus system 6–9 8–9 68-bus system 10–11 10–13 23–24 17–27 30–32 8 8 6 6 7 7 16 16 33 (KP, KI) 0.8, 3.0 0.8, 12.0 0.8, 18.0 0.8, 18.0 1.8, 7.0 1.8, 11.0 0.1, 10.0 2.0, 11.0 2.0,12.0 Fault cleared by line isolation

g
1.0193 1.0325 1.1068 1.1068 1.0243 1.0243 1.0139 1.0157 1.0229

Faulty line 8–7 8–9 6–4 6–9 7–6 7–6 16–21 16–21 33–34

(KP, KI) 0.8, 3.0 0.8, 12.0 0.8, 18.0 0.8, 18.0 1.8, 7.0 1.8, 11.0 0.1, 10.0 2.0, 11.0 2.0,12.0

g
1.0353 1.0743 1.0895 1.1146 1.0237 1.0270 1.0663 1.0341 1.0343

538

D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539

let us check the actual stability condition of the system. At ?rst, the system without STATCOM is considered. The fault clearing time is increased till the system becomes unstable. Then the STATCOM is placed at the preferable location with controller constants chosen from Table 4. For example, for a fault in lines 33–34 of the 68bus system, system is found to become unstable for tcl = 0.21 s. However, when a STATCOM is placed in lines 30–32 (the corresponding PPL) with chosen controller constants the system retains stability for the same tcl. The plots of relative rotor angle delta 11– 16 for the two cases (with and without STATCOM) are shown in Fig. 9a. It can be seen that delta 11–16 diverges and becomes unbounded in case of system without STATCOM, whereas it remains bounded and stable when STATCOM is placed in the system. Similarly, Fig. 9b shows the plots of delta 3–1 for a fault in lines 6–9 of the 9-bus system for the system without STATCOM and system with STATCOM in lines 8–9. The value of tcl is 0.42 s in both the cases. Here also, the system with STATCOM retains stability.

5.2. Effect of STATCOM on bus voltages The primary goal of a power system operator is to maintain stability following a disturbance. However, a fault and subsequent disconnection of the corresponding line often results in voltage sag at or near the fault location even in the post-fault stable system. The use of a STATCOM-controller may help to overcome this problem also. As described in Section 4.2, the STATCOM is controlled in such a way that it should hold the voltage of its point of connection with the system (i.e. line mid-point) at the pre-fault value. Fig. 10a shows the variation of the mid-point voltage of lines 10–13 for the uncompensated system and for the system with STATCOM in that line. The fault is in lines 7–6 (of the 68-bus system) in both the cases. Fig. 10b shows the same for a fault in lines 8–7 of the 9-bus system. Here the STATCOM is placed in lines 6–9. The pre-fault voltages are extended as a dashed straight line in both the ?gures for comparison. It can be observed that in both the cases the post-fault voltages are lower than the pre-fault voltages for uncompensated system whereas this fall of voltages are restricted when a STATCOM is connected at the line mid-point. Another important point is the variation of voltages at different load buses of the system. Table 5 shows the post-fault steady state voltage magnitudes at some of the load buses of the two systems

Fig. 10. Variation of line mid-point voltage in an uncompensated system and a compensated system for a fault in (a) lines 7–6 of the 68-bus system and (b) lines 8–7 of the 9-bus system.

with and without compensation. For the 9-bus system, voltages of buses 5, 6 and 8 are shown. For the 68-bus system, voltages of buses 15, 16, 18, 21, 24 are shown for a fault in lines 16–21 and voltages of buses 4, 7, 9, 12 are shown for a fault in lines 7–6. The pre-fault voltage is also shown for comparison. It can be observed that the application of STATCOM reduces the amount of post-fault voltage dip from the uncompensated system in all the cases.

6. Discussions and conclusions Assessment of transient stability condition of a multi-machine power system compensated by FACTS-controllers becomes complex when energy function method is used. This paper presents trajectory sensitivity analysis as a suitable alternative to assess stability condition of STATCOM compensated power systems. The effects of the STATCOM-controller on the transient stability condition of a power system are studied for different fault and compensator locations. The inverse of the maximum value of the norm of the sensitivities (g) of state variables like generator rotor angle and rotor speed deviation is used as a measure of transient stability margin. The choice of the controller constants is very important for proper functioning of the STATCOM. Trajectory sensitivity has been used to make that choice. A fault in a power system can be either of self-clearing type or it is cleared by line isolation. The suitable controller parameters for these two types of faults may be different. A technique has been discussed to choose optimized controller parameters so that the STATCOM can be effective for both types of faults. The transient stability margin of a power system is found to improve in most of the cases with the application of a STATCOM with suitably chosen controller constants. A system without STATCOM becomes unstable when the fault duration exceeds a certain value. However, STATCOM in suitable location with parameters chosen using TS may help the system to retain stability even beyond that fault clearing time. In addition to stability improvement, the controller is also found to maintain the post-fault steady state voltage of the point of connection (i.e. line mid-point) at the pre-fault value. The dip in load bus voltages in post-fault conditions are also restricted by the application of the STATCOM.

Fig. 9. Plots of relative rotor angles (a) delta 11–16 for a fault (with tcl = 0.21 s) in lines 33–34 of the 68-bus system, and (b) delta 3–1 for a fault (with tcl = 0.42 s) in lines 6–9 of the 9-bus system.

D. Chatterjee, A. Ghosh / Electrical Power and Energy Systems 33 (2011) 531–539 Table 5 Variation of load bus voltages with fault and STATCOM locations. Fault in lines 6–4, STATCOM in lines 8–9 Load bus Pre-fault voltage Post-fault voltage Without STATCOM 5 6 8 0.9958 1.0128 1.0162 0.9977 0.9552 1.0066 With STATCOM 1.0022 0.9705 1.0242 5 6 8 0.9958 1.0128 1.0162 Fault in lines 8–7, STATCOM in lines 6–9 Load bus Pre-fault voltage Post-fault voltage Without STATCOM 0.9750 0.9986 0.9694

539

With STATCOM 0.9798 1.0184 0.9862

Fault in lines 16–21, STATCOM in 23–24 Load bus Pre-fault voltage Post-fault voltage Without STATCOM 15 16 18 21 24 1.0171 1.0334 1.0338 1.0325 1.0386 1.0100 1.0250 1.0285 1.0294 1.0264 With STATCOM 1.0156 1.0318 1.0325 1.0354 1.0365

Fault in lines 7–6, STATCOM in lines 10–13 Load bus Pre-fault voltage Post-fault voltage Without STATCOM 4 7 9 12 1.0059 0.9995 1.0391 1.0553 1.0024 0.9776 1.0353 1.0532 With STATCOM 1.0075 0.9834 1.0366 1.0609

Acknowledgement The authors are thankful to Prof. M.A. Pai of University of Illinois at Urbana-Champaign for his valuable suggestions and constructive criticism on the paper. References
[1] Laufenberg MJ, Pai MA. A new approach to dynamic security assessment using trajectory sensitivities. IEEE Trans Power Syst 1998;13(3):953–8. [2] Pai MA, Nguyen TB. Trajectory sensitivity theory in nonlinear dynamical systems: some power system application. Stability and control of dynamical systems with applications: A tribute to Anthony N. Michel, D. Liu and P. J. Antsaklis, Eds. Birkhauser Boston; 2003. [3] Hiskens IA, Pai MA. Trajectory sensitivity analysis of hybrid systems. IEEE Trans Circ Syst – Part 1: Fund Theory Appl 2000;47:204–20. [4] Hiskens IA, Akke Magnus. Analysis of the Nordel power grid disturbance of January 1, 1997 using trajectory sensitivities. IEEE Trans Power Syst 1999;14(3):987–94.

[5] Shubhanga KN, Kulkarni AM. Determination of effectiveness of transient stability controls using reduced number of trajectory sensitivity computations. IEEE Trans Power Syst 2004;19(1):473–82. [6] Soman SA, Nguyen TB, Pai MA, Vaidyanathan Rajani. Analysis of angle stability problems, a transmission protection systems perspective. IEEE Trans Power Deliver 2004;19(3). [7] Chatterjee D, Ghosh A. TCSC control design for transient stability improvement of a multi-machine power system using trajectory sensitivity. accepted for publication in Electric Power System Research. [8] Padiyar KR, Lakshmi Devi A. Control and simulation of static condenser. In: Proceedings of the 9th annual applied power electronics conference and exposition. 13–17th Febraury; 1994. [9] Canizares CA. Power ?ow and transient stability models of FACTS controllers for voltage and angle stability studies. In: Proceedings of IEEE PES winter meeting Singapore. 2 (January); 2000. p. 1447–1452, 23–27. [10] Sauer PW, Pai MA. Power system dynamics and stability. Indian branch, Delhi, India: Pearson Education (Singapore) Pvt. Ltd., [11] Padiyar KR. Power system dynamics, stability and control. Hyderabad: BS Publications; 2002. [12] Hingorani NG, Gyugyi L. Understanding FACTS. Delhi: Standard Publishers Distributors; 2001.


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