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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005

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Application of a New Sensitivity Analysis Framework for Voltage Contingency Ranking

Nima Amjady, Member, IEEE, and Masoud Esmaili

Abstract—In this paper, a new sensitivity analysis framework for voltage contingency ranking has been presented. The proposed sensitivity analysis is a combination of linear sensitivities and eigenvalue analysis. The sensitivity analysis framework can determine the voltage stability status of the power system due to the occurrence of each contingency. Moreover, stability margin or instability depth of the post-contingency state is determined in the framework. In other words, a severity index is obtained for each voltage contingency and so the contingencies can be ranked. This rank shows bottlenecks of the power system in the priority order, a property that is a key issue for both planners and operators of the power system. The proposed method can also evaluate islanding contingencies as well as the nonislanding ones. Moreover, the method can consider the generator contingencies in addition to the branch contingencies in a unique framework. The proposed method has been tested on the New Zealand test system and Iran’s power network. Obtained results, discussed comprehensively, con?rm the validity of the developed approach. Index Terms—Contingency ranking, sensitivity analysis, voltage contingency.

I. INTRODUCTION OR the secure operation of a power system, all forms of security constraints must be met in any considered operating point. These include static constraints such as thermal limits of circuits, as well as dynamic constraints such as voltage, transient, and small-signal stability limits. Violating these constraints may result in severe consequences, even system wide blackout. In this paper, we focus on the voltage stability limit. Voltage instability is a load driven instability, which constitutes an important subset of power system instabilities. Nowadays, in many countries, the introduction of competitive supply and corresponding organizational separation of supply, transmission, and system operation has resulted in more highly stressed and unpredictable operating conditions, more vulnerable networks, and an increased need to monitor the operational security level of the transmission system [1]. These conditions, brought on by natural load growth especially for developing countries coupled with a signi?cant increase in long-distance transmission usage, often result in heavy transmission circuit loadings, depressed bus voltage magnitudes, and closer proximity to voltage instability [1].

F

Manuscript received February 13, 2004; revised April 16, 2004. Paper no. TPWRS-00071-2004. N. Amjady is with the Department of Electrical Engineering, Semnan University, Semnan, Iran (e-mail: amjady@tavanir.org.ir). M. Esmaili is with the National Dispatching Department, Tavanir Company, Tehran, Iran (e-mail: esmaili@iust.ac.ir). Digital Object Identi?er 10.1109/TPWRS.2005.846088

As electric utilities attempt to maximize the usage of their transmission system capacities to transport real power, voltage collapse acts as a limiting factor [2]. IEEE de?nitions for voltage collapse, instability and security can be found in [3] and [4], where it is concluded that voltage collapse may be caused by a variety of single or multiple contingencies known as voltage contingencies [5] in which voltage stability of the power system is threatened. Voltage contingencies such as a sudden removal of real and reactive power generation, loss of transmission line or a transformer or an increase in the system load without an adequate increase in the reactive power can decrease voltage stability margin of the power system [6]. Thus, a key issue in voltage stability studies is to determine the severity of each voltage contingency, i.e., evaluate the risk level of each contingency. The ranking of insecure contingencies according to their severity is known as contingency ranking [6], which is vital for both operators and planners of the power system. In this paper, we focus on the problem of voltage contingency ranking. However, contingency ranking for other kinds of stabilities such as transient stability and small signal stability is also important and was considered in the previous works [7], [8]. Early methods for voltage contingency ranking are based on the severity of MVA branch ?ow overloads [9], [10]. However, these methods cannot see other factors of voltage stability such as reactive power reserves [4] and so are not accurate. Others consider the severity of voltage violations due to contingencies [11], [12]. However, although voltages decline during voltage collapse, it is possible for a system to encounter it near nominal voltage levels [13]. The ?rst and second order eigenvalue sensitivity analyzes have been proposed for voltage contingency ranking in [14] and [15], respectively. These sensitivity analyzes are based on the dominant eigenvalue, an eigenvalue with the least absolute value. It can be shown that severe voltage contingencies can change the dominant eigenvalue and singular value position [16]. In other words, the dominant value of the precontingency can be replaced by another one in post-contingency. Thus, monitoring the dominant eigenvalue/singular value of the base case in the sensitivity analysis can result in ranking errors for severe voltage contingencies. In [1], a probabilistic method is proposed for risk evaluation and contingency ranking. The method requires not only a large amount of data to compose probability distribution functions (PDFs) for each contingency but also high computation burden to combine PDFs. Thus, the implementation of the method in a practical power system with many possible voltage contingencies, from hundreds to thousands, is very time consuming. Besides, in [17], linear and quadratic estimates of voltage stability margin change have been used for contingency ranking.

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These methods, using load domain voltage stability margin or loadability margin, are based on the algebraic Jacobian or static Jacobian. It is concluded that the linear estimate lacks accuracy for practical power systems and the quadratic estimate can only screen severe contingencies, but its rank is not correct. Indeed, to rank correctly this method requires higher order estimates (theoretically in?nite order). However, the computation burden of higher order estimates rapidly increases so that the estimates more than quadratic are not applicable for practical power systems. More recent approaches considering the combination of linear sensitivities with eigenvalue analysis [18], [19] are more ef?cient in comparison with the previous ranking methods. However, not only their application is limited to branch or nonislanding contingencies [18], [19] but also their estimation error is still high. In this paper, a composite sensitivity analysis that can modify imperfections of the previous methods is proposed for voltage contingency ranking. The proposed method can determine stability margin or instability depth of the post-contingency state, which is used as a severity index to rank the contingencies. The method is able to analyze and rank both nonislanding and islanding branch contingencies as well as generator contingencies. II. PROBLEM DESCRIPTION The goal of our study is the quantitative evaluation of the risk level of voltage contingencies to rank them. Our study is based on the following assumptions. 1) We consider only single contingencies (N-1 study), which is a practical assumption in voltage security studies. Probability of higher order voltage contingencies such as double contingencies is much lower than single contingencies. Moreover, considering higher order contingencies usually results in more expensive defense plans, which are almost not practically feasible [20]. 2) In this paper, we focus on the voltage contingencies. Thus, contingencies of the small signal stability are not considered. To more illustrate this assumption, consider that the dynamics of a large class of physical systems can be modeled by the following parameter dependent differential algebraic equations [21], [22]: (1) (2) (3) , dynamic state variables and In the state space algebraic variables are distinguished. The dynamics of states is de?ned by (1), and the behavior of variables is satis?ed by (2). In the above equations, is an array of parameters de?ning a speci?c system con?guration and the operating condition. Voltage stability and small signal stability problems are involved with some dynamic variables or , such as those of generators, and some algebraic variables or , such as those of power ?ow balance. Also, these stabilities depend on the power

system con?guration, generation pattern and load scenario, which can be represented by parameters. Therefore, the above model is comprehensive for voltage and small signal stability studies of the power system, and so it is used here. For a ?xed value of , an equilibrium point (EP) is a solution of the system (4) To determine the stability margin of the EP, by means of linearization, the above model can be represented as [15], [21] (5) where is called the unreduced Jacobian, augmented Jacobian, or augmented system state matrix [15], [21], [22]. Matrix is the partial Jacobian, the partial derivative of vector with respect to vector . Similarly, the other components of the are is nonsingular, it can be readily partial Jacobians. Assuming shown that (6) where is called reduced Jacobian or reduced system matrix; is called algebraic Jacobian or static Jacobian [15], [21], [22]. It can be shown that one of the eigenvalues of in small in signal instability point and one of the eigenvalues of voltage instability point reaches the origin [15], [18], [22]. Thus, the eigenvalue with the least absolute value, the closest eigenvalue to the origin, is the dominant eigenvalue for stability studies. Although the dominant eigenvalue is an attractive index for contingency ranking, but this index can unpredictably change due to large load variations and severe contingencies [16], [23]. Thus, the evaluation of the behavior of the existent dominant eigenvalue can result in wrong ranking, since another eigenvalue becomes the dominant. However, we can obtain useful insights from its characteristics. To study the voltage and small signal stabilities, we can focus on the Jacobian matrices and , respectively. According to the second assumption, we . Moreover, we propose a new formulation of the consider sensitivity analysis employing the zero crossing characteristic of the dominant eigenvalue to balance the equations. Then, the formulation is extended to handle generator contingencies and islanding branch contingencies. III. PROPOSED SENSITIVITY ANALYSIS In previous works, the dominant eigenvalue/singular value [14], [15], combination of voltage drops due to load buses [5], [25], and energy functions [13] have been used as the severity index for voltage contingencies. However, in the proposed framework, the load domain voltage stability margin has been considered as the severity index. This margin is the maximum loadability of the power system constrained by the voltage stability limit. Unlike the dominant eigenvalue/singular value, the margin is not disturbed by the severe contingencies. Voltage drops cannot accurately re?ect the risk level of voltage instability, as described in Section I. Energy functions as the severity index are usually highly nonlinear. Moreover, the load domain margin gives a better view than the energy margin. Finally, we

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can extend the load domain margin for unstable contingencies, as it will be described in Section III-B. In the proposed framework, sensitivity of the voltage stability margin with respect to the voltage contingencies is evaluated. Besides, the adaptive continuation technique [3], [4] is employed as the benchmark method to obtain the voltage collapse point and stability margin. The adaptive continuation method considers the reactive power reserves of the power system, PV-PQ transitions, and on-load tap changer (OLTC) control. This method has adaptive steps and can ?nd the collapse point more accurately and rapidly in comparison with the normal continuation method. The benchmark method is based on the contingency analysis (CAN) utility of the PN40 software package [29], developed by the ABB Company and widely used by the industry. We enhanced the CAN utility to the adaptive continuation for voltage contingency analysis. More details can be found in [3] and [4]. To introduce the proposed sensitivity analysis, we begin with the simplest case. A. Non-Islanding Branch Contingencies We simply begin with the load ?ow equations (7) (8) where is state vector including voltage magnitude and phase; and are net active and reactive power injections, respectively; and indicate power balance equations. Voltage contingencies can change the power system con?guration and so they affect the voltage stability. However, voltage contingencies are usually represented by discrete values: 1 means in service or no contingency and 0 means out of service or the occurrence of contingency. Mathematically, we cannot differentiate with respect to discrete variables. We solve this problem by means of an auxiliary parameter . Consider classical model for branches (transmission line or and indicate shunt and transformer) as shown in Fig. 1. series admittances, respectively. In this model

and indicate active and reactive power injections where of bus at initial operating point (initial loading), respectively; and are injections at current operating point; is scaling factor; and represent participation factors of bus in active and reactive scenarios, respectively. If bus does not and/or participate in the active and/or reactive scenarios, its become zero. These factors are selected according to the voltage stability scenario and then the scaling factor increases until the maximum transfer capability or voltage collapse point is obtained [25]. It is noted that for each scenario, a speci?c collapse point and thus a speci?c stability margin is obtained. By considering effect of the voltage contingency in the scenario, i.e., (10), we have (11) (12) (13) (14) In the base case, (11) to (14) are the same as (10) due to . In the post-contingency state, active and reactive powers of the transmission component are removed from buses and . It is necessary to consider the effect of contingency on both initial ) and voltage stability scenario (such as loading (such as ). For the other buses not involving in the contingency, the original equation (10) holds. Now we consider effects of the contingency and scenario in the load ?ow equations (15) (16) In the Appendix, it is shown that is dependent on the contingency or , and so the left hand side of (15) and (16) is a function of the state vector and contingency effect. The right hand side is a function of the scenario and contingency according to (11)–(14). Now, we apply the sensitivity analysis with respect to contingency to (15) and (16)

Out of Service In Service

(17) (9) (18) We ?lter out ?xed parameters of the state vector : voltage magnitude and phase of slack bus and voltage magnitude of PV buses (sensitivity with respect to a ?xed value is meaningless). including Thus, the dimension of X becomes phase angle of PQ and PV buses plus voltage magnitude of PQ buses. By combining (17) and (18), the result can be written as (19) where dimension of the upper and lower subvectors is and , respectively. Let

Thus, the auxiliary parameter can model occurrence of the is a factor voltage contingency. Moreover, since of admittance, it can continuously vary and we can differentiate with respect to it. Another bene?t of this technique is that we can model the partial outages, for instance, the outage of one circuit of a double circuit transmission line. In addition to the con?guration change or contingency effect, load scenario, including load level and distribution, and generation pattern of the base case are also effective on the voltage stability status of the post-contingency state [6], [24]. The two factors of load scenario and generation pattern are commonly referred to as voltage stability scenario, usually represented as follows [3], [4], [18], [24] (10)

. The

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Fig. 1.

Classical model for transmission components.

coef?cient matrix of is an matrix including partial derivatives of the load ?ow equations with respect to state variables. This is algebraic or static Jacobian [15], [22], previously explained as the key matrix for voltage staconsists of sensitivity of the bility studies. The vector state vector elements with respect to the contingency. indicates sensitivity of the loading margin. By means of and , we can determine effect of the contingency on the states and margin, respectively. Computation of coef?cients of (19) is described in the Appendix. These equations constitute a new ?rst-order sensitivity analysis framework for voltage contingency ranking. There are ef?cient software packages such as and Eurostag [26] to compute algebraic or static Jacobian the other coef?cients of (19) can be easily computed. Thus, the introduced sensitivity analysis is fast and ef?cient. However, this sensitivity analysis has a de?ciency. Number of unknowns ( variables of and one variable of in (19) is ) with only equations. Instead of using any approximation such as linear or quadratic estimates, we ?nd another equation by the zero crossing characteristic of the dominant eigenvalue introduced in Section II. The reference conditions for the sensitivity analysis are the base case status of the power system. By sensitivity analysis, we can determine deviation of each variable from the reference conditions due to each voltage contingency. The base case continuation curve (see Fig. 2) indicates the reference conditions. In other words, we have still a degree of freedom in selecting a point of the base case continuation. This degree of freedom is spent according to the zero crossing characteristic of the dominant eigenvalue, i.e., we select the voltage collapse point of the base case continuation as the reference point

equations can be easily solved (for instance, by LU factorization) and its computation burden is low. It is noted that the coef?cients of (19) and (21) are computed at the reference conditions, i.e., the collapse point of the base case continuation. Finally, the proposed sensitivity analysis framework for nonislanding branch contingencies can be summarized as the following step-by-step algorithm (we call it base algorithm). 1) Analyze voltage stability status of the power system in the base case (precontingency) using the adaptive continuation method, which determines the voltage collapse point and stability margin of the power system under the applied scenario [3], [4]. The collapse point of the base case is considered as the reference conditions of the sensitivity analysis. Now, compute the left and right eigenvectors of the dominant eigenvalue. linear algebraic equations of Constitute the set of (19). Constitute the single linear algebraic equation of (21). Solve the set of (19) and (21). At the output, we have two important results: Sensitivity of the loading margin with respect to the . Using the ?rst-order approxicontingency mation of Taylor series (?rst-order sensitivity), it can be written that

2)

3) 4) 5) a)

(22) (23) (24) and indicate value where at the collapse point of the base case and post-contingency states, respectively. For most voltage conis positive ( is negative), a tingencies property which decreases the or stability margin of the post-contingency state. Each and so has its specontingency has its own ci?c effect on the margin of the power system. More decrease in the margin results in a more severe contingency. So we can rank the voltage contingencies. Sensitivity of the state vector including voltage magnitude and phase with respect to the contingency

(20) where and are dominant eigenvalue and its corresponding right eigenvector, respectively. In the Appendix, it is shown that (20) results in the following equation:

(21) variables of . This is a single equation in terms of linear algebraic Thus, (19) and (21) constitute a set of variables. Now, the number of equaequations in terms of tions and unknowns are balanced. This set of linear algebraic

b)

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. Thus, we can determine variation of each state due to the contingency (25) indicates state . Using this sensitivity vector, where we can estimate the post-contingency status without analyzing it. It is noted that the voltage contingency changes the con?guration and operating point of the power system. So the post-contingency status requires a separate analysis [3], [4], [6], which can be avoided in the proposed sensitivity framework. Due to the space limitation and subject of the paper, only the obtained will be further discussed in the results for the paper. Now we extend the framework to generator and islanding branch contingencies. B. Extension to Islanding and Unstable Branch Contingencies Branch contingencies topologically include two important categories: nonislanding and islanding. The nonislanding contingencies affect the loading margin, while the islanding contingencies affect both topology and loading margin of the power system. Moreover, based on the stability margin, voltage contingencies consist of two categories: stable and unstable, with positive and negative post-contingency margin, respectively. It is noted that the unstable contingencies can be different in severity. From state space viewpoint, we can de?ne instability depth like the stability margin. More stability margin results in a less risky stable contingency and more instability depth results in more severity. As the stability margin is essential to perform preventive and corrective actions, the instability depth is important to design modi?cation plans [6]. Thus, four cases of branch contingencies must be considered: stable nonislanding, unstable nonislanding, stable islanding, and unstable islanding. For stable nonislanding branch contingencies, the base algorithm of the previous subsection is performed to obtain the post-contingency stability margin (see Fig. 2). In unstable nonislanding contingencies, no feasible operating point is obtained in the post contingency state. In such cases, the benchmark method applies the scenario in the opposite direction (see Fig. 2). In other words, we decrease the load level of the base case (by negative ) until a feasible operating point in the post-contingency is obtained. The required load curtailment to obtain the ?rst stable post-contingency operating point indicates the voltage instability depth (see Fig. 2). The benchmark method requires separate scenarios to obtain stability margin and instability depth. However, the proposed sensitivity framework can determine the instability depth like the stability margin using the unique procedure of the base algorithm. According to (22)–(24), if is less/greater than then the contingency is stable/unstable, respectively. For islanding contingencies, a virtual network including both islanded parts is considered. If both parts are stable in the postcontingency state, then the contingency and virtual network are considered stable. However, if one subnetwork or both become unstable, the contingency and virtual network are unstable. For

Fig. 2. Voltage continuation curves of the base case (1), stable contingency (2), and unstable contingency (3).

stable islanding contingencies, the benchmark method determines the collapse point of each islanded part separately according to the scenario of the original network. The total load increment of the both parts up to the ?rst collapse point indicates the stability margin of the virtual network and the post-continMin indicates gency state. In other words, value of the post-contingency state, where and are the scaling factor at the collapse point of each islanded part, respectively. In unstable islanding contingencies (one or both unstable part), we apply the voltage stability scenario in the opposite direction to the original network until both islanded parts (in the post-contingency state) stabilize. The ?rst stable point of the virtual network including both parts indicates stability boundary. Total interrupted load of the original network up to this point is the instability depth of the virtual network and post-contingency state. It is noted that in islanding contingencies, total load increment/decrement of the islanded parts is equal to those of the base case (original network). As seen, the benchmark method requires different models for islanding and nonislanding contingencies. However, the proposed sensitivity framework evaluates both cases by the same model of the base algorithm. This is due to the fact that the sensitivity framework, represented by (19) and (21), depends on the voltage collapse point of the base case where the post-contingency information is not required. This matter will be shown more clearly in the Numerical Results section. However, before numerical examination of the framework, we extend it to generator contingencies, too. C. Extension to Generator Contingencies Generator outage contingencies, which require a different model compared with branch contingencies [19], constitute an important part of voltage contingencies. Most voltage contingency ranking methods cannot evaluate generator contingencies [11]–[18]. One important reason for this fact is due to the lack of an admittance model (such as model) for generators. However, we have solved this problem in the proposed sensitivity analysis framework by the aid of practical considerations. In

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005

Fig. 3. Single line diagram of the New Zealand test system.

a practical power system, generators are generally connected to the transmission network with a step-up transformer commonly known as the main transformer (see Fig. 3). Only small generators (such as diesel generators) may be connected to the distribution network without the main transformer, but their outage contingency is not important for voltage stability. Another practical consideration is due to the structure of the main transformer. Practical main transformers are usually large transformers with low excitation current in comparison with their nominal current. Thus, the shunt admittance of these transformers can be ignored. It is noted that this assumption is not necessarily valid for transmission lines especially long transmission lines, since the reactive charge of these branches and so their shunt admittance can be considerable. However, the assumption is acceptable for the main transformers. Thus, the circuit model of these transformers reduces to series impedance and consequently the outage of the generator can be equivalent to the outage of its main transformer (see Fig. 3). Consequently, we can replace generator contingency with outage of its main transformer as an islanding branch contingency. The islanded parts include a single node for generator bus and the remaining network. Thus, we can evaluate the stability margin/instability depth of the post-contingency state as described in the previous subsection, though the single generator node is not usually determinant for voltage stability limit. Now the sensitivity analysis framework is completed and can be summarized as the ?owchart of Fig. 4. It is noted that in the block “Run the Base Algorithm,” steps 1 and 2 related to the analysis of the precontingency state are performed once. Moreover, in the steps 3 and 4 of the base algorithm, the contingency independent coef?cients of (19) and (21), are only computed for the ?rst case. Computation of the contingency dependent coef?cients and step 5, i.e., solving of (19) and (21), are iterated for each contingency. This matter will be described in more details in the following section. An advantage of the proposed framework is its ability to analyze generator and branch contingencies using a unique model. Note: To evaluate the whole security of a power system due to a contingency, the rotor angle stabilities must be analyzed in addition to the load driven stabilities. For instance, a nonislanding contingency may disturb the active balance of the power system more than the reactive balance and so it affects the fre-

Fig. 4.

Flowchart of the whole method.

quency stability more than the voltage stability. As another example, after an islanding contingency, the islanded parts may save their voltage stability, but one or both parts cannot save their frequency and a kind of rotor angle instability such as transient or frequency instability occurs. In the more complicated cases such as cascading events a combination of the load driven and rotor angle instabilities can occur, a phenomenon that can be seen in large and practical power networks. However, due to the subject of the paper, we only evaluate the contingencies from the voltage stability viewpoint. IV. NUMERICAL RESULTS The proposed contingency ranking method has been examined on two practical networks of the New Zealand test system, and Iran’s power network. The New Zealand test case, a wellknown small practical test system, includes 17 buses, 20 trans-

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mission lines, and six transformers, which all are main transformers (see Fig. 3). This test system, which is really the New Zealand South Island’s network, has voltage levels of 11 kV, 14 kV, 16 kV, and 220 kV and its data can be found in [27]. Another practical test system used in this paper is the transmission network of Iran, a test system with 397 buses, 569 transmission lines, 219 transformers, and 35 shunt devices. This test case, which can be considered as a large and complicated test system, mainly includes voltage levels of 132 kV, 230 kV, and 400 kV and its data can be found in [28]. The proposed method obtains the stability margin/instability depth as the severity index for voltage contingencies and then ranks them as shown in Fig. 4. The index and rank of the 20 most severe contingencies of the New Zealand test system obtained by the proposed and benchmark methods are shown in Table I. The benchmark value of each nonislanding contingency has been obtained by a separate execution of the adaptive continuation method. In islanding contingencies, two executions of the benchmark method are required. The index values in Table I are related to the entire power system. Thus, the index is the product of and load increment/decrement step of the entire power system. The load increment and decrement is due to the normal and opposite directions of the voltage stability scenario (see Fig. 2), respectively. It can be written that (26) from (24), the post-contingency severity Using index and rank of the voltage contingencies can be obtained. A less severity index results in a more severe contingency with the higher rank. The last column of Table I represents error of the post-contingency severity index Error Proposed Value Benchmark Value Benchmark Value (27)

RESULTS

OF THE

TABLE I CONTINGENCY RANKING TEST SYSTEM

FOR THE

NEW ZEALAND

and y or n indicates the occurrence of misranking (yes or no). The base case voltage stability margin is 542.6 MVA and all post-contingency margins are less than that of the base case. Six single-ended items appearing in the To column are generator contingencies and 14 double-ended ones are branch contingencies in which the 15th item (INVER220 to ROXB-220) is islanding. The other 13 branch contingencies are nonislanding. Each generator contingency has been replaced with its equivalent islanding branch contingency on the corresponding main transformer as described in Section III-C. Out of 20 contingencies, 17 cases are stable and three cases are unstable. Here, three unstable cases are generator contingencies. In this test system, generator contingencies are generally more severe than branch contingencies from voltage stability viewpoint. In the single islanding branch contingency, the network is divided into four-bus and 13-bus subnetworks where both are voltage stable. In the base case, active and reactive powers especially reactive power is transferred from four-bus subnetwork to 13-bus part. Thus, in the post-contingency state, the ?rst voltage collapse point is related to the 13-bus part due to less support especially reactive. Total load increment of the both parts in the collapse point of the 13-bus subnetwork (stability margin of the virtual network) is computed as 508.1 MVA. The decrease in the

margin of the post-contingency compared with the base case is relatively small. It results in the rank of 15 for this contingency. This is due to the fact that the transmission line of INVER220 to ROXB-220 carries relatively low power in the base case and is not a critical branch. As seen from Table I, the estimation accuracy of the proposed method for the severity index of stable/unstable, islanding/nonislanding and generator/branch contingencies is acceptable (2.52%). However, the estimation accuracy of the index slightly decreases for contingencies in the stability border (both marginally stable and unstable). The method can accurately detect 20 selected contingencies. However, small ranking errors are seen. Misranking of contingencies 13, 14 and 15 is due to relatively high estimation error of the islanding branch contingency. Misranking of contingencies 17 and 18 is due to close values of the index (519.2 and 520.2). Unlike the New Zealand test system, Iran’s transmission network is a complicated and large-scale test system. Obtained results for the 20 most severe contingencies of the Iran’s power network are presented in Table II. To analyze the post-contingency status of each contingency in Iran’s power system, entire network is considered and no model reduction is applied. However, number of all probable contingencies for Iran’s power network is very high. Thus, for illustration purposes, we consider only voltage contingencies of one Regional Electric Company (REC) out of 16 RECs of Iran’s power network. We empirically know that this REC named Khuzestan has voltage stability problems especially in the summer season. The Khuzestan REC has 37 buses and 80 branches including 72 lines and 8 transformers (excluding main transformers of generators).

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RESULTS

OF THE

TABLE II CONTINGENCY RANKING NETWORK

FOR IRAN’S

TRANSMISSION

TABLE III REQUIRED EXECUTION TIMES FOR THE PROPOSED AND BENCHMARK METHODS

For better illustration of the voltage stability problems, the most stressed case, i.e., annual peak load (9:22 pm, August 9, 2003), of the Iran’s power network has been considered for the contingency ranking. Single line diagram of the Khuzestan REC can be found in [28]. This REC consists of seven generating plants with 35 units. In the small test system of New Zealand, each plant has one equivalent generating unit. However, in the large practical test system of Iran, each plant can have a few similar or different units (for instance, a few similar steam units with a few similar gas turbines). Practically, in this power system, the single generator contingency consists of outage of one generating unit, since the other units of the plant can remain in service. Even similar generators of a plant can result in different contingencies with different severities. Some generators may be derated due to the operational constraints. For instance, Ramin power plant in the Khuzestan REC has four similar steam units; each with 305 MW installed capacity. However, at the test time, units 2 and 3 have been limited to 240 MW and 270 MW due to boiler and condenser problems, respectively. Moreover, bidding strategy of generators in the power market can result in different output powers and so different generator contingencies. For instance, Iran has a pool power market without clearing price (pay as bid). Each generator can offer its capacity in different steps with different prices (ten steps are allowed). Moreover, the generator owners usually monitor the load demand of the network and offer different bids, even for similar units. So, the generators can be selected with different output powers. The similar situation is seen for parallel transmission lines especially in congestion times. Thus, due to the operational limits and market effects, all 115 cases including 80 transmission and 35 generator

contingencies are considered and analyzed by the exact and proposed methods, respectively. The base case stability margin is 1639.2 MVA. In Table II, code numbers 8 and 9 indicate 230-kV and 400-kV buses, respectively. Only one generator contingency is seen in the Table. This unit has the largest active and reactive output power and also supplies a relatively vulnerable area. All 19 branch contingencies are related to transmission lines. Out of 20 most severe contingencies of Table II, ?ve cases are unstable, which all are islanding branch contingencies. In all ?ve cases, the smaller part islanded from the original network cannot supply its demand especially reactive load and so it encounters with the voltage collapse. The other 14 branch contingencies are nonislanding. For this test system, the branch contingencies are generally more severe than the generator contingencies. Estimation accuracy of the proposed method for Iran’s test case (5.44%) is slightly more than the New Zealand test system, although the former is about 24 times as large as the later. The estimation error in Table II slightly increases at the stability boundary like Table I, but it cannot affect the ranking results. Only small ranking errors are seen in Table II. The errors are due to close values of the severity index such as cases 15 to 18. Thus, the overall accuracy of the method is acceptable for the large test system of Iran. In Table III, execution times, which have been measured on the Pentium III 500 MHz personal computer, required for running the benchmark and proposed method for two test systems are shown. As seen from the last column, the proposed method works 3.04 times in the New Zealand test system and 36.06 times in Iran’s test case as fast as the exact method. The time saving advantage of the method for large power systems is much better than the small systems. By increasing the dimension of the power system, both number of voltage contingencies and computation burden for each contingency in the benchmark method increase. Thus, the total computation burden of the method rapidly grows with the dimension of the power system. However, the proposed method involves only with a set of linear algebraic equations, which can be easily and effectively solved by the numerical routines. The relatively time consuming effort of the proposed method is formation of the linear algebraic equations. However, the coef?cient matrices of these equations

and

(28)

are constant for all contingencies and so are calculated once. Only, the vectors of known values

and

(29)

AMJADY AND ESMAILI: APPLICATION OF A NEW SENSITIVITY ANALYSIS FRAMEWORK

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are dependent on the voltage contingency. But, according to the Appendix, the vectors of (29) can be easily computed. In other words, the complex part of the linear equations is computed once and the simple part of these equations must be separately calculated for each contingency. This characteristic highly reduces the computation burden of the proposed method, especially when the number of voltage contingencies increases. It is noted that if all voltage contingencies of Iran’s power network are considered, the time saving advantage of the proposed method becomes much better (our experience shows ?gures more than 100). Thus, the overall ef?ciency of the method is enhanced for large power systems. V. CONCLUSION In this paper, a new composite sensitivity analysis framework has been presented for voltage contingency evaluation and ranking. The proposed formulation considers the voltage stability margin/instability depth of the entire power system as the severity index for voltage contingencies. The base algorithm of the proposed framework is a new combination of the ?rst-order sensitivities and eigenvalue analysis. This algorithm is used for estimation of severity index for nonislanding branch contingencies. Then, the algorithm is extended to diverse kinds of voltage contingencies including unstable, islanding and generator contingencies. The proposed method has been tested on the New Zealand test system and Iran’s transmission network. Obtained results indicate that the proposed method can highly reduce the computation time while the accuracy is saved. Moreover, the ef?ciency of the method increases for large power systems with a large number of credible voltage contingencies. Here, the proposed framework is con?ned to single contingencies due to the practical considerations. However, the method can be easily extended for multiple contingencies. Moreover, the framework is capable of modeling the small signal stability contingencies. To do this, the applied static Jacobain for voltage contingencies should be replaced with the unreduced Jacobian for small signal stability. Finally, we used only a portion of the capabilities of the proposed sensitivity analysis related to the sensitivity of the loading margin with respect to the voltage contingency. However, the method can also calculate sensitivity of the state variables, which can be a matter of the future research. APPENDIX Consider Fig. 1 in which the transmission component between buses and trips. Then the varied elements of bus adare as follows: mittance matrix or (30) (31) (32) , , , : Self admittances; , : Mutual ad, , : self and mutual admittances without mittances; considering the component; , , : the admitcomponent. Indeed, in the tances with considering the

relations, we convert and to and , respectively. The second term in the left hand side of (19) is an dimensional vector. According to (30)–(32), at most four elements of this vector are nonzero if if and and are PV or PQ buses are PQ buses (33) (34)

, It is noted that the contingency affected admittances ( , and ) appears in , , and . If or is PV bus or slack bus then the number of nonzero elements is less than four. Similarly, according to (11)–(14), the second term in the right hand side of (19) has at most four nonzero elements if if and and are PV or PQ buses are PQ buses (35) (36)

The ?rst coef?cient matrix in the right hand side of (19) has as many nonzero elements as the number of buses contributing in active or reactive scenario if bus if bus contributes in active scenario contributes in reactive scenario (37) (38)

The nonzero elements of (33)to (38) are as follows: (39) (40) indicates the real part function (extracts real part of a where complex term) and superscript represents the complex conjugate. By means of (39) and (40) it can be written that (41) According to (30) and (32), we have (42)

(43) For partial derivatives of , it can be written that (44) (45)

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005

where indicates imaginary part function. By means of (44) and (45), we have (46) (47)

At voltage collapse point, we have

(59) Combination of (58) and (59) results in (21). REFERENCES

(48) . According to Similar results can be obtained for (11)–(14), nonzero elements of (35) and (36) are as follows: (49) (50) For nonzero elements of (37) and (38), it can be written that

(51)

(52)

(53) Up to this point, coef?cients of (19) have been calculated. Now we obtain (21) by taking derivative from (20)

(54) is a function of and . The coefIt is recalled that ?cients of (54) can be mathematically computed as follows: (55) where and indicate row and column number, respectively (56) (57) There is a new set of variables in (54). Thus, this equation in the current form is not useful, since the total number of variables will be increased. To solve this problem, we multiply (54) by the left eigenvector (58)

[1] M. Ni, J. D. McCalley, V. Vittal, and T. Tayyib, “Online risk-based security assessment,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 258–265, Feb. 2003. [2] B. Venkatesh, G. Sadasivam, and M. A. Khan, “Optimal reactive power planning against voltage collapse using the successive multiobjective fuzzy LP technique,” Proc. Inst. Elect. Eng., Gen., Trans., Distrib., vol. 146, no. 4, pp. 343–348, Jul. 1999. [3] N. Amjady and M. Esmaili, “Voltage security assessment and vulnerable bus ranking of power systems,” Int. J. Elect. Power Syst. Res., vol. 64, no. 3, pp. 227–237, Mar. 2003. , “Improving voltage security assessment and ranking vulnerable [4] bus with consideration of power system limits,” Int. J. Elect. Power Energy Syst., vol. 25, no. 9, pp. 705–715, Nov. 2003. [5] T. Jain, L. Srivastava, and S. N. Singh, “Fast voltage contingency screening using radial basis function neural network,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1359–1366, Nov. 2003. [6] N. Amjady, “Dynamic voltage security assessment by a neural network based method,” Int. J. Elect. Power Syst. Res., vol. 66, no. 3, pp. 215–226, Sep. 2003. [7] D. Ernst, D. R. Vega, M. Pavella, P. M. Hirsch, and D. Sobajic, “A uni?ed approach to transient stability contingency ?ltering, ranking and assessment,” IEEE Trans. Power Syst., vol. 16, no. 3, pp. 435–443, Aug. 2001. [8] C. Y. Chung, L. Wang, F. Howell, and P. Kundur, “Generation rescheduling methods to improve power transfer capability constrained by small-signal stability,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 524–530, Feb. 2004. [9] V. Brandwajn, “Ef?cient bounding method for linear contingency analysis,” IEEE Trans. Power Syst., vol. 3, no. 1, pp. 38–43, Feb. 1988. [10] F. D. Galiana, “Bound estimates of the severity of line outages in power system contingency analysis and ranking,” IEEE Trans. Power App. Syst., vol. 103, no. 9, pp. 2612–2624, Sep. 1984. [11] N. Hadjsaid, M. Benahmed, J. Fandino, J. C. Sabonnadiere, and G. Nerin, “Fast contingency screening for voltage-reactive considerations in security analysis,” IEEE Trans. Power Syst., vol. 8, no. 1, pp. 144–151, Feb. 1993. [12] V. Brandwain, Y. Liu, and M. G. Lauby, “Prescreening of single contingencies causing network topology changes,” IEEE Trans. Power Syst., vol. 6, no. 1, pp. 30–36, Feb. 1991. [13] N. Amjady, “Application of a new neural network to on-line voltage stability assessment,” Can. J. Elect. Comput. Eng., vol. 25, no. 2, pp. 102–110, Apr. 2000. [14] T. Smed, “Feasible eigenvalue sensitivity for large power systems,” IEEE Trans. Power Syst., vol. 8, no. 2, pp. 555–561, May 1993. [15] H. K. Nam, Y. K. Kim, K. S. Shim, and K. Y. Lee, “A new Eigen-sensitivity theory of augmented matrix and its applications to power system stability analysis,” IEEE Trans. Power Syst., vol. 15, no. 1, pp. 363–369, Feb. 2000. [16] A. C. Zambroni, “Identifying a vanishing eigenvalue in voltage collapse analysis with consideration of limits,” Proc. Inst. Elect. Eng., Gen., Trans., Distrib., vol. 148, no. 2, pp. 263–267, Mar. 2001. [17] S. Greene, I. Dobson, and F. L. Alvarado, “Contingency ranking for voltage collapse via sensitivities from a single nose curve,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 232–240, Feb. 1999. [18] A. J. Flueck, R. Gonella, and J. R. Dondeti, “A new power sensitivity method of ranking branch outage contingencies for voltage collapse,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 265–270, May 2002. [19] N. Yorino, H. Q. Li, S. Harada, A. Ohta, and H. Sasaki, “A method of voltage stability evaluation for branch and generator outage contingencies,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 252–259, Feb. 2004. [20] H. Wan, J. D. McCalley, and V. Vittal, “Risk based voltage security assessment,” IEEE Trans. Power Syst., vol. 15, no. 4, pp. 1247–1254, Nov. 2000. [21] B. Long and V. Ajjarapu, “The sparse formulation of ISPS and its application to voltage stability margin sensitivity and estimation,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 944–957, Aug. 1999.

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[22] G. M. Huang and N. C. Nair, “Voltage stability constrained load curtailment procedure to evaluate power system reliability measures,” in Proc. IEEE Power Eng. Soc. Winter Meeting, vol. 2, New York, Jan. 2002, pp. 761–765. [23] A. C. Z. de Souza, C. A. Canizares, and V. H. Quintana, “New techniques to speed up voltage collapse computations using tangent vectors,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1380–1387, Aug. 1997. [24] F. Capitanescu and T. V. Cutsem, “Preventive control of voltage security margins: a multicontingency sensitivity-based approach,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 358–364, May 2002. [25] B. Milosevic and M. Begovic, “Voltage-stability protection and control using a wide-area network of phasor measurements,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 121–127, Feb. 2003. [26] [Online]. Available: http://www.eurostag.be [27] M. A. Pai, Computer Techniques in Power System Analysis. Norwell, MA: Kluwer, 1986. [28] [Online]. Available: http://www.tavanir.com [29] PN40 Operator Manual, vol. 2, ABB Network Control Ltd., Turgi, Switzerland, 1993.

Nima Amjady (M’97) was born in Tehran, Iran, on February 24, 1971. He received the B.S.c, M.Sc., and Ph.D. degrees in electrical engineering in 1992, 1994, and 1997, respectively, from Sharif University of Technology, Tehran. At present, he is an associate professor with the Electrical Engineering Department, Semnan University, Semnan, Iran. He is also a consultant with the National Dispatching Department of Iran. His research interests include security assessment of power systems, reliability of power networks, load forecasting, and arti?cial intelligence and its application to the problems of power systems.

Masoud Esmaili was born in Sarab, Iran, in 1971. He received the B.Sc. and M.Sc. degrees in electrical engineering in 1994 and 1997, respectively, from IUST University, Tehran, Iran. At present, he is a senior engineer with the National Dispatching Department of Iran. His research interests include voltage security, contingency analysis, and insulation coordination.

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