A Fast Continuation Load Flow Analysis for an Interconnected Power System

Transcription

1 A Fast Continuation Load Flow Analysis for an Interconnected Power System D. Hazarika Assam Engineering College, Gauhati-13, Assam, India Abstract- The paper provides an algorithm for a fast continuation load flow to determining critical load for a bus with respect to its voltage collapse limit of an inter connected multi-bus power system using the criteria of singularity of load flow Jacobian matrix. For this purpose, load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected bus by incorporating the effect of all the other buses of the system on the target bus. The validity of the proposed method has been investigated for the IEEE 30 and IEEE 118 bus system. Keywords- Voltage Stability; Load Margin; Voltage Collapse; Power System List of symbols N = Total number of buses in the system. NG = Total number of generation buses in the system. npv = Total number of PV buses (including slack bus) in the system. Pi = Injected active power at ith bus. Qi= Injected reactive power at ith bus. PDi = Active power demand at ith bus. QDi= Reactive power demand at ith bus. PGi = Active power generated by ith generator. QGi= Reactive power generated by ith generator. Vi = Magnitude of voltage at ith bus. δi = Angle of the bus voltage at ith bus. Cos Øi = Load power factor of the ith load bus.. Gij + jbij = Element of Y-BUS matrix at ith row and jth column. I. INTRODUCTION Information/knowledge of load margin of a bus with respect to its voltage collapse limit constitutes important criteria for load pick-up step during power system restoration planning and operation. In a situation, when a load bus has to be energized through network building process (i.e., connecting the load to energized bus through transmission line(s)) and several energized buses are available for the connection, it becomes essential for power system planner/operator to select a proper energized bus for this purpose. Under such circumstances, the bus having highest load margin appears to be the best choice [1] and magnitude of load to be picked-up must be less than the load margin for the connected bus. Voltage collapse of a bus is characterized by a slow variation in system operating point, due to increase in the loads, in such a way that the voltage magnitude gradually decreases until a sharp accelerated change occurs. It has been observed that voltage magnitudes in general, do not give a good indication of proximity to voltage stability limit [2]. In recent literature many voltage stability and voltage collapse prediction methods have been presented. Some of the important ones are: voltage collapse index based on a normal load flow solution [3, 4, 5], voltage collapse index based on closely located power-flow solution pairs [6], voltage collapse index based on sensitivity analysis [7], and minimum singular value of Newton - Raphson power flow Jacobian matrix [8, 9]. Continuation power flow analysis is based on locally parametrized continuation technique [10]. The key idea here is to avoid the singularity of the Jacobian by slightly reformulating the power flow equations, and applying the locally parametrized continuation technique [11, 12]. The method uses predictor-corrector approach and some parameter (continuation parameter) as additional condition to overcome ill conditioning of Jacobian matrix. Network partitioning technique has been employed in determination of load margin at a load bus [13]. A two parameter continuation technique has been proposed for evaluating branch outage contingencies introducing new branch parameter [14]. A voltage stability index VMPI (voltage margin proximity index) is proposed considering voltage limits, especially lower voltage limits [15]. A simple, computationally very fast local voltage-stability index has been proposed using Tellegen s theorem [16]. It is easy to implement in the wide-area monitoring and control center or locally in a numerical relay. A new node voltage stability index called the equivalent node voltage collapse index ENVCI), which is based on ESM and uses only local voltage phasors, is presented [17]. Well-known ZIP model has been used to represent loads having components with different power to voltage sensitivities [18] and also, the choice of voltage stability index in the context of load modeling has been suggested. These methods assess the closeness to the critical loading by looking into the voltage stability sensitivity indices or the smallest Eigen-value or singular value of load flow Jacobian matrix. The performance indices are used to provide the measure of the severity of static voltage stability problem. However, the disadvantages with the performance indices are that they cannot directly provide the margin of load at a load bus with respect to its voltage collapse limit

2 The proximity of voltage collapse in a power system is indicated by the condition of Newton-Raphson power flow Jacobian matrix. At the point of voltage collapse, the determinant of the Jacobian matrix of a Newton-Raphson power flow analysis becomes zero. As such, the elements of load flow Jacobin matrix can be used to indicate voltage instability problem of a power system. This paper proposes a continuation load flow for determining the load margin of a bus with respect to its voltage collapse limit. For this purpose, load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected bus. The elements of the transformed two by two matrix and the bus voltage of the target bus are used to develop an algorithm to determine load margin of a target bus of the system. II. ANALYSIS OF VOLTAGE STABILITY PROBLEM FOR A MULTI- BUS INTER CONNECTED POWER SYSTEM It is established that at the point of voltage collapse in a power system, the determinant of load flow Jacobian matrix becomes zero. To quantify the condition of singularity of load flow Jacobian matrix of a multibus system at the point of voltage collapse, it is transformed into a two by two matrix relating Δδk, ΔVk, ΔPk and ΔQk for a target kth bus. The change in voltage magnitudes and voltage phase angles of a multi-bus inter connected power system are related to change in real and reactive power injection at the buses as follows: Load Flow Jacobian matrix of an interconnected system into a two by two matrix with respect to a target bus k. The paper presents a sensitivity analysis approach to relating change in real and reactive power of the target bus due to change in real and reactive power in other buses. Recognizing the fact that the change in bus voltage angle is primarily influenced by the change in real power injection at the buses [19], the change in real power injection at busbars can be expressed as [20] : Similarly, for other buses the relation among voltage phase angle and real power injections at the buses can be expressed as: Where H, N, M, and L are the elements of Jacobian matrix used for the load flow analysis of Newton Raphson method. Equation (1) can be expressed as: Equation (7) can be rearranged as given below: For the convenience of representing the system (load flow) equation, the following assumptions are made: (i) bus Number 1 is considered as the slack bus; (ii) the system has npv number of voltage control buses which includes the slack bus also, i.e., the system has (npv 1) number of PV buses. Now, using the elements of W, X, Y and Z of Equation (2) the change in voltage phase angle and voltage magnitude of a kth target bus can be expressed as: where; From Equation (6) and Equation (8) we have, The right hand side of the Equations (3) and (4) contain change in real and reactive power terms for all buses including the kth target load bus. To relate Δδk and ΔVk of Equations (3) and (4) to only ΔPk and ΔQk, it is required to transform the

3 The Equation (10) can be rearranged as follow: voltage phase angle and real power injections at the buses can be written as: Equation (16) can be rearranged as given below: where, is a row matrix, therefore, [ΔP] values are to be [ץ] calculated using pseudoinverse technique i.e. [20], From Equation (15) and Equation (17) we have, Equation (12) is solved for terms ΔPi for i = 2...N for change in ΔPk and expressed as: Similarly, recognizing the fact that the change in bus voltage angle is primarily influenced by the change in real power injection at the buses [19], the change in real power injection at busbars can be expressed as: The Equation (19) can be rearranged as follow: The change in slack bus reactive power in terms of change in voltage phase angles can be expressed as: Similarly, for other buses (excluding PV buses, where voltage magnitudes are assigned as fixed value) the relation among [α] is a row matrix, therefore, [ΔQ] values are to be calculated using pseudoinverse technique i.e. [20],

4 Substituting values of ΔPi and ΔQi from Equations (13) and (22) in Equations (3) and (4) we have where, Where, Gkk, Bkk, GkG and BkG are the elements of admittance matrix [Y] for the equivalent two bus system. But, the values of Gkk, Bkk, GkG and BkG depend on the system bus voltages and their phase angles. Therefore, while developing the mathematical model for load margin for the kth bus, parameters Gkk, Bkk, GkG and BkG are represented in terms of the elements of transformed two by to matrix represented by Equation (29). The elements of Jacobian matrix for the equivalent two bus system can be expressed as Therefore, At the point of voltage collapse the determinant of the Jacobian matrix becomes zero, i.e., At the point of voltage collapse The change in determinant value of the two bus equivalent system of the multibus system with respect to change in Vk and δk can be expressed as: The element of the transformed two by two matrix for the kth target bus are function of network state namely bus voltages and their phase angles. III. DETERMINATION OF LOAD MARGIN In this section, a procedure for determination of load margin of the target kth bus has been developed using the elements of the modified two by two Jacobian matrix given by the Equation (28) and the bus voltage of the target bus. Equation (28) can be represented as two bus system with the target kth load bus, Y-Bus elements and an equivalent source as depicted in Figure 1. Again, the expression for JD in terms of The terms are derived utilizing equations from (32) to (35). It is to be ensured that Fig. 1 Equivalent two bus system with target kth bus, Y-Bus elements and an equivalent source Now, for this equivalent two bus system, the expression for real and reactive power injections at the target kth bus are as follows: the terms contain only the elements of the modified two by two Jacobian matrix and the bus voltage of the target bus. Because, they are the known values for the transformed two bus system

5 and ΔQk as follows: Now, applying values of of Equations (40), (41), (42) and (43) in Equation (39) we have, The system collapse occurs when J D becomes zero. Now, if is the determinant value corresponding to the defined operating condition of the system, then the required change in ΔJ D of Equation (53) to force J D zero is and corresponding change in injection at kth bus will be Similarly, the term can be expressed as: Again, utilizing equations from (32) to (35), we have Therefore, load margin at kth load bus, i.e., the additional load that can be supplied to the kth load bus to push it to the proximity of voltage collapse is: Therefore, predicted critical load for kth load bus at the point of voltage collapse is: Processing Equation (33) and Equation (34), Gkk can be expressed as: Processing Equation (32) and Equation (35), Bkk can be expressed as: Now, applying values of of Equations (46), (47), (48), (49), (50) and (51) in Equation(45) we have, The terms of Equations (44) and (52) are the elements of the modified Jacobian matrix of a multibus system represented by Equation (28) and Vk is the bus voltage of the kth target bus. Now, to relate change in real power injection at target kth bus to the change in determinant of the modified Jacobian matrix of a multibus system, variables Δδk and ΔVk of Equation (37) are replaced by ΔPk But, ΔP margin Dk is determined using linear relation between ΔJ D and ΔPk. As such, load margin for the kth load bus (P margin Dk ) would be more than the actual load margin of the bus. Further, for wide change in J D, the predicted load value will be considerably high compared to that of actual critical load value at the point of voltage collapse. Slight over prediction of load margin could mislead system planner and operator in the decision making related to allowing more load in the load buses which are somewhat close to voltage stability limit. Therefore, it is required to confirm the actual load margin or critical load of a load bus through an iterative load flow analysis using the load margin determined by the Equation (55). To ensure convergence of the load flow analysis in the proposed iterative procedure, it is required to predt normalize the predicted critical load value P Dk in such a way that the modified predicted critical load value remains below the actual critical load of the bus. For this purpose, a reduction factor (RF) and a distance factor (DF) are used to normalize the predicted critical load value for the kth load bus as follows: The distance factor is expressed as

6 IV. The distance factor (DF) will be higher when prediction is done for wide change of ΔJ D, i.e., far from the proximity of voltage collapse limit. Thus, it will normalize the effect of over prediction due to wide change of ΔJ D. Therefore, as the iterative procedure approaches the proximity of voltage collapse limit P Dk margin will become very small; thus, the term will also approach to 1. When the value of J D becomes negative the iterative procedure has to be terminated. PROCEDURE FOR DISTRIBUTION OF NORMALIZED PREDICTED LOAD MARGIN TO THE GENERATORS Equation (11) represents sensitivity relation for all the buses with respect to the target kth bus. It is justified to supply the normalized predicted load margin ( ) of the kth target bus from the generators having higher sensitivity factor values. Therefore, to distribute the normalized predicted load margin of kth target bus, the sensitivity relation for generator buses with respect to target kth bus are utilized. Equation (11) can be expressed for generator buses only as follows; must be less than actual critical load of the bus. In case, P Dk crt determined using Equation (57) becomes slightly more than actual critical load of the bus (due to improper selection of Reduction Factor (RF), the load flow analysis of the iterative procedure will not converge. To take into account of such a situation, the proposed algorithm is equipped with a step after the load flow analysis. This step reloads the loads and generations of the system of the previous iteration values and reduces the reduction factor (RF) as RF = RF/3.0 and load flow analysis is carried out again before proceeding to the other portion of the algorithm. This step helps in changing the value of reduction factor (RF) to ensure proper normalization (by Reduction Factor (RF)) of critical load governed by Equation (53). The algorithm of the proposed iterative procedure is as follows: 1. Initialize the load flow data for the system and set iteration count K = 0, RF = (0.3 to 0.5). 2. Conduct base case load flow analysis of the system. crt K+1 3. Determine P Dk using Equation (57) and assign P Dk crt k+1 = P Dk, Q Dk = tan Øk P K+1 Dk and distribute the additional load (RF/DF) P margin Dk among the generating stations based on defined criteria subjected to generation limits of the generating stations as described in Section-III. In case of reactive power limit violation, a PV bus has to be changed into a PQ bus by assigning reactive power at its limit. 4. Conduct the load flow analysis and Check for load flow convergence criteria. If load flow has converged, then go to Step-5. Otherwise, reduce reduction factor RF =RF/ and reload P K+1 Dk = P K Dk, Q K+1 Dk = Q K+1 Dk, V [i] K+1 = V [i] K for i = 1...N and δ[i] K+1 = δ[i] K for i = 1...N, also the generating stations output with their Kth iteration values and go to Step Check for J D < 0.0 go to Step-6, otherwise, set K = K + 1 and go to Step Stop. VI. SIMULATON, RESULTS AND DISCUSSIONS Therefore, modified generation values are Where, P Gi max is the maximum limit on generation for ith generating station. V. ALGORITHM OF THE ITERATIVE PROCEDURE Reduction factor (RF) is used to normalize the over prediction of load margin (which is determined using linear relation between ΔP Dk margin and ΔJ D governed by Equation (53)) with the objective of ensuring convergence of load flow analysis used in the proposed iterative procedure. As such, load at a bus must be always less than actual critical load of the bus, i.e. to say that P Dk crt determined using Equation (57) To verify the validity and applicability of the proposed method, simulations were carried out on IEEE 30 bus and IEEE 118 bus systems. The aim of the simulations was to examine the nature of change of J D for different load bus of IEEE 30 and IEEE 118 bus system with respect to change in load at the target bus. For this purpose, load at a target bus is increased gradually manually using an interactive load flow program and the change in J D values were recorded. It is found that the J D reduces parabolically with increase in load and at the point of voltage collapse it becomes zero. Figures 2 and 3 illustrate the variation of J D value of two bus equivalent system of the IEEE 30 bus system with respect to change in load at load buses 21 and 29 for load power factors 0.8 and 0.9. Figures 4 and 5 illustrate the variation of J D value of two bus equivalent system of the IEEE 118 bus system with respect to change in load at load buses 88 and 118 for load power factors 0.8 and

7 Fig. 2 Variation of J D for load change at load bus 21 of IEEE 30 bus system for load power factors pf=0.8 & 0.9 Fig. 3 Variation of JD for load change at load bus 29 of IEEE 30 bus system for load power factors pf=0.8 & 0.9 Fig. 4 Variation of JD for load change at load bus 88 of IEEE 118 bus system for load power factors pf=0.8 & 0.9 Fig. 5 Variation of JD for load change at load bus 118 of IEEE 118 bus system for load power factors pf=0.8 & 0.9 It is observed that J D value of two bus equivalent system of IEEE 30 and 118 bus systems have different initial values for different target buses, as such, J D value for a bus reflects the voltage stability characteristic of the bus. The proposed algorithm is used to determine critical load of a bus with different reduction factors (RF). It has been found that RF values between 0.1 to 0.5 ensures convergence of load flow analysis in the iterative process for IEEE 30 and IEEE 118 bus system for any target bus of the system. But, with RF = 0.3 to 0.5, the iterative procedure terminates with less number of iterations. During iteration, the additional load assigned on the target (k) bus was distributed among all the generating stations based on their load contribution (subjected to their limits) as described in Section-IV. Continuation power flow analysis technique is also used to determine the critical load at the target buses with step size σ= to ensure that it arrives the point of voltage collapse. In the continuation power flow analysis technique the point of voltage collapse is arrived when the predicted continuation parameter (Δλ) becomes negative. However, in this case predicted normalized load is not distributed among the generators. Therefore, the proposed algorithm is also used to determine the critical load at the target buses without distributing predicted normalized load among the generators. Table 1 and Table 2 represent the simulation results for some of the buses of IEEE 30 bus system with RF = 0.4 with power factor 0.9 and 0.8 respectively. Table 3 and Table 4 represent the simulation results for some of buses of IEEE 118 bus system with RF = 0.4 with power factor 0.9 and 0.8 respectively. Tables represent the target bus selected for determination of critical load with respect to its voltage collapse limit, initial value of the determinant of the two by two Jacobian matrix for the target bus, initial value of predicted continuation parameter (Δλ), initial load at the target bus, final value of the determinant of the two by two Jacobian matrix,final value of predicted continuation parameter (at the point of collapse), critical load for the target bus and remarks about the methods. The abbreviation used in the remarks column of the tables stand for: PMWGD = Proposed method with generation distribution

8 PMWOGD = Proposed method without generation distribution. CLFWOGD = Continuation power flow without generation distribution

9 It is observed that predicted continuation parameter (Δλ) of IEEE 30 and 118 bus systems has different initial values for different target buses, as such, initial value of Δλ reflects the voltage stability characteristic of the target bus. Further, the

10 simulation results for the IEEE systems indicate that the negative when the continuation power flow analysis is terminated. It signifies that the systems are at the point of voltage collapse. Also, the determinant value of the transformed two by two matrix of the system becomes slightly negative at the point of voltage collapse for the IEEE systems. It is to be noted that sensitivity based distribution of normalized predicted load margin of a target bus among generators ensures higher critical load of a target bus in comparison to that of continuation power flow analysis without distribution of normalized predicted load margin among generators. It is due to the fact that the distribution of normalized predicted load is carried out among generators using sensitivity relation governed by Equation (61). This ensures that the higher portion of the normalized predicted load would be assigned to the generator having higher sensitivity factor i.e., the generator electrically closed to the target load bus.it has been observed that the distance factor becomes close to 1, when the iterative process terminates. But, at the beginning of the iterative process it appears to be high depending upon change of ΔJ D used for the prediction of load margin for the load bus. It normalizes the effect of over prediction of load margin due to wide change of ΔJ D and ensures convergence of the load flow analysis of the system during the iterative process. To examine the effect of limits on real and reactive power supply from generator(s) on the load margin, simulations were carried out on IEEE 30 bus system, since; it has only six generating stations. It is observed that Bus-29 is more sensitive to generator located at Bus-8. Different limits on real and reactive power supply are set this generator to examine the value of load margin at Bus-29. Case I: The maximum real and reactive power limits on max max generator located at Bus-8 are set as PG 8 = 0.55 pu QG 8 = 0.34 respectively and the bus is treated as PQ bus. Table-5 represents the load margin of bus-29 of IEEE 30 bus system with limits mention above. TABLE 5: CRITICAL LOAD OF BUS-29 OF IEEE 30 BUS SYSTEM FOR LOAD PF 0.8 (APPLYING LIMITS ON REAL AND REACTIVE POWER SUPPLY ON GENERATOR LOCATED AT BUS-8) With the limits on Generator 8, the contribution of real and reactive power supply from the generator located at Bus-1, Bus-2, Bus-5, Bus-11, Bus-13 are PG1 = pu QG1 = pu, PG2 = pu QG2 = pu, PG5 = pu QG5 = pu PG11 = pu QG11 = pu and PG13 = pu QG13 = pu respectively. predicted continuation parameter (Δλ) becomes slightly Case II: The maximum real and reactive power limits on max generator located at Bus-8 are set as as PG 8 = 0.55 pu QG max 8 = 0.24 respectively and the bus is treated as PQ bus. Table-6 represents the load margin of Bus-29 of IEEE 30 bus system with limits mention above. TABLE 6: CRITICAL LOAD OF BUS-29 OF IEEE 30 BUS SYSTEM FOR LOAD PF 0.8 (APPLYING LIMITS ON REAL AND REACTIVE POWER SUPPLY ON GENERATOR LOCATED AT BUS-8) With the limits on Generator 8, the contribution of real and reactive power supply from the generator located at Bus-1, Bus-2, Bus-5, Bus-11, Bus-13 are PG1 = pu QG1 = , PG2 = pu QG2 = pu,pg5 = pu QG5 = pu PG11 = pu QG11 = pu and PG13 = pu QG13 = pu respectively. Table 6 indicates that the load margin of Bus-29 of IEEE 30 bus system is found to be pu without any violation on real and reactive power supply of all the six generators of the system. Whereas, with limits on real and reactive power supply on generator located at Bus 8, the load margin decreased for the bus as indicated by Table-5 and Table-6. For Case I, the load margin is found to be pu with limits on Generator-8 as PG max 8 = 0.55 pu QG max 8 = 0.34 pu. But, the load margin decreases to pu with limits on Generator-8 as PG max 8 = 0.55 pu QG max 8 = 0.24 pu. It indicates that the both real and reactive supply limits on generating station(s) influence the load margin of a bus. Further, case study Case I and Case II indicate that real and reactive supply limits on generating station(s) influence the contribution from the generating station(s) supplying loads within their limits. Bifactorization matrix solution technique with optimal ordering of the sparse load flow jacobian matrix has been incorporated in the load flow analysis used for proposed method and that of continuation power flow analysis. The programs were executed on a P C with Pentium-4 processor having processor speed of 1.5 GHz and LINUX operating system. The simulation results show that the proposed method requires considerably less CPU time compared to that of continuation power flow analysis. Continuation power flow analysis requires on an average 2-3 times more CPU time than that of proposed method for IEEE systems. VII. CONCLUSION The paper proposed algorithm for a fast continuation load flow analysis to determine load margin of a target/selected load bus using the singularity condition of the load flow Jcaobian matrix. The load flow Jacobian matrix of an interconnected multi-bus power system is transformed into a two by two elements Jacobian matrix with respect to a target/selected load bus by incorporating the effect of all the

11 other buses of the system to the target bus. To incorporate the effect of other buses to the target bus, sensitivity relations have been formulated to relate change in injection at the 118 bus system indicated that the determinant of two bus equivalent system of a multibus system reduces with increase in load at the target bus and it becomes zero (near to zero) when load flow solution does not converge i.e., voltage collapse takes place. Therefore, it could be concluded that when the modified two by two matrix become singular, actual load flow Jacobian matrix also becomes singular. But, it is observed that determinant value of two bus equivalent system of IEEE 30 and 118 bus systems have different initial values for different target buses, as such, it shows that the transformed two by two elements Jacobian matrix reflects the property/quality of the target bus. The algorithm proposed for the determination of critical load of a bus with respect to its voltage collapse limit of a power system works for all buses of IEEE 30 and IEEE 118 bus system, as such, it could be used for any inter connected power system. The use of reduction factor (RF) and distance factor (DF) ensures convergence of the load flow analysis of the system during the proposed iterative procedure. These two factors effectively normalize the prediction of load margin, which is carried out using the linear relation between ΔJ D and ΔPk governed by Equation (55). The simulation results show that the proposed method requires considerably less CPU time compared to that of continuation power flow analysis. REFERENCES [1] D. Hazarika and A.K. Sinha Power system restoration: Planning and simulation. Int. Journal of Electrical Power & Energy Systems, Vol- 25, 2003, pp [2] Clark. H. K. New challenges: Voltage stability IEEE Power ENG. Rev (April 1990), pp [3] P. Kessel and H. 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Distrib., Vol. 145, No. 1, January, 1998, pp Durlav Hazarika received the degree in Electrical Engineering from Jorhat Engineering College, Jorhat, Assam, India in 1983, the Master degree from Indian Institute of Technology (IIT) Bombay, Mumbai, India in 1986 and the Ph.D degree from Indian Institute of Technology (IIT) Kharagpur, Kharagpur, West Bangle, and India in He is currently working as a Professor with the Department of Electrical Engineering, Assam Engineering College, Guwahati, Assam, India