1 Abstract— Reactive power plays a significant rule in power system reliability and security. Reactive power is considered as the network constraint in conventional reliability evaluation techniques. The impact of the failures of reactive power sources such as synchronous condensers and compensators on system reliability has not been considered in the existing reliability techniques. This paper presents a technique to evaluate system and load point reliability indices of power systems considering reactive power shortages due to the failures caused by reactive power sources. The reliability indices due to the reactive power shortages are separated from those due to the real power shortages. Two reliability indices related to reactive power shortage are proposed. The IEEE 30-bus system is modified and analyzed to illustrate the proposed technique. The results provide very important information for system planners and operators for reactive power management. Index Terms— Reactive power, power system reliability. I. I NTRODUCTION he objective of reactive power provision is to maintain power system reliability and security. Reactive power reserve is a basic requirement for maintaining voltage stability. The adequate reactive power reserve is expected to maintain system integrity under both normal and post contingency operation. As one of the well established ancillary services, reactive power support and voltage control plays a vital role in the operation of both conventional and restructured power systems. The effect of reactive power on system stability and security has been well investigated [1]- [8]. Large area blackout usually occurs in a heavily loaded system which does not have adequate local reactive reserves. Heavily loaded systems not only have high reactive power demand but also high reactive power losses in the lines. During a disturbance, the real power component of line loadings does not change significantly, whereas the reactive power flow can change dramatically [1]. The reason is that the voltage drop resulting from a contingency reduces the reactive power generation from line charging, thereby increasing reactive power losses. Sufficient reactive reserves should be available to meet the Var changes following a disturbance. How much more reactive power a power system can deliver depends on the operating condition The authors are with the School of EEE, Nanyang Technological University 1 , Singapore (e-mail: epwang@ntu.edu.sg) and Power Engineering School, Taiyuan university of Technology 2 , China and the location of the reserves. Many approaches have been developed in reactive power management and monitoring in order to improve the reliability of the system with respect to voltage stability/security problems [1]-[8]. Therefore it is important to consider the impact of reactive power in power system reliability evaluation in order to obtain more accurate reliability indices. Reliability evaluation techniques have been well developed [9]-[12]. The reactive power is usually considered as the network constraints in those techniques. During post contingency load shedding, the network violation is alleviated through the proportional or priority load shedding without considering the rule of reactive power. The estimation of post-contingency voltages and reactive power generation and flows was discussed using sensitivities [13]. Though employing piecewise linear estimates, the effect of equipment limits on the estimates was captured. The effect of shunt capacitor on distribution system reliability was studied [14]. The composite system reliability was investigated from the standpoint of voltage limits and generator real/reactive power constraints in [15]. The expected value of curtailed kWh due to lack of reactive power generation or due to exceeding of voltage limits and the expected value of voltage irregularity were also investigated [15]. However the following problems are either ignored or seldom considered in the existing reliability evaluation techniques. Firstly most existing techniques for power system reliability evaluation ignored the impact of the failures of reactive power resources such as synchronous condensers and various compensators on system reliability. Secondly most reliability evaluation techniques concerned more on the problems caused by real power shortage rather than those caused by reactive power unbalance during post contingency load shedding. Thirdly, the reliability indices due to the reactive power shortage were seldom considered separately with those due to the real power losses. System operators could not find the information related to the reliability problems caused by reactive power shortage from the existing reliability indices provided by the conventional reliability evaluation techniques. Therefore there is a need to find a relationship between the reactive power and system reliability with respect to voltage violations and system reliability. This paper presents a technique to evaluate system and load point reliability of a power system considering reactive power shortage due to failures caused by reactive power sources such as generators, synchronous condensers and Reliability Assessment of Power Systems Considering Reactive Power Sources Peng Wang 1,2 , Member, IEEE, Wenping Qin 2 , Xiaoqing Han 2 , Yi Ding 1 , Xinghui Du 2 T 978-1-4244-4241-6/09/$25.00 ©2009 IEEE 2 compensators. The reliability indices due to the reactive power shortages are separated from those due to the real power losses. The reliability indices related to reactive power shortages are proposed to provide more information to system operators and planners. The IEEE test system is modified and analyzed to illustrate the proposed technique. Section II discusses the important characteristics of reactive power sources and load. The real and reactive power models of generator, transmission line, compensator and load are also presented. The reliability evaluation techniques and load shedding methods will be discussed and the reliability indices associated with Var shortage during the post contingency are proposed in Section III. The modified IEEE 30-bus system has been analyzed using the proposed techniques and the results are presented in IV. Section V concludes the paper. II. R EACTIVE P OWER C HARICTERISTICS A ND M ODELING There are three aspects that differentiate reactive power from active power in power system operation. Firstly, it is not efficient to transfer reactive power over a long distance because the reactive power loss in transmission system is significant. Reactive power losses are typically about ten times of the active power losses due to the inductive nature of transmission lines. Therefore it is better to compensate the reactive power locally. Secondly, the main role of reactive power is to maintain voltage stability/security of power systems. The effect of reactive power on system reliability in terms of energy not supplied is indirect and should be calculated based on reactive power shortage and voltage violation. Thirdly, the total reactive power loss in the transmission network often exceeds the total reactive power load. The reactive power loss changes with system configuration and operation condition [7]-[8]. Reactive power requirement for releasing voltage violation after a contingency are heavily dependent on reactive power reserve distributions in the system. In order to determine and re-dispatch real and reactive power reserve for post contingency restoration, the models of various real and reactive power sources have to be studied. A. Generator A generator can provide both capacitive and inductive reactive power. According to a NERC planning standard guideline [4], reactive capability within 0.9 lagging and 0.95 leading should be available. A physical constraint in Var provision by a generator is its generation capability constraint which represents the hard physical limitation of a generator's capability for the simultaneous production of real and reactive power. A typical generation capability curve is shown in Fig. 1. The real power output of a generator is usually limited to a value within the MVA rating by the capability of its prime mover [16]. When real power and terminal voltage of a generator is fixed, the armature and field winding heating limits restrict its reactive power output. The armature heating limit is a circle with radius at IVR = 1 centered on the origin C 1 (0, 0) and given by the following equation: 2 at 22 )IV(QP ≤+ (1) Fig. 1 Typical generation PQ curves The field heating limit follows a circle with radius d it 2 X EV R = , centered at ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − d 2 t 2 X V ,0C and given by Equation 2. 2 d it 2 d 2 t 2 X EV X V QP ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ++ (2) where V t is the voltage magnitude at the generator bus, I a is the steady state armature current, E i is the excitation voltage magnitude, X d is the synchronous reactance, P and Q are real and reactive power output, respectively. The machine rating S R is the intersection point of the two circles. The corresponding rated real power output is denoted by P R . The reactive power capability limits of generator can be determined by: () d 2 t 2 2 d it max X V P X EV PQ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≤ for P<P R (3) () 22 atmaxg P)IV(PQ −≤ for P>P R (4) In real time operation, actual reactive power reserve Q r from a generator can be determined using the capability curves as () currentmaxr QPQQ −= (5) where Q current is the reactive power dispatched in the normal operation. In most conventional reliability techniques, constant maximum and minimum reactive power limits Q max and Q min are used in AC power flow analysis. The reactive power model for synchronous condenser is similar to generator. In this paper the maximum reactive power provided by a generator is assumed to be Q R when the real power output is P R. 3 B. Transmission line An equivalent π model is used to represent a long line in Fig. 2. Y ’ /2 Z ’ I s I r V s V r Y ’ /2 + + - - Fig. 2. Equivalent π model of a transmission line. The real and reactive power losses along a transmission line depend on line parameters such as series impedance Z ’ , and parallel admittance Y’, and bus voltages at the sending and receiving ends [16]. The equations to calculate the real and reactive power losses for short, medium and long lines can be found in [16]. Heavily loaded transmission lines lead to large inductive reactive power losses. In this case generators and compensators need to produce sufficient reactive capability to maintain the reactive power balance and to keep voltages within the specified limits. C. Var Compensators Compensation devices can provide both capacitive and inductive reactive power for a power system. A reactive power compensator is usually connected between a bus and ground. There are static and dynamic Var compensators in power systems. A basic dynamic Var generation arrangement using a fixed capacitor with a thyristor-controlled inductor is shown in Fig.3. The constant capacitive Var generation Q C of the fixed capacitor is opposite to the variable Var absorption Q L ( γ ) of the thyristor-controlled inductor to yield the total variable Var output Q T ( γ ) =Q L ( γ )-Q c . For a specific voltage V, the required positive or negative Q T ( γ ) can be obtained by changing turn off angle γ of thysistor. It is assumed Var compensators can provide both capacitive and inductive reactive power. Fig.3. A typical thyristor-controlled Var compensator and its Q output D. Load Load at a bus of a bulk power system is the aggregation of the associated sub-transmission network and various loads. Therefore modeling such an aggregate load requires the network parameters of sub-transmission system and the profile of each individual load. A typical PV curve is shown in Fig. 4 which addresses the proper voltage profile at a load bus in a power system [17]. It can be seen from the curve that bus voltage is reduced as load increases. The real P b and reactive Q b power at point b correspond to values at the low limit of the bus voltage. c is the point of system voltage collapse when the reactive power provided by system is less than the required reactive power Q c to maintain voltage level at V c . The reactive safety margin can be defined using the curve as: bcurrentr QQQ −= (6) Fig.4. A typical PV curve at a load bus III. R ELIABILITY I NDICES AND E VALUATION T ECHNIQUE A. Component reliability models A power system component such as generator, synchronized condenser, transmission line and reactive device can be represented using the two-state reliability model [18] as shown in Fig. 5. The availability A and unavailability U can be calculated based on failure rate and repair rate using the following equations. Up λ Down μ Fig. 5 Two-state model of a component A μλ μ + = (7) U μλ λ + = (8) B. Basic system reliability indices Considering an N-component power system, the basic reliability parameters such as the probability i p , departure rate i λ , frequency F i , the total real power generation capacity P i and reactive power generation capacity Q i for state i with M failed components can be determined using the following equations respectively: ∏∏ =+= = M 1j j N 1Mj ji UAp (9) ∑∑ += +== N Mj M j jji 11 μλλ (10) V + I L SW I c I Q L C Q L ( γ ) Q T ( γ ) Q c γ Q V P(Q) O a b c P b (Q b ) P current (Q current ) P c (Q c ) 4 iii PF λ = (11) ∑ = = Ngi 1k ki PP (12) ∑ = = Nqi 1k ki QQ (13) where A j , U j , j λ and j μ are the availability, the unavailability, the failure rate and the repair rate of component j respectively, P k is the available real power capacity of generator k, Q k is reactive power capacity of compensator or generator k, and N gi is the total number of generators and N qi is the total number reactive power sources. C. Load point and system reliability indices Two conventional annualized reliability indices of the expected load curtailment (ELC) and the expected energy not supplied (EENS) can be calculated using the proposed technique. The EENS due to the real power shortage (EENS P ) and reactive power shortage (EENS Q ) are separately calculated. The expected MVarh shortage (EVarS) considering the failures of reactive power sources and the expected Var not supplied EVNS are defined to provide more reliability information for system planners and operators. The annualized reliability indices for annual constant load can be calculated using the following equations: ∑ = ×= NC 1i ii FLCELC (14) ∑ = ××= NC 1i iPiP 8760pLCEENS (15) ∑ = ××= NC 1i iQiQ 8760pLCEENS (16) ∑ = ××= NC 1i iQiQ 8760pQCEVNS (17) 8760pVarSEVarS NC 1i iQi ××= ∑ = (18) where NC is the total number of considered contingencies, Pi LC and Qi LC are the real power load curtailment due to real power and reactive power shortage for state i respectively, Qi QC are the reactive power load curtailment due to reactive power shortage for state i , Qi VarS is the Var shortage which cause the voltage drop, and LC i =LC Pi +LC Qi . D. Load shedding Network violations for a contingency can be released using load shedding and real and reactive power re-dispatch. Load shedding technique is more complicated when considering both real and reactive power shortage. In real time operation, different power systems may have different load curtailment strategies such as proportional, priority and wheeling load shedding, and the corresponding direct load control equipments are required for the implementation of those strategies, which should and can be considered in reliability evaluation. A two step curtailment strategy is presented in this section to illustrate load shedding with considering reactive power shortage. For each contingency state, the corresponding total real power available capacity is firstly compared with the total real power demand (load plus estimated transmission loss). If the total real power capacity is less than total real power demand, the real and related reactive power of the load at each bus in the system is curtailed using the proportional load shedding technique. After the load shedding due to the real power shortage, AC power flow is performed to check network violations and generation adequacy. If the total real power generation is sufficient for the total demand and the voltage violations exist, the violations are due to reactive power shortage. Because the reactive power cannot be delivered efficiently through a long distance due to the transmission loss, the reactive power is usually compensated locally. Therefore the load shedding related to the reactive power shortage should be in the local buses with the low voltage, which is different with the load shedding approach due to the real power shortage. In this technique, the load shedding due to reactive power shortage is usually performed at the nodes with voltage violation. If the voltage violation still exists after the load shedding at those buses, the load shedding is expanded to the surrounding nodes. E. Determination of reactive power shortage In order to determine reactive power shortage under a contingency state, reactive power is injected step by step at the node with the low voltage to raise the voltage. When the voltage reaches its low limit, the corresponding reactive power injected is the reactive power shortage VarS Qi in equation (18). F. Procedure of reliability evaluation Based on the proposed models, equations and techniques, the procedure of the AC power flow based reliability evaluation considering reactive power include the following steps: Step1: Input basic network data, load data, generation data, and basic reliability data. Step2: Determine system state using contingency analysis and calculate basis system reliability indices for state i. Step3: Calculate total system real and reactive power capacity P i and Q i respectively. Step4: If P i is less than the total real power demand, cut real and reactive load proportionally at each load bus and update EENS P , EVNS Q and ELC. Otherwise go to next step Step5: Determine line overflow violations using AC power flow analysis. Step6: If there is the line overflow, gradually cut the load at the ending bus of the line and go to Step5. Otherwise update EENS P , EVNS Q and ELC, and go to next step. Step7: Determine voltage violations using AC power flow analysis. Go to next step if there is voltage violation. Otherwise go to Step14. Step8: If the real power capacity is sufficient, gradually increase reactive power injection at the node with low voltage. Step9: Perform AC power flow analysis. Go to Step8 if there is voltage violation. Otherwise update EVarS and go to next 5 step. Step10: Remove the reactive power injected at Step8. Step11: Gradually cut the load at the bus with low voltage. Step12: Perform AC power flow and check the voltage violations. Step13: Go to Step11 if there is voltage violation. Otherwise update EENS Q , EVNS Q and ELC and go to Step14. Step14: If all contingencies are considered go to next step. Otherwise go to Step2 for next state. Step15: Calculate system reliability indices. IV. S YSTEM S TUDIES The IEEE 30-bus system [19] as shown in Fig. 6 is analyzed to illustrate the proposed technique. The system is selected due to its requirements of the reactive power compensation caused by special radial configuration from two generation stations to the remote load centers. Fig. 6 The single line diagram of IEEE 30-bus system There are two generation buses in the system. Generation bus 1 consists of 4×60MW units. Generation bus 2 consists of 3×40MW units. In order to illustrate the effect of reactive power reserve on system reliability, the system has been modified. The reactive power limits of the generators, synchronous condensers and static Var compensators have been changed. The reactive power limits and the reliability parameters for each generator and condenser are shown in Table A1. The reliability parameters of the transmission lines are shown in Table A2. Annual constant load is used in analysis. The real and reactive power load is bundled together using fixed power factor, which means that real and reactive power must be curtailed at the same percentage during the real or reactive power shortage. The failures up to second order have been considered in the analysis. The proposed annual load point and system EENS P , EENS Q and ELC have been calculated and the results are shown in Table 1. It can be seen from Table 1 that load point at bus 26 has highest EENS P followed by the load point at bus 5. The highest EENS P at bus 26 is because of the single transmission line connected to the bus and the high probability of the first order failure. The higher EENS P at bus 5 is due to the highest load for the second order failure. Unlike EENS P the load point at bus 29 has highest EENS Q followed by the load point at bus 30. This is because the long transmission line from reactive power compensators to the two buses. The results also show that the system EENS Q caused by the voltage violation is about 28 percent of EENS P due to real power shortage and should be considered in reliability evaluation. T ABLE 1 L OAD POINT AND SYSTEM EENS P , EENS Q AND ELC Bus # EENS P (MWh/yr) EENS Q (MWh/yr) ELC (MW/yr) 2 29658.65 423.13 1902.36 3 3414.77 46.80 239.57 4 10387.36 148.19 666.26 5 139311.10 92880.21 23155.98 7 33718.61 7112.26 3476.45 8 45207.55 9358.23 4756.47 10 7927.20 113.10 508.46 12 15307.69 5663.05 1751.07 14 9256.00 2550.78 1056.31 15 11207.42 159.89 718.86 16 6696.84 68.25 720.86 17 16085.15 1987.96 1999.76 18 5584.61 62.40 542.97 19 17777.68 781.66 2000.93 20 3561.90 51.17 314.94 21 48446.32 7938.02 8470.37 23 5719.17 712.83 712.35 24 11890.79 714.91 880.86 26 1124030.00 34429.42 147960.80 29 11319.81 182601.40 25045.13 30 49995.85 105795.30 22653.80 System 1606505.00 453599.00 249534.50 The proposed annual load point and system EVNS Q and EVarS are also calculated and the results are shown in Table 2. The EVarS results show that the voltage violations caused by Var shortage can be released using the reactive power injection at those nodes instead of shedding the load. Therefore EVarS provide the system operator with very important information for post contingency restoration. Both 6 real and reactive power shortage will cause the reactive power curtailments because real and reactive power loads are bundled by the fixed power factor. T ABLE 2 L OAD POINT AND SYSTEM EVNS Q AND EVarS Bus # EVNS Q (Mvarh)/yr EVarS (Mvarh)/yr 2 17605.47 0 3 1730.79 0 4 2218.01 0 5 46832.64 73617.88 7 19520.02 9238.56 8 54565.78 10443.33 10 2272.51 0 12 14042.91 6356.84 14 3046.91 2009.08 15 3465.64 0 16 3479.19 0 17 11647.11 2373.76 18 1588.22 0 19 6642.29 560.42 20 1149.61 15.31 21 36085.98 9261.80 23 3216.00 587.33 24 9707.84 764.81 25 0 0 26 761273.60 39626.82 27 0 0 29 72720.46 199034.20 30 27924.83 90334.29 System 1101236.00 444224.40 V. C ONCLUSIONS This paper presents a technique to evaluate system and load point reliability of power systems considering reactive power shortage due to failures caused by reactive power sources such as synchronous condensers and compensators. The reliability indices due to the reactive power shortage are separated from those due to the real power shortage. The reactive power shortage is calculated through the reactive power injection at the nodes with low voltage. The IEEE 30- bus system is modified and analyzed to illustrate the proposed technique. The results provide very important information and different ways for system operator to alleviate the network violations and for system planner to determine the optimal location for installing new reactive power compensators. VI. R EFERENCES [1] B. Leonardi, V. 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Appendix T ABLE A1 R ELIABILITY PARAMETERS AND REACTIVE POWER LIMITS Bus # Q min Q max λ μ Generator 1 -20 25 6 194.67 2 -20 20 4.5 219 Compensator 5 -20 25 6 194.67 8 -10 25 6 194.67 11 -6 20 6 194.67 13 -6 20 6 194.67 7 T ABLE A2 R ELIABILITY PARAMETERS OF T RANSMISSION L INES From Bus To Bus λ μ 1 2 1 876 1 3 1 876 2 4 1 876 3 4 1 876 2 5 1 876 2 6 1 876 4 6 1 876 5 7 1 876 6 7 1 876 6 8 1 876 6 9 1 876 6 10 1 876 9 11 1 876 9 10 1 876 4 12 1 876 12 13 1 876 12 14 1.5 876 12 15 1.5 876 12 16 1.5 876 14 15 1.5 876 16 17 1.5 876 15 18 1.5 876 18 19 1.5 876 19 20 1.5 876 10 20 1.5 876 10 17 5 876 10 21 5 876 10 22 5 876 21 22 5 876 15 23 5 876 22 24 1.5 876 23 24 1.5 876 24 25 1.5 876 25 26 5 876 25 27 5 876 28 27 1.5 876 27 29 5 876 27 30 5 876 29 30 5 876 8 28 1.5 876 6 28 1 876 Peng Wang (M’00) received his B.Sc. degree from Xian Jiaotong University, China, in 1978, the M. Sc. degree from Taiyuan University of Technology, China, in 1987, and the M. Sc. and Ph.D. degrees from the University of Saskatchewan, Canada, in 1995 and 1998, respectively. Currently, he is an associate professor of Nanyang Technological University, Singapore. Qin Wenping received her B.S. (1995) and M.S. (2001) degrees in College of Electrical & Power Engineering of Taiyuan University of Technology (TUT), Taiyuan, China (e-mail: qinwenping@tyut.edu.cn). Currently, she is a lecturer in TUT and a visiting scholar in Nanyang Technological University, Singapore. Her research interests include power system reliability analysis, security assessment, stability analysis and protection.