Flexible AC Transmission Systems6[1]

7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems This chapter discusses the recent developments in steady state unbalanced three phase voltage stability analysis and control with FACTS. The objectives of this chapter are: 1. to review steady state voltage stability analysis methods in unbalanced threephase power systems; 2. to introduce the continuation three-phase power flow technique that can be used for steady state unbalanced three-phase voltage stability analysis; 3. to examine the PV curves of unbalanced three-phase power systems; 4. to reveal the interesting phenomena of voltage stability of unbalanced threephase power systems; 5. to investigate the impact of FACTS controls on voltage stability limit of unbalanced three-phase power systems. 7.1 Steady State Unbalanced Three-Phase Power System Voltage Stability Voltage stability has been recognized as a very important issue for operating power systems when the continuous load increase along with economic and environmental constraints has led to systems to operate close to their limits including voltage stability limit. In the past, various methodologies have been proposed for voltage stability analysis [1]-[4]. Among the voltage stability analysis methods, the continuation power flow methods have been considered as one of the useful tools [5]-[11]. However, in the literature only the application of the continuation power flow methods in voltage stability analysis of positive-sequence power systems has been described. Due to the following reasons, a continuation three-phase power flow may be required: (a) there are unbalances of three-phase transmission lines in high voltage transmission networks; (b) there are unbalanced three-phase loads; (c) in addition, there are single-phase or two-phase lines in distribution networks; (d) there are single-phase or two-phase loads; (e) there may also be possible unbalanced threephase structures and control of transformers and FACTS-devices. In addition to the reasons above, with the recent integration of large amount of distributed generation into power networks, new voltage stability analysis tools, which should 218 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems have the modeling capability of unbalanced networks, become increasingly important. Furthermore, it is recognized that voltage stability analysis should be able to deal with asymmetrical contingencies such as single-phase and two-phase transmission line outages, etc. It is known that the single-phase continuation power flow is not able to deal with unbalanced networks and loads and can not deal with single-phase and two-phase outages of unbalanced transmission lines. In the light of the above considerations, in this chapter, a continuation threephase power flow approach for voltage stability analysis of unbalanced threephase power systems [12] is presented. In addition, voltage stability control by FACTS is also discussed. 7.2 Continuation Three-Phase Power Flow Approach 7.2.1 Modeling of Synchronous Machines with Operating Limits The modeling of synchronous machines in three-phase power flow analysis has been discussed in chapter 5. The operating limits of synchronous machines, which play very important role in voltage stability analysis, should be considered. In the following, the operating constraints of synchronous machines are presented and incorporation of the limits in three-phase power flow and continuation three-phase power flow analysis is discussed. In Fig. 5.1, Vi a = Vi a ∠θ ia , Vib = Vib ∠θ ib , V ic = Vic ∠θ ic , which are the threephase voltages at the generator terminal bus, are expressed in phasors in polar coordinates. Similarly, the voltages at the generator internal bus may be given by E ia = E ia ∠δ ia , E ib = E ib ∠δ ib , E ic = E ic ∠δ ic . In fact the voltages at the generator internal bus ° are ° balanced, we have E ia = E ib = E ic and δ ia = δ ib + 120 = δ ic − 120 . Therefore, in the following derivation of the power flow equations of the generator, δ ia and E ia can be considered as independent state variables of the internal generator bus while δ ib and E ib , δ ic and E ic are dependent state variables and can be represented by δ ia and E ia . For a PV machine, the total reactive power Qg i at its terminal bus should be within its operating limits: Qg imin ≤ Qg i ≤ Qg imax (7.1) where Qg imin and Qg imax are the lower and upper reactive limits, respectively. In addition, due to the limitation of the field current, the following constraint should hold 7.2 Continuation Three-Phase Power Flow Approach 219 E ia ≤ E imax (7.2) where E imax is the maximum limit of the internal voltage of the machine, which corresponds to the maximum filed current. Eia is the actual voltage magnitude at the internal bus. For a PQ machine, the positive-sequence voltage Vi1 at its terminal bus should be within its operating limits: Vi min ≤ Vi1 ≤ Vi max (7.3) where Vi min and Vi max are the upper and lower voltage limits, respectively. In addition, the field current constraint as given by (7.2) is also applicable. The basic constraint enforcement principle of a synchronous machine is that, when an inequality constraint, such as a current or voltage or reactive power inequality constraint, is violated, the constraint is enforced by being kept at its limit, while the voltage or reactive power control constraint of the synchronous machine is released. In other words, enforcing an inequality constraint and releasing an equality constraint must form a pair. In case there are two or more inequality constraints of a synchronous machine being violated in the same time, the strategy proposed in [16] can be used. The reactive power constraint in (7.1) and current constraint in (7.2) of a machine are considered as internal constraints while the voltage constraint in (7.3) is considered as external constraint. Generally, an internal constraint has priority to be enforced if both the internal and external constraints are violated simultaneously. In case the internal and external constraints cannot be enforced within the limits simultaneously, the external constraint should be released. 7.2.2 Three-Phase Power Flow in Polar Coordinates The power mismatch equations at buses except generator internal buses, which are given by (5.51) and (5.52), are presented as follows: ∆Pi p = − Pd ip − Vi p ¦ j∈i m = a ,b,c pm pm pm pm m ¦ V j (Gij cos θ ij + Bij sin θ ij ) = 0 (7.4) ∆Qip = −Qd ip − Vi p ¦ j∈i m = a ,b,c pm pm pm pm m ¦ V j (Gij sin θ ij − Bij cos θ ij ) = 0 (7.5) where i = 1, 2, …, N. Pd ip and Qd ip are the active and reactive load powers of phase p at bus i, respectively. The power mismatch equations at generator internal buses (for the case of PQ machine), which are given by (5.53) and (5.54), are presented as follows: 220 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems ∆Pg i = − Pg i p m pm pm pm pm − ¦ ¦ [Vi Vi (Gg i cos θ i + Bg i sin θ i ) + p = a ,b ,c m = a ,b,c p = a ,b,c m = a ,b ,c ¦ pm p p p pm p m m ¦ [Vi E i (Gg i cos(θ i − δ i ) + Bg i sin(θ i − δ i )) (7.6) ∆Qg i = −Qg i p m pm pm pm pm − ¦ ¦ [Vi Vi (Gg i sin θ i − Bg i cos θ i ) + p = a ,b ,c m = a ,b,c p = a ,b,c m = a ,b ,c ¦ pm p p p pm p m m ¦ [Vi E i (Gg i sin(θ i − δ i ) − Bg i cos(θ i − δ i )) (7.7) where i = 1, 2, …, Ng. Ng is the number of generators. In three-phase power flow calculations, Pg i and Qg i , which are specified, are the active and reactive generation powers of the generator at bus i, respectively. For the case of PV and slack machine, two constraint equations can also be obtained. Modeling of other power system components is referred to [14][15]. A number of three-phase power flow methods [17]-[26], etc. have been proposed since 1960s. In the following, the three-phase Newton power flow algorithm in polar coordinates, which is similar to that proposed in [19], will be used. The nonlinear equations (7.4)-(7.7) can be combined and expressed in compact form: F ( x) = 0 (7.8) where F(x) represents the whole set of power flow mismatch and machine terminal constraint equations. x is the state variable vector and given by x = [ a , V a , b , V b , c , V c , a , E a ]t . The Newton equation is given by: J (x)∆x = −F(x) where F(x) = [∆P a , ∆Q a , ∆P b , ∆Q b , ∆P c , ∆Q c , ∆Pg a , ∆Qg a ]t , J (x) = system Jacobian matrix. (7.9) ∂F(x) is the ∂x 7.2.3 Formulation of Continuation Three-Phase Power Flow Predictor Step. To simulate three-phase load change, Pd ip and Qd ip , which are shown in (7.4) and (7.5), may be represented by: Pd ip = Pd 0 ip (1 + λ * KPd ip ) Qd ip = Qd 0 ip (1 + λ * KQd ip ) (7.10) (7.11) 7.2 Continuation Three-Phase Power Flow Approach 221 where Pd 0 ip and Qd 0 ip are the base case active and reactive load powers of phase p at bus i. λ is the loading factor, which characterize the change of load. The ratio of KPd ip / KQd ip is constant to maintain constant power factor. Similarly, to simulate generation change, Pg i and Qg i , which are shown in (7.5) and (7.6), are represented as functions of λ and given by: Pg i = Pg 0 i (1 + λ * KPg i ) Qg i = Qg 0i (1 + λ * KQg i ) (7.12) (7.13) where Pg 0 i and Qg 0 i are the total active and reactive powers of the generator of the base case. The ratio of KPg i / KQg i is constant to maintain constant power factor for a PQ machine. For a PV machine, equation (7.13) is not required. For a machine, when the reactive limit is violated, Qg i should be kept at the limit and equation (7.13) is also not required. The nonlinear equations (7.9) are augmented by an extra variable λ as follows: F ( x, λ ) = 0 (7.14) where F(x, λ ) represents the whole set of power flow mismatch equations. The predictor step is used to provide an approximate point of the next solution. A prediction of the next solution is made by taking an appropriately sized step in the direction tangent to the solution path. To solve (7.14), the continuation algorithm with predictor and corrector steps can be used. Linearizing (7.14), we have: dF (x, λ ) = F x dx + Fλ dλ = 0 (7.15) In order to solve (7.15), one more equation is needed. If we choose a non-zero magnitude for one of the tangent vector and keep its change as ±1 , one extra equation can be obtained: t k = ±1 (7.16) where t k is a non-zero element of the tangent vector dx . Combining (7.15) and (7.16), we can get a set of equations where the tangent vector dx and dλ are unknown variables: ªF x Fλ º ª dx º ª 0 º » «dλ » = «± 1» « e k ¼¬ ¼ ¬ ¼ ¬ (7.17) where ek is a row vector with all elements zero except for K th , which equals one. In (7.17), whether +1 or –1 is used depends on how the K th state variable is 222 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems changing as the solution is being traced. After solving (7.17), the prediction of the next solution may be given by: ªx * º ª x º ª dx º « *» = « » +σ« » «λ » ¬λ ¼ ¬dλ ¼ ¬ ¼ (7.18) where * denotes the estimated solution of the next step while σ is a scalar, which represents the step size. Corrector Step. The corrector step is to solve the augmented Newton power flow equation with the predicted solution in (7.18) as the initial point. In the augmented Newton power flow algorithm an extra equation is included and λ is taken as a variable. The augmented Newton power flow equation may be given by: ªF (x, λ ) « x − ¬ k º ª0 º » = «0 » ¼ ¬ ¼ (7.19) where , which is determined by (7.18), is the predicted value of the continuation parameter xk . The determination of the continuation parameter is shown in the following solution procedure. The corrector equation (7.19), which consists a set of augmented nonlinear equations, can be solved iteratively by Newton’s approach as follows: ªFx Fλ º ª ∆x º ªF( x, λ ) « e » «∆λ » = − « x − k ¬ ¬ k ¼¬ ¼ º » ¼ (7.20) 7.2.4 Solution of the Continuation Three-Phase Power Flow The general solution procedure for the Continuation Three-Phase Power Flow is given as follows: Step 0: Run three-phase power flow when Pd ip , Qd ip , Pg i and Qg i are set to Pd 0ip , Qd 0ip , Pg 0i and Qg 0i , respectively. The initial point for tracing the PV curves is found. Step 1 - Predictor Step: (a) Solve (7.17) and get the tangent vector [ dx, dλ ]t ; (b) Use (7.18) to find the predicted solution of the next step. (c) Choose the continuation parameter by evaluating xk : tk = max(| dxi |) . (d) Check whether the critical point (maximum loading point) has been passed by evaluating the sign of dλ . If dλ changes its sign from positive to negative, then the critical point has just passed. 7.2 Continuation Three-Phase Power Flow Approach 223 (e) Check whether λ* <0 (Note 0 ≤ λ ≤ λ max ). If this is true, go to Step 3. Step 2 - Corrector Step: (a) According to the chosen continuation parameter to form the augmented equation (7.19); (b) Form and solve the Newton equation (7.20); (c) Update the Newton solution and continue the iterations until the corrector step converges to a solution with a given tolerance; (d) Go to Step 1. Step 3: Output solutions of the PV curves. 7.2.5 Implementation Issues of Continuation Three-Phase Power Flow 7.2.5.1 The Structure of Jacobian Matrix The structures of the Jacobian matrix (7.17) and the Jacobian matrix (7.20) are very similar. In comparison to the 4 by 4 Jacobian blocks in single-phase power flow analysis, the Jacobian matrix blocks of Fx in three-phase power flow analysis become 12 by 12 matrix blocks for all buses except internal buses of generators while the Jacobian blocks of the internal buses of generators are 4 by 4, 4 by 12, 12 by 4 matrix blocks. Similar to that of single-phase power flow analysis, the equations (7.17) and (7.20) of three-phase power systems can be solved by sparse matrix techniques. 7.2.5.2 Improvement of Computational Speed In order to improve the computational speed for tracing the PV curves, in the implementation, the three-phase power flow calculations may be used with gradually increasing system load until the three-phase power flow cannot converge. Then the above continuation three-phase power flow approach can be used to trace the remaining parts of the PV curves. Using the three-phase continuation power flow, a small predictor step may be used at the vicinity of the point of voltage collapse while a large step may be used otherwise. In the present implementation of the continuation three-phase power flow algorithm, the tangent method is used at the predictor step. It has been recognized that the tangent method may be more reliable than the secant method. In addition, the tangent method can produce an approximate left eigenvector at the saddle node bifurcation point. However, as far as computational time is concerned, the secant method may be more attractive [10] since using the method, solution of (7.17) is not needed. On the other hand, the solution of (7.17) may be significantly improved by using the sparse vector method [27] in the implementation. Since the only one nonzero element of the right-hand vector in (7.17) is at the bottom, the forward substitution is not needed at all. 224 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems 7.2.5.3 Comparison of Balanced Three-Phase Systems and SinglePhase Systems It should be pointed out that for a balanced system, a three-phase power flow solution is, in principle, exactly identical to that of the equivalent positive-sequence power flow while positive-and zero-sequence voltage components of the former are zero and at any bus, phase a, b and c voltages are balanced and interdependent except 120 $ shifting between them. In other words, for a balanced three-phase power system, any phase voltage at a bus can completely characterize the positivesequence voltage at that bus. Mathematically, the balanced three-phase system can be decoupled into the equivalent positive-, negative-, and zero-sequence networks, and the singularity resulting from the equivalent positive-sequence network can be solved by choosing any phase voltage at the bus as the continuation parameter. This means, in nature, the reason and solution of the singularity of the balanced three-phase power flow are exactly the same to that of the equivalent positivesequence power flow. The difference between them is whether the system is represented in three-phase or single-phase coordinates. 7.2.6 Numerical Results In this paper, numerical results are carried out on a 5-bus system and a modified IEEE 118-bus system. The single-line diagram of the 5-bus system I and the system data are presented in the Appendix of Chapter 5. For the modified IEEE 118bus system, 54 three-phase Wye-Grounded/Delta transformers are inserted between the original network and 54 generators, and negative- and zero-sequence parameters of transmission lines are amended. The modified IEEE 118-bus system consists of 172 three-phase buses (or 516 single-phase buses). In the studies, loads are represented by P and Q powers. 7.2.6.1 Results for the 5-Bus System without Line Outages The following cases on the 5-bus network have been studied: Case 1: Balanced network and the whole system with balanced load. Case 2: Balanced network with unbalanced load at Bus 3 with 6.0+j3.0 p.u., 6.3+j2.7 p.u., 5.7+j3.3 p.u. for phase a, b, c loading, respectively. Case 3: Unbalanced network and the whole system with balanced load. Case 4: Unbalanced network with unbalanced load at Bus 3 with 6.0+j3.0 p.u., 6.3+j2.7 p.u., 5.7+j3.3 p.u. for phase a, b, c loading, respectively. The PV curves of cases 1 - 4 are shown in Fig. 7.1 to Fig. 7.4. It is known that the tracing direction of the PV curves of a single-phase or positive-sequence system is clockwise. From Fig. 7.1, it can be seen that for the balanced three-phase power 7.2 Continuation Three-Phase Power Flow Approach 225 system, the three PV curves at any bus are exactly the same and the tracing direction of these PV curves is clockwise. As expected, the three PV curves for phase a, phase b and Phase c at Bus 3 of case 1 are exactly the same. Furthermore, these PV curves have very similar pattern to that of a single-phase or positive-sequence power system. That is each PV curve consists of a high voltage portion and a low voltage portion. As the loading factor λ is increasing between 0 and λ max , the operating point of the system is moving from the initial point to the maximum loading point or the point of voltage collapse, which is corresponding to the higher voltage portion of the PV curve. After the point of voltage collapse, the loading factor λ is decreasing from λ max to 0, which is corresponding to the lower voltage portion of the PV curve. It is known that any points on the lower voltage portion are unstable. Having discussed the PV curves for single-phase and examined also the PV curves of balanced three-phase systems, the PV curves of unbalanced three-phase systems are to be discussed here. It has been found that the three PV curves for phase a, phase b and Phase c at Bus 3 for any of case 2 - 4, are not the same. Examining the PV curves at Bus 3 of case 2 shown in Fig. 7.2, it was interestingly found: • In the PV curves of Phases a and c, the voltages are decreasing when λ is increasing between 0 and λ max . The tracing direction of these two PV curves is clockwise and the patterns of these two PV curves are very similar to that of single-phase or balanced three-phase power systems. • However, in the PV curve of phase b, the voltage is decreasing till at a point close to the point of λ max , then the voltages become increasing. The tracing direction of the PV curve is anti-clockwise. In this PV curve, the ‘higher voltage’ portion is corresponding to the unstable power flow solutions while the ‘lower voltage’ portion is corresponding to the stable power flow solutions. Further examining the PV curves of case 3 and 4 shown in Fig. 7.3 and Fig. 7.4, respectively, it can be found: • The patterns of the PV curves of the unbalanced three-phase systems are quite different from that of the balanced three-phase systems. At least one of the PV curves at a bus has the clockwise tracing direction and the voltage of the phase is much lower than that of the other phases while the PV curves of the other phases may have the anti-clockwise tracing direction. • However, when the network and load are balanced, the three PV curves at any bus merge into one as seen in Fig. 7.1. Then a positive-network analysis is sufficient. • Voltage stability analysis of unbalanced three-phase power systems are much more complex than that of single-phase positive sequence power systems or balanced three-phase power systems. 226 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems 1.4 1.2 Voltage Magnitude in P.U. 1 0.8 Phase a of bus 3 Voltage Magnitude in P.U. 1.4 1.2 1 0.8 Phase a of bus 3 0.6 0.4 0.2 0 Phase b of bus 3 Phase c of bus 3 0.6 0.4 0.2 0 0 Phase b of bus 3 Phase c of bus 3 0.5 1 1.5 0 Loading Factor 0.5 1 Loading Factor 1.5 Fig. 7.1. PV curves of bus 3 for case 1 Fig. 7.2. PV curves of bus 3 for case 2 1.2 1.2 1 1 Voltage Magnitude in P.U. Voltage Magnitude in P.U. 0.8 Phase a of bus 3 0.6 Phase b of bus 3 Phase c of bus 3 0.4 0.8 Phase a of bus 3 Phase b of bus 3 Phase c of bus 3 0.6 0.4 0.2 0.2 0 0 0.5 1 1.5 Loading Factor 0 0 0.5 1 Loading Factor 1.5 Fig. 7.3. PV curves of bus 3 for case 3 Fig. 7.4. PV curves of bus 3 for case 4 The maximum loading factors for cases 1-4 are shown in Table 7.1. From Table 7.1, it can be clearly seen that unbalanced network and load can significantly affect the system loading capability. 7.2 Continuation Three-Phase Power Flow Approach Table 7.1. Maximum loading factors without line outages Case No. 1 2 3 4 Maximum loading factor λmax 1.4530 1.3127 1.0864 1.0374 Lowest voltage magnitude (in p.u.) at maximum loading point 0.5818 0.6095 0.6465 0.6299 Bus no of the lowest voltage 3 3 3 3 227 Phase of the lowest voltage a, b, c a c c 7.2.6.2 Results for the 5-Bus System with Line Outages In order to investigate the voltage stability of unbalanced three-phase systems where there are transmission line outages, the following cases were carried out: Case 5: This is similar to case 4 but Phase a of one of the double lines between Bus 1 and Bus 3 is open-circuited. Case 6: This is similar to case 4 but Phase b of one of the double lines between Bus 1 and Bus 3 are open-circuited. Case 7: This is similar to case 4 but the Phase c of one of the double lines be tween Bus 1 and Bus 3 are open-circuited. Case 8: This is similar to case 4 but Phase a and Phase b of one of the double lines between Bus 1 and Bus 3 is open-circuited. Case 9: This is similar to case 4 but Phase b and Phase c of one of the double lines between Bus 1 and Bus 3 are open-circuited. Case 10:This is similar to case 4 but Phase c and Phase a of one of the double lines between Bus 1 and Bus 3 are open-circuited. Case 11:This is similar to case 4 but Phase a, Phase b and Phase c of one of the double lines between Bus 1 and Bus 3 are open-circuited. The maximum loading factors for cases 5-11 are shown in Table 7.2. From this table, it can be seen: • Surprisingly the maximum loading factor of case 6 (with line 1-3 outage on phase b) is greater than that of case 4 without any line outage. This means that for the unbalanced three-phase system studied, case 6 with single–phase line outage is less serious than case 4 without line outages in the point of view of voltage stability. 228 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems Table 7.2. Maximum loading factors with line outages Case No. Type of line outage Maximum loading factor λmax Lowest voltage magnitude (in p.u.) at maximum loading point 0.6183 0.6329 0.6182 0.6067 0.6329 0.6147 0.6407 Bus no of the lowest voltage Phase of the lowest voltage 5 6 7 8 9 10 11 Phase a Phase b Phase c Phases a and b Phases b and c Phases a and c Phases a, b and c 0.6878 1.0627 0.5317 0.5681 0.5987 0.5176 0.5665 3 3 3 3 3 3 3 a c c a c c c • Among the single-phase line outages of cases 5-7, case 6 is less serious than the other two cases. The maximum loading factor of case 6 is about two times that of the other two cases, respectively. • Cases 8 and 9 with two-phase line outages have larger loading factors than case 7 with one-phase line outage. This means that for the unbalanced three-phase system studied, the two-phase line outages of cases 8 and 9 are less serious than the one-phase line outage of case 7 in the point of view of voltage stability. • Case 11 with three-phase line outage has larger loading factor than case 7 with one-phase line outage and case 10 with two-phase line outages. This means that the three-phase line outage of case 11 is less serious than the one-phase line outage of case 7 and the two-phase line outage of case 10 in the point of view of voltage stability. The above observations are very interesting phenomena from the unbalanced three-phase systems, which are quite different from that of single-phase systems or balanced three-phase systems. Due to the combinations of the complexity of unbalanced load and network, it is not easy to explain qualitatively the above observations. Instead, we try to show numerical results to reveal the possible reasons for the above phenomena. The three-phase power flow results of cases 4 - 11 at particular loading levels are shown in Table 7.3 and Table 7.4 respectively. In the Tables, V31 , V32 and V30 are the positive-, negative-, zero-sequence voltage magnitudes at bus 3, respectively while voltage sensitivities, which are the largest in magnitude for the corresponding cases, are shown in the last column of these two Tables. 7.2 Continuation Three-Phase Power Flow Approach Table 7.3. Power flow results at λ =0.5 (voltage and power in P.U.) Case No. 4 5 6 7 8 9 10 11 V3 1 229 V3 2 V3 0 Ploss Qloss Normalized Qloss ∂V ∂λ -0.21 -0.41 -0.20 -1.14 -0.74 -0.59 -1.51 -0.70 0.9117 0.8837 0.8983 0.8602 0.8672 0.8621 0.8373 0.8452 0.0378 0.0567 0.0276 0.1037 0.0636 0.0727 0.0910 0.0788 0.0403 0.0857 0.0154 0.1133 0.1193 0.0599 0.1312 0.0946 1.33 1.62 1.45 1.76 1.85 1.74 2.02 1.99 8.89 11.09 10.00 12.77 13.04 12.51 14.74 14.22 100% 125% 113% 144% 147% 141% 166% 160% Table 7.4. Power flow results at λ =1.03 (voltage and power in P.U.) Case No. 4 6 V3 1 V3 2 V3 0 Lowest phase voltage 0.6629 0.6971 Ploss Qloss ∂V ∂λ -1.94 -0.88 0.8042 0.0887 0.1087 0.7966 0.0540 0.0333 3.18 3.38 24.70 26.39 The voltage sensitivities can be considered as an indicator of voltage instability. In principle, the larger the voltage sensitivity is, the lower the maximum loading factor will be. The voltage sensitivities in Table 7.3 and Table 7.4 correlate well with the maximum loading factors of case 4 – 11. In addition, it has been found that for most of the cases, high power losses and negative- and zero-sequence voltage components are associated with large voltage sensitivities. Comparing the results of case 4 and case 6 in Table 7.3 and Table 7.4, it can be seen: • At λ = 0.5, the largest voltage sensitivity of case 4 is larger than that of case 6. • As λ is increased to 1.03, the voltage sensitivity of case 4 becomes much larger than that of case 6. The voltage sensitivities indicate that case 4 is more vulnerable to voltage instability and hence a lower maximum loading factor is expected for this case. 7.2.6.3 Results for the Modified IEEE 118-Bus System The following four cases were carried out on the modified IEEE 118-bus system with balanced network and loads: Case 12: This is the base case system. Case 13: This is similar to case 12 but one phase of the line between Bus 68 and Bus 81 is open-circuited. Case 14: This is similar to case 12 but two phases of the line between Bus 68 and Bus 81 are open-circuited. 230 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems Case 15: This is similar to case 12 but three-phases of the line between Bus 68 and Bus 81 are open-circuited. The maximum loading factors for cases 12-15 on the modified IEEE 118-bus system are shown in Table 7.5. Table 7.5. Maximum loading factors on the modified IEEE 118-bus system Case No. 12 13 14 15 Maximum loading factor λmax 0.6910 0.6599 0.5998 0.5010 Type of line outage None One phase outage of line 68-81 Two phase outage of line 68-81 Three-phase outage of line 68-81 7.2.6.4 Reactive Power Limits For case 4, with generator reactive power limits applied, the PV curves of bus 3 are shown in Fig. 7.5. From Fig. 7.5 it can be found: • In comparison of Fig. 7.5 to Fig. 7.4, as expected, the maximum loading factor is decreased when the generator reactive power limits are applied. • The significant reduction in the voltage magnitudes of phases a and b can be seen when the generator reactive power limits are encountered. Phase c voltage magnitudes are actually reduced as well. 1.2 1 Voltage Magnitude in P.U. 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Phase a Phase b Phase c Loading Factor Fig. 7.5. PV curves of bus 3 for case 4 with generator reactive power limits 7.2 Continuation Three-Phase Power Flow Approach 231 • The effect of the reactive power limits is the reduction in bus voltage magnitudes. In other words, the three PV curves move down since the generator terminal bus voltage cannot hold up to the setting point any longer. Noting the tracing direction of the PV curves of phases a and b is anti-clockwise, the effect of the reactive power limits on the voltage magnitudes of these two phases is more significant and delays the voltage rise of phases a and b. Hence, the effect causes the “higher portion” of the PV curves fall and cross the “lower portion” of these. A continuation three-phase power flow approach for voltage stability analysis of unbalanced three-phase power systems has been proposed. The approach can take into account the unbalances of both network and load. In addition, it can also deal with various transmission line outages. Numerical examples have demonstrated the approach proposed is effective. Some very interesting results using PQ loads have been obtained: • When network and load are balanced, the PV curves of phase a, phase b and phase c at a bus are identical as expected and the pattern of these is very similar to that of the PV curves of single-phase power systems. In this situation, a single-phase positive network analysis is sufficient. • However, when network and load are unbalanced, the patterns of the PV curves are very interesting, which have not been observed and discussed in the past. It has been found that with unbalanced network and load, at least one of the PV curves at a bus is very similar to that of single-phase or balanced three-phase systems and the tracing direction of the PV curve is clockwise while the rest of the PV curves (or curve) at the bus have the anti-clockwise tracing direction. For those PV curves with anti-clockwise tracing direction, the higher voltage portion of the PV curves is corresponding to the unstable power flow solutions while the lower voltage portion of the PV curves is corresponding to the stable power flow solutions. The characteristic is unique to unbalanced three-phase power systems. • It has been found for unbalanced power systems that (a) the maximum loading factor of the system with a single-phase line outage may be greater than that of the system without any line outages; (b) the maximum loading factor of the system with a two-phase line outage may not be necessarily less than that of the system with a single-phase line outage; (c) similarly the maximum loading factor of the system with a three-phase line outage may not be necessarily less than that of the system with a single-phase line outage or a two-phase line outage. The phenomena have been explained based on numerical analyses. Basically, the maximum loading factor is dependent on the degree of unbalance, which is characterized by the magnitudes of negative- and zero-sequence voltages. The degree of unbalance itself is determined by the combination of unbalanced network and loading conditions. The phenomena observed above reveal that the voltage stability mechanisms of three-phase power systems are much more complex than that of single-phase 232 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems power systems. This clearly indicates that a continuation three-phase power flow is needed when there are unbalanced network and load existing in a power system. Otherwise, the results may be unrealistic and could not be able to characterize accurately the voltage stability problem of unbalanced power systems. Similar to that in conventional continuation power flow analysis, reactive power limits of generators play a very important role in the determination of PV curves of three-phase power systems. Basically consideration of the reactive power limits will decrease the maximum loading factor and affect the shape of the PV curves. When the reactive power limits are taken into account, there is the reduction in voltage magnitudes, and subsequently this will affect the shape of the PV curves. It has been found that the effect of the reactive power limits on the PV curves whose tracing direction is anti-clockwise is more significant than on those whose tracing direction is clockwise. The present results are based on the PQ load model. Further research is needed to investigate the effect of voltage dependent load models on the voltage stability of unbalanced three-phase power systems. The continuation three-phase power flow approach will be a very useful tool for voltage stability of unbalanced three-phase power systems. The approach can also be used to investigate multiple power flow solutions of unbalanced threephase power systems. As distributed generators are increasingly connected to power systems, the continuation three-phase power flow approach may become an important tool to evaluate the unbalanced system operation conditions including contingencies. 7.3 Steady State Unbalanced Three-Phase Voltage Stability with FACTS In this section, the effects of FACTS controls on the steady state voltage stability limit of unbalanced three-phase power systems are investigated. FACTS-devices considered here are STATCOM, SSSC and UPFC [28][29]. The modeling of these FACTS in three-phase power flow analysis is referred to chapter 5. 7.3.1 STATCOM Cases 1-4 with a STATCOM installed at the middle of transmission line 1-3, which are corresponding to case 1-4 in section 7.2.6.2, respectively, have been studied: Case 1: Balanced network and the whole system with balanced load. Case 2: Balanced network with unbalanced load at Bus 3 with 6.0+j3.0 p.u., 6.3+j2.7 p.u., 5.7+j3.3 p.u. for phase a, b, c loading, respectively. Case 3: Unbalanced network and the whole system with balanced load. 7.3 Steady State Unbalanced Three-Phase Voltage Stability with FACTS 233 Case 4: Unbalanced network with unbalanced load at Bus 3 with 6.0+j3.0 p.u., 6.3+j2.7 p.u., 5.7+j3.3 p.u. for phase a, b, c loading, respectively. The STATCOM models used in the studies are the three-phase model with symmetrical components control, and three single-phase units with independent phase control. The former is referred to model 1 while the latter is referred to model 2. Assuming that the voltage control reference of the STATCOM is 1.05 p.u. for both models. The maximum loading factors for cases 1-4 are shown in columns 3 and 4 in Table 7.6. For the sake of comparison, in Table 7.6 the maximum load factors for cases 1-4 without STATCOM as discussed in section 7.2.6.1 are listed in column 2. From the table, for both model 1 and model 2, the maximum loading factors have been improved significantly. When both the network and load are balanced, the maximum loading factors are the same for both model 1 and model 2. However, when either the network or load is unbalanced, the maximum loading factors for model 1 are bigger than that for model 2. The reason is that model 1 can balance the bus voltages well in comparison to model 2 since the former can simultaneously control three single-phase voltages while the latter can only control positive sequence voltage. For case 4, the relationship between voltage control reference of the STATCOM and the maximum loading factor is investigated, which is given by Fig. 7.6. From this figure, it can be found that the higher the voltage control reference, the bigger the system maximum loading factor. 2.5 2.4 2.3 2.2 Loading Factor 2.1 2 1.9 1.8 1.7 1.6 1.5 0.95 1 1.05 1.1 Voltage Control Reference in P.U. model 1 model 2 Fig. 7.6. The relationship between the voltage control reference and the maximum loading factor 234 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems Table 7.6. Maximum loading factors with STATCOM Case No. Maximum loading factor λmax without STATCOM 1.4530 1.3127 1.0864 1.0374 Maximum loading factor λmax with STATCOM (Model 1) 2.6962 2.3695 2.1913 2.1517 Maximum loading factor change in percentage with STATCOM (Model 1) 86% 81% 102% 107% Maximum loading factor λmax with STATCOM (Model 2) 2.6962 2.5351 2.3630 2.3313 Maximum loading factor change in percentage with STATCOM (Model 2) 86% 93% 117% 125% 1 2 3 4 7.3.2 SSSC Cases 5-8 with a SSSC installed at the middle of transmission line 1-3, which are corresponding to case 1-4 in section 7.3.1, respectively, have been studied. The SSSC models used in the studies are the three-phase model with symmetrical components control, and the three single-phase units with independent phase power flow control. The models have been discussed in section 5.3 of chapter 5. The three-phase model with symmetrical components control is referred to model 1 while the three single-phase units with independent phase power flow control is referred to model 2. Assuming that the total three-phase power flow control reference for model 1 is 6.5*3 p.u. while the single phase power flow control reference is 6.5 p.u. The maximum loading factors with SSSC control are shown in Table 7.7. From this table, it can be found that proper power flow control using SSSC can increase the maximum loading factors. From Table 7.7, it can be seen that as the unbalance of network and load increases, the maximum loading factor in percentage change increases using the SSSC power flow control. The relationship between the power flow control reference for SSSC model 1 and the maximum loading factor is shown in Fig. 7.7. Table 7.7. Maximum loading factors with SSSC Case No. Maximum loading factor λmax without SSSC Maximum loading factor λmax with three-phase SSSC (Model 1) 1.8884 1.8155 1.6918 1.6746 Maximum loading factor change in percentage with SSSC (Model 1) 30% 38% 56% 61% Maximum loading factor λmax with single phase SSSC (Model 2) 1.8884 1.8000 1.6969 1.7277 Maximum loading factor change in percentage with SSSC (Model 2) 30% 37% 56% 67% 5 6 7 8 1.4530 1.3127 1.0864 1.0374 7.3 Steady State Unbalanced Three-Phase Voltage Stability with FACTS 235 3 2.5 Loading Factor 2 1.5 1 0.5 0 0 10 20 30 40 50 Power Flow Control Reference in P.U. Fig. 7.7. The relationship between the power flow control reference and the maximum loading factor 7.3.3 UPFC Cases 9-12 with a UPFC installed at the middle of transmission line 1-3, which are corresponding to case 1-4 in section 7.3.1, respectively, have been studied. The UPFC model used in the studies is the three-phase model with symmetrical components control. The model has been discussed in section 5.3 of chapter 5. Assuming that the total three-phase power flow control reference for model 1 is 7.5*3 p.u. while the single phase power flow control reference is 7.5 p.u. The voltage control reference for both models is 1.05 p.u. The maximum loading factors with the UPFC are shown in Table 7.8. Table 7.8. Maximum loading factors with UPFC Case No. 9 10 11 12 Maximum loading factor λmax without UPFC 1.4530 1.3127 1.0864 1.0374 Maximum loading factor λmax with three-phase UPFC 1.8654 1.8468 1.8676 1.8478 Increase of maximum loading factor in percentage 28% 41% 72% 78% 236 7 Steady State Voltage Stability of Unbalanced Three-Phase Power Systems From Table 7.8, it can be found that proper power flow and voltage control using UPFC can increase the maximum loading factors. In particular, when the network and load are unbalanced, the UPFC control can effectively increase the maximum loading factor in percentage change with respect to that without UPFC control. It has been found that the voltage stability limit can be improved significantly using FACTS such as STATCOM, SSSC and UPFC. In particular, when the unbalance of network and load increases, the maximum loading factor change in percentage can increase significantly using FACTS control. The continuation three-phase power flow approach with FACTS control will be a useful tool to investigate the voltage stability control of unbalanced three-phase power system. The continuation three-phase power flow approach can be also used to determine the security operating limits in terms of voltage, thermal and voltage stability limits. With the integration of distribution generation into power grids, such a tool will pay an increasingly important role in operation, planning and control of distributed power grids. 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