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Review of reduction techniques in the determination of composite system adequacy equivalents
Electric Power Systems Research 80 (2010) 1385–1393
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Review
Review of reduction techniques in the determination of composite system adequacy equivalents A. Akhavein a , M. Fotuhi Firuzabad b,∗ , R. Billinton c , D. Farokhzad d a
Department of Engineering, Islamic Azad University, Science & Research Branch, Tehran, Iran Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran c Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, Canada d Iran Grid Management Company, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 17 January 2010 Received in revised form 27 May 2010 Accepted 4 June 2010 Available online 7 July 2010 Keywords: Reliability equivalent Adequacy equivalent External network modeling Network reduction
a b s t r a c t Reliability evaluation of a large composite power system involves the consideration of numerous outage events and consequently extensive calculations. Therefore, applying justifiable simplifications such as the determination of an equivalent networks can be very useful in reliability evaluation of large systems. This paper presents a review of procedures which are directly or indirectly applicable to the determination of composite system adequacy equivalents. Following a brief discussion on the basic characteristics of the various methods, their limitations are presented. Concluding points regarding the methods and modeling schemes related to adequacy equivalent are presented at the end of the paper. © 2010 Elsevier B.V. All rights reserved.
Contents 1. 2.
3.
4. 5.
6. 7.
8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 Equivalent networks in load-flow studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.1. Ward equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.2. Radial, equivalent and independent (REI) network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.3. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387 Adequacy equivalents in the form of a capacity-probability table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 3.1. Vertical integrated environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 3.2. Restructured or deregulated environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 3.3. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 Probabilistic load flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Methods based on graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 5.1. Binary and capacitated-flow graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 5.2. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Intelligent-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Limitations of the reviewed methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.1. Equivalent networks in load-flow studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.2. Adequacy equivalents in the form of a capacity-probability table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.3. Methods based on graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392
∗ Corresponding author. Tel.: +98 21 66165901/21; fax: +98 21 66023261. E-mail address: [email protected] (M.F. Firuzabad). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.06.002
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1. Introduction
2. Equivalent networks in load-flow studies
There are many instances in which studies are of more concern in one area of a power system and are not of direct concern in the remaining parts. The phrases “study area” (SA) and “external area” (EA) are used in this paper to describe the area in question and the remaining regions, respectively. Fig. 1 shows the partitioning of a power system into SA and EA. Boundary buses and tie lines can be defined in each SA or EA. It is usually intended to perform detailed studies in the SA. However, the EA is important to the extent where it affects SA analyses. The attraction to apply an equivalent for the EA increases as the dimensions of the power system and the complexity of the required analysis increases. This equivalent network reduces the dimensions of the EA and subsequently leads to reduction in the required time for analysis. Reliability evaluation in a composite generation and transmission system is a problem with massive and time-consuming calculations and hence, it is an obvious candidate for the application of equivalent networks. Reliability analysis is conventionally categorized into adequacy and security problems. Adequacy is interpreted as the existence of sufficient facilities to supply the system loads with consideration of equipment constraints under static conditions. Security is a measure of system capability to withstand dynamic or transient disturbances [1]. The problem of finding a reliability equivalent (often adequacy equivalent) was first addressed by researchers in the 1970s. In this problem, it is intended to find an equivalent (simplified) model for an EA which facilitates adequacy evaluation studies. As the EA is usually the larger portion of the system, the connection of an EA adequacy equivalent to the SA reduces the calculation burden and makes the detailed reliability evaluation in the SA become more feasible. This paper presents a review of methods, which are directly or indirectly related to the determination of adequacy equivalents. These methods can generally be divided into two types. The first type deals with finding an equivalent network for a portion of a system. Simplification of the reliability evaluation is the main focus in the second type. Continuous enhancement in computer capabilities cannot generally obviate the need for adequacy equivalents in composite system reliability evaluation. This is because of concurrent increases in the sizes of interconnected power systems and in the complexity of their operation procedures. Power transfer agreements between power systems, deregulation and system restructuring and open access to transmission networks are such examples. In addition, limitations and costs of technical PC-based software usually limit the size of a system under study to a few hundred buses. It is therefore useful for a user to apply adequacy equivalents for the purpose of system size reduction. Since a significant common concept exists between reliability and risk, these terms have been used interchangeably with the same meaning in this paper.
The issue of determining static equivalents for EA in load-flow studies dates back over 60 years. The use of equivalent networks proved to be successful and equivalent network concepts have been extended in other fields such as security and steady-state stability analysis. An equivalent network is generally obtained for an operating point (called the base condition) in the power system. It is often assumed that in the base condition, loads are at peak values, circuit breakers are in their normal state and no element outages exist [2–5]. Some papers utilize the average load condition [3,4]. Multiple base conditions corresponding to different loading levels have also been utilized [6]. It should be pointed out that equivalent networks are only exact at the base condition and greater deviation from this condition, leads to less precision [2,7,8].
Fig. 1. Partitioning a power system into external and study areas.
2.1. Ward equivalent Determination of a Ward equivalent for an EA contains three main stages [2,7,9–11]. In the first stage, with consideration of the base condition, power injections in the EA buses are converted into current injections. Gaussian reduction is used to omit EA buses in the second stage where the outcomes are equivalent current injections and fictitious admittances in the boundary buses between the EA and the SA. In the third stage, equivalent current injections are converted to corresponding equivalent power injections in the base condition. Fig. 2 shows an example of the Ward equivalent network of an EA in the boundary buses of a SA. The older version of the Ward equivalent model is the admittance version in which power injections are converted to admittances in the base condition. This version has lower precision especially for PV buses [7,9–12]. 2.2. Radial, equivalent and independent (REI) network In a REI network, some or all of the power injections at the EA buses are substituted by one equivalent power injection toward the boundary buses. At least one equivalent generation node and one equivalent load node are usually assigned for generators and loads, respectively. Fig. 3 shows a typical REI conversion process in which m generators are transformed to a single equivalent REI generation node.
Fig. 2. Ward equivalent network of an external area with equivalent power injections in the boundary buses.
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Fig. 3. Stages in the determination of a REI network for an external area in which generators in this area are substituted with a passive R node (shunt branches among boundary buses and ground are not shown for simplicity).
As stated in [2–4,6,7,9,10,12,13], the determination of a REI includes three stages. In the first stage, REI node R and common node 0 are established. The equivalent complex power injection in node R is equal to the sum of the complex power of the generators omitted. The voltage of the 0 node is arbitrary and often considered to be zero. In the second stage, fictitious admittances among nodes 1 to m and 0 are calculated in such a way that power injections toward the remaining buses are the same as the base-condition injections. Gaussian elimination of the passive nodes among node R and boundary buses constitutes the third stage. The negative injection powers at load buses are accommodated by changing the sign of powers and currents (direction of the powers and currents) in Fig. 3 [2,4]. 2.3. Further examples In [14] composite system reliability is evaluated using a DC Ward equivalent of the EA. In order to simulate branch outages in the EA, the statistical mean of the positive and negative injections and the admittance values of the boundary branches are used in the EA Ward equivalent. Shunt branches are not considered in the process of AC Ward equivalent extraction for the EA in [15]. Electrical distance between buses, based on mutual admittance, is considered as a guide in partitioning a system into a SA and an EA. As contingencies do not result in considerable admittance changes in the EA, it is suggested in [15] that the statistical mean of the admittances to be used in the equivalent network. As shown in Fig. 4, papers [16,17] separate a power system into three regions of equipment outage area, optimization area and external area with the intention of increasing the calculation performance in a composite system reliability evaluation. In the equipment outage area, the probabilistic behavior of generators and transmission lines are fully considered. In the optimization area it is assumed that equipments are fully reliable and participate in optimization problems associated with remedial actions. Using the assumption of fixed generators and loads, the external area is substituted by its DC Ward equivalent. Capacity limitations of the transmission lines in the external area are neglected.
In [4], reliability evaluation for two interconnected systems is accomplished by determining one generation and one load REI node for each system. This paper creates a REI network based on DC loadflow equations. Ref. [13] applies an AC REI network of the EA for analysis of power transactions between an EA and a SA and system security. Two REI nodes are introduced for the generators and loads in the EA. Transaction patterns are simulated through power injection variations in the REI nodes by the user. In [18,19], a DC Ward equivalent is determined for passive EA network in which all generators and loads of EA are deactivated. Then, as shown in Fig. 5, the virtual slack bus and branches are added to the remaining system. Generation and load balance in EA is assigned to the introduced slack node. Reactances of the fictitious branches among the slack node and boundary buses are calculated such that power entries in the boundary buses are the same as the base condition. It is assumed that the EA and SA do not assist each other during emergency conditions. The method mentioned in [18,19] is very sensitive to the selection of the EA and SA and direction of power flow between them. Reliability evaluation of the entire power network in Iran is difficult and in [20], the Iran network is divided into five areas. Separation of areas is performed such that a minimum number of tie lines remain between them. As shown in Fig. 6, each tie line is cut at its mid point and virtual loads and generators are added at
Fig. 4. System partitioning into three areas of equipment outage, optimization and external.
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Fig. 5. Determination of an equivalent for active and passive elements of EA in boundary buses with SA.
Fig. 7. Sample system to illustrate finding an equivalent for the EA. Table 1 Reliability data for the generators and transmission lines in Fig. 7. Component
Failure rate (occ./year)
Repair rate (occ./year)
Rating (MW)
Generating units Transmission lines
1 2
99 998
30 35
3. Adequacy equivalents in the form of a capacity-probability table 3.1. Vertical integrated environment
Fig. 6. Tie lines bisection and addition of virtual generators.
this point in order to simulate real and reactive power flows under normal conditions. Generators with zero real generation, as shown in Fig. 6, are added to increase the accuracy of the contingency analysis.
Refs. [21–26] present a method in which an adequacy equivalent of an EA is obtained in the form of equivalent generation and load (available capacity) for a desired contingency level in the EA. Fig. 7 shows a sample system to explain this approach. The associated data is given in Table 1. Table 2 shows the available capacities or powers at the boundary bus 2 of the EA with their probabilities and frequencies for contingencies up to the second level. It has been assumed that all the generators and lines are the same. During the process of tabulating, the tie line between buses 2 and 3 is considered to be in an open state. Once the contingencies or different system failure states are analyzed, states with the same effect (same available capacity) are combined [21–24,27]. The equivalent generator column in Table 2 represents the total available generation in bus 2 if all the loads in the EA are neglected. Subtraction of the equivalent generator and available capacity columns produces the equivalent load column in Table 2. Consequently, as shown in Fig. 8, the EA can be modeled for each outage state j, as an equivalent generator and load at boundary bus 2. Introducing an equivalent load facilitates modeling of load shedding philosophies [24].
Table 2 Adequacy equivalent for the EA at boundary bus 2 in Fig. 7. State no.
Contingency level
Equivalent generator at bus 2 (MW)
Available capacity at bus 2 (MW)
Equivalent load at bus 2 (MW)
Probability
Frequency (occ./year)
1 2 3 4 5 6
0 G L 2G G&L 2L
60 30 35 0 30 0
35 5 15 −25 5 −20
25 25 20 25 25 20
0.976184 0.019721 0.003913 0.000010 0.000079 0.000004
5.857101 2.050971 3.920384 0.020119 0.086946 0.007833
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Fig. 8. Replacement of the EA with an equivalent generator and load for each outage state j.
Refs. [21–24] apply the network flow method (as described in [28]) for determining the maximum flow from a source node to a sink node. The network flow method solves a transportation problem with only constraint of branch capacity limitation. This method has the advantage of simplicity and low calculation burden [25,26]. In general, determination of available capacity leads to an optimization problem subject to generation and transmission constraints. 3.2. Restructured or deregulated environment Adequacy equivalents at each bulk load point (BLP) have been determined in [29–35] in order to evaluate the reliability in a restructured power system. A distribution entity with customers is considered as a BLP [32]. An adequacy equivalent is in the form of a table containing different service states (available capacities) with their probabilities and frequencies from a BLP view. This model is designated as an equivalent multi-state service provider (EMSP). Fig. 9 depicts the concept of EMSP at a BLP [31]. In subsequent works, the EMSP has been divided into two equivalents representing a generation provider and transmission provider according to a model of the power market [30,32,33]. Equivalents are determined at each BLP for generation and transmission service providers considering the transactions agreed between producers and consumers (market model). These equivalents are designated as an equivalent multi-state generation provider (EMGP) and an equivalent multi-state transmission provider (EMTP). The EMGP is a table containing outage states with their probabilities and frequencies and the real power that producers can deliver to the BLP in each outage state [30,32]. In the power pool operation mode of the market, one EMGP is created for all Gencos, as shown in Fig. 10. In the bilateral transaction market operation mode, each Genco has an EMGP as displayed in Fig. 11 [30,32,33].
Fig. 9. Determination of a composite system equivalent for a bulk load point in the form of an equivalent multi-state service provider (EMSP).
Fig. 10. Replacement of producers with equivalent multi-state generation provider (EMGP) in power pool operation mode.
In addition to the type of agreement between producer and load entity (the parties), the load point reliability also depends on transmission configuration between these parties. Refs. [32,33] propose a multi-state EMTP model for the transmission network between the EMGP and the related BLP. During the determination of the EMTP model, power transfer possibilities between the EMGP and related load points are evaluated for each outage state [30]. Figs. 12 and 13 illustrate modeling the transmission network between the producers and the load points [30,32–34]. After finding the adequacy equivalents for the generation and transmission service providers, the load point reliability indices are calculated in three stages. In the first stage, indices corresponding to convolution of the generation and load models are obtained. Combination of the transmission and load models provides reliability indices for the second stage. The calculated indices in the previous stages are aggregated in the third stage. This summation gives an over-estimation of the reliability indices [30,32,36]. 3.3. Further examples In the first step of the procedure proposed in [37], maximum power injections from boundary buses of EA to SA are determined for selected contingencies. In the next step, power injection states of the EA are categorized in different groups based on their available power levels. Finally, the states in each group are merged and therefore the rank of the transition-rate matrix, or M, is reduced.
Fig. 11. Replacement of producers with equivalent multi-state generation provider (EMGP) for each Genco in bilateral transaction operation mode.
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4. Probabilistic load flow The determined table in the adequacy equivalent models, described in the previous section, is a variant form of the probability density function of the available power at the boundary buses. This reasoning could lead to the conclusion that it is possible to apply probabilistic load flow to determine the adequacy equivalent. Although probabilistic load flow obtains probability density functions for voltages and power flows, it is not suitable for finding the adequacy equivalent as remedial actions are not included in probabilistic load-flow assessments [39,40]. 5. Methods based on graph theory 5.1. Binary and capacitated-flow graphs
Fig. 12. Equivalent for the transmission network (EMTP) between multi-state generation provider (EMGP) and bulk load points (BLPs) in the power pool operation mode.
If the number of EA injection states and the power categories are respectively n and c, then the rank of the matrix M is reduced from n × n to c × c. Ref. [38] uses the fact that outage of the farther elements in the EA has less impact on the risk indices in the SA. As shown in Fig. 14, the system is partitioned into several areas. The first area is the SA. The second area includes the SA and all the elements with one transmission line farther to SA and so on. The last area includes the whole system. The effect of elements in each area on the SA risk indices, defines up to which area, the outage of elements should be considered. No noticeable changes in the SA indices were observed in [38] beyond area 2.
If connectivity of supply and demand nodes is the principle factor in a reliability evaluation, then the graph under consideration is called a binary network [41]. Reliability evaluation methods using binary networks are mostly applicable to reliability calculations in the networks where branch ratings or limitations are not of direct concern. A more comprehensive criterion in a reliability evaluation is the probability of service continuity in a graph liable to element failures and route traffic or congestion [42]. This topic is related to an important type of graph with branch current limitations or to capacitated-flow graphs. Graphs with current limitations are more consistent with the operating conditions in composite power systems. Cut-sets and tie-sets are two conventional tools for reliability evaluation in graphs [43–46,47]. The time required to determine the cut-sets and tie-sets increases rapidly with graph size and hence it is normal to apply graph simplifying methods. In this regard, reduction and factoring are the two major simplification methods [42,43,45,48,49]. Simplifying transformations are used in the reduction method and omission of graph vertices and edges are applied in the factoring method. 5.2. Further examples A comprehensive discussion on graph reduction is provided in [42,50] with the purpose of application in k-terminal reliability evaluation in communication networks. Equations for parallel, series, delta–star and star–delta transformations in capacitated-flow graphs are presented in [51]. These equations have questionable ability to cope with large systems. In order to facilitate reliability evaluation in composite power systems, Ref. [52] applies graph reduction techniques for EA simplification. Two methods are proposed to extract important generation and load nodes in the EA which are not candidates for graph reduction. In the first method, the generation nodes in the EA with more generation and less distance (impedance) from SA consumers are considered as important nodes. In the same manner, load nodes with higher consumption and less distance to the SA generators are essential nodes. Bus risk index errors in this method are noticeable. In the second method important generation and load nodes in the EA are those with higher impact on the risk indices of all the SA buses. In spite of the increased precision, this method suffers from immense variations in sensitivity analysis between buses. 6. Intelligent-based methods
Fig. 13. Equivalent for transmission network among each multi-state generation provider (EMGP) and related bulk load points (BLPs) in bilateral transaction operation mode.
Since intelligent-based methods have been used in variety of power system problems, it seems necessary to give an answer in this section for the question that whether there are instances
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Fig. 14. Network areas surrounding the SA in the IEEE-RTS.
for applicability of these methods for determination of adequacy equivalent? Based on the reviewed references, it can be argued that intelligent-based methods have not been applied for this purpose in the relevant literatures. For instance, Refs. [53–58] have used intelligent-based methods to categorize system states into success and failure states (or state filtering). Filtering system states decreases the necessary execution duration of the load curtailment optimization problem and hence reduces the computation time for risk index calculation. It is evident that this kind of system filtering is not suitable for obtaining adequacy equivalent.
7. Limitations of the reviewed methods 7.1. Equivalent networks in load-flow studies • The models are highly dependent on the base condition or load point. • Contrary to PQ buses, the elimination of PV buses has a noticeable negative effect on voltage and reactive power flow precision [12]. • Neglecting shunt branches during the determination of an equivalent network introduces no considerable error in real (active) power flows [11,12]. • Inserting a buffer zone between the SA and the EA increases the result accuracy. High capacity PV buses and transmission lines can be retained in the buffer zone [6,12,59]. • In a Ward equivalent network, it is difficult to designate PV or PQ behavior to the remaining buses which have power injections and the user cannot conveniently simulate generation or load changes in the EA [3,5,7,9,10,12,59–61].
• Due to the existence of at least one generation and one load node in the REI network, the simulation of generation and load variation in the EA is more straightforward [2–4,6,8,9,13]. • Admittances of fictitious branches in a REI network may be more unusual in comparison to these admittances in a Ward equivalent [2,4]. 7.2. Adequacy equivalents in the form of a capacity-probability table The main feature of the adequacy equivalent tabular method is the combination of states with similar available capacity at the boundary buses and not real network reduction. For example, in the system of Fig. 7 the total number of events in the EA up to the N-2 level is:
4 0
+
4 1
+
4 2
= 11
Due to the merging of states with similar capacity at boundary bus 2, the number of events is decreased to 6, as shown in Table 2. The adequacy equivalent tabular method loses its efficiency in large systems, where the number of combinable states at boundary buses is not considerable or the number of boundary buses is high. 7.3. Methods based on graph theory Methods based on graph theory and related simplifying transformations are often suitable for distribution and small-scale systems. These methods are less useful in composite (generation
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and transmission) systems where several power sources connect to the transmission network with branch limitations. In composite systems, branch flows considerably affects the reliability evaluation results while, graph-based simplifications have difficulty to cope with branch ratings [51]. In addition, the following limitations can be stated for the methods based on graph theory [6,47,51]: • Reduction formulas for binary graphs are not valid in capacitatedflow graphs. • Applying the transformation method in capacitated-flow graphs leads to numerous flow states with different probabilities. • Nearly all reliability evaluation methods in capacitated-flow graphs depend on cut-sets and tie-sets, which have questionable performance in large graphs. 8. Conclusions A review of the methods, which are directly or indirectly applicable to the determination of composite system adequacy equivalents is presented in this paper. Classification of the methods with a brief discussion about their specifications and limitations is provided. The following conclusions relating to adequacy equivalents can be drawn based on previous sections of this paper: • Heuristic methods for finding adequacy equivalents in large systems could be more practical than circuit-based or mathematicalbased methods [2,59]. • Partitioning of EA buses into essential and nonessential sets have been applied in some publications. Essential buses (for instance high capacity PV or large PQ buses in an EA) have considerable impact on the operating conditions at the boundary buses. It is preferred to retain these buses and elements that are connected to them in the equivalent network of an EA [3,4,6,9]. • There is no general rule for partitioning areas and it is usually done based on engineering judgment [17]. The definition of areas depends on the power transfer agreements or assistance between the areas and the capacity or the number of tie lines [18,19]. • From time or calculation saving view point, it is less efficient to determine an AC equivalent network and hence most of the proposed methods are based on DC load-flow equations [11,21,38]. • It is an advantage in developing a probabilistic equivalent model, to observe the following points [3–6,9–13,18,22,24,37,39,62]: - Keeping the identity or effects of important or determinant elements. - Possibility of including generation and load changes without wide variation in the model parameters. - Efficiency in simulating outage events. - Less dependency on the size of the EA. - Practicability and no need for a very large number of input data or modeling parameters. - Reduction of the time and computation burden compared to more conventional methods. References [1] R. Billinton, W. Li, Reliability Assessment of Electric Power Systems Using Monte Carlo Method, Plenum Press, 1994. [2] W.F. Tinney, W.L. Powell, The REI approach to power network equivalents, in: PICA Conference, 1977, pp. 314–320. [3] T.E. Dy Liacco, S.C. Savulescu, K.A. Ramarao, An on-line topological equivalent of a power system, IEEE Trans. Power Syst. 97 (5) (1978) 1550–1563. [4] A.D. Patton, S.K. Sung, A transmission network model for multi-area reliability studies, IEEE Trans. Power Syst. 8 (2) (1993) 459–466. [5] K.I. Geisler, A. Bose, State estimation based external network solution for online security analysis, IEEE Trans. Power Syst. 102 (8) (1983) 2447–2454. [6] S.C. Savulescu, Solving open access transmission and security analysis problems with the short-circuit currents method, Paper Presented at the Latin America Power Conference, Controlling and Automating Energy Session, Mexico, 2002.
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Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Review
Review of reduction techniques in the determination of composite system adequacy equivalents A. Akhavein a , M. Fotuhi Firuzabad b,∗ , R. Billinton c , D. Farokhzad d a
Department of Engineering, Islamic Azad University, Science & Research Branch, Tehran, Iran Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran c Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, Canada d Iran Grid Management Company, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 17 January 2010 Received in revised form 27 May 2010 Accepted 4 June 2010 Available online 7 July 2010 Keywords: Reliability equivalent Adequacy equivalent External network modeling Network reduction
a b s t r a c t Reliability evaluation of a large composite power system involves the consideration of numerous outage events and consequently extensive calculations. Therefore, applying justifiable simplifications such as the determination of an equivalent networks can be very useful in reliability evaluation of large systems. This paper presents a review of procedures which are directly or indirectly applicable to the determination of composite system adequacy equivalents. Following a brief discussion on the basic characteristics of the various methods, their limitations are presented. Concluding points regarding the methods and modeling schemes related to adequacy equivalent are presented at the end of the paper. © 2010 Elsevier B.V. All rights reserved.
Contents 1. 2.
3.
4. 5.
6. 7.
8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 Equivalent networks in load-flow studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.1. Ward equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.2. Radial, equivalent and independent (REI) network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 2.3. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387 Adequacy equivalents in the form of a capacity-probability table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 3.1. Vertical integrated environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 3.2. Restructured or deregulated environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 3.3. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 Probabilistic load flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Methods based on graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 5.1. Binary and capacitated-flow graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 5.2. Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Intelligent-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1390 Limitations of the reviewed methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.1. Equivalent networks in load-flow studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.2. Adequacy equivalents in the form of a capacity-probability table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 7.3. Methods based on graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392
∗ Corresponding author. Tel.: +98 21 66165901/21; fax: +98 21 66023261. E-mail address: [email protected] (M.F. Firuzabad). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.06.002
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1. Introduction
2. Equivalent networks in load-flow studies
There are many instances in which studies are of more concern in one area of a power system and are not of direct concern in the remaining parts. The phrases “study area” (SA) and “external area” (EA) are used in this paper to describe the area in question and the remaining regions, respectively. Fig. 1 shows the partitioning of a power system into SA and EA. Boundary buses and tie lines can be defined in each SA or EA. It is usually intended to perform detailed studies in the SA. However, the EA is important to the extent where it affects SA analyses. The attraction to apply an equivalent for the EA increases as the dimensions of the power system and the complexity of the required analysis increases. This equivalent network reduces the dimensions of the EA and subsequently leads to reduction in the required time for analysis. Reliability evaluation in a composite generation and transmission system is a problem with massive and time-consuming calculations and hence, it is an obvious candidate for the application of equivalent networks. Reliability analysis is conventionally categorized into adequacy and security problems. Adequacy is interpreted as the existence of sufficient facilities to supply the system loads with consideration of equipment constraints under static conditions. Security is a measure of system capability to withstand dynamic or transient disturbances [1]. The problem of finding a reliability equivalent (often adequacy equivalent) was first addressed by researchers in the 1970s. In this problem, it is intended to find an equivalent (simplified) model for an EA which facilitates adequacy evaluation studies. As the EA is usually the larger portion of the system, the connection of an EA adequacy equivalent to the SA reduces the calculation burden and makes the detailed reliability evaluation in the SA become more feasible. This paper presents a review of methods, which are directly or indirectly related to the determination of adequacy equivalents. These methods can generally be divided into two types. The first type deals with finding an equivalent network for a portion of a system. Simplification of the reliability evaluation is the main focus in the second type. Continuous enhancement in computer capabilities cannot generally obviate the need for adequacy equivalents in composite system reliability evaluation. This is because of concurrent increases in the sizes of interconnected power systems and in the complexity of their operation procedures. Power transfer agreements between power systems, deregulation and system restructuring and open access to transmission networks are such examples. In addition, limitations and costs of technical PC-based software usually limit the size of a system under study to a few hundred buses. It is therefore useful for a user to apply adequacy equivalents for the purpose of system size reduction. Since a significant common concept exists between reliability and risk, these terms have been used interchangeably with the same meaning in this paper.
The issue of determining static equivalents for EA in load-flow studies dates back over 60 years. The use of equivalent networks proved to be successful and equivalent network concepts have been extended in other fields such as security and steady-state stability analysis. An equivalent network is generally obtained for an operating point (called the base condition) in the power system. It is often assumed that in the base condition, loads are at peak values, circuit breakers are in their normal state and no element outages exist [2–5]. Some papers utilize the average load condition [3,4]. Multiple base conditions corresponding to different loading levels have also been utilized [6]. It should be pointed out that equivalent networks are only exact at the base condition and greater deviation from this condition, leads to less precision [2,7,8].
Fig. 1. Partitioning a power system into external and study areas.
2.1. Ward equivalent Determination of a Ward equivalent for an EA contains three main stages [2,7,9–11]. In the first stage, with consideration of the base condition, power injections in the EA buses are converted into current injections. Gaussian reduction is used to omit EA buses in the second stage where the outcomes are equivalent current injections and fictitious admittances in the boundary buses between the EA and the SA. In the third stage, equivalent current injections are converted to corresponding equivalent power injections in the base condition. Fig. 2 shows an example of the Ward equivalent network of an EA in the boundary buses of a SA. The older version of the Ward equivalent model is the admittance version in which power injections are converted to admittances in the base condition. This version has lower precision especially for PV buses [7,9–12]. 2.2. Radial, equivalent and independent (REI) network In a REI network, some or all of the power injections at the EA buses are substituted by one equivalent power injection toward the boundary buses. At least one equivalent generation node and one equivalent load node are usually assigned for generators and loads, respectively. Fig. 3 shows a typical REI conversion process in which m generators are transformed to a single equivalent REI generation node.
Fig. 2. Ward equivalent network of an external area with equivalent power injections in the boundary buses.
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Fig. 3. Stages in the determination of a REI network for an external area in which generators in this area are substituted with a passive R node (shunt branches among boundary buses and ground are not shown for simplicity).
As stated in [2–4,6,7,9,10,12,13], the determination of a REI includes three stages. In the first stage, REI node R and common node 0 are established. The equivalent complex power injection in node R is equal to the sum of the complex power of the generators omitted. The voltage of the 0 node is arbitrary and often considered to be zero. In the second stage, fictitious admittances among nodes 1 to m and 0 are calculated in such a way that power injections toward the remaining buses are the same as the base-condition injections. Gaussian elimination of the passive nodes among node R and boundary buses constitutes the third stage. The negative injection powers at load buses are accommodated by changing the sign of powers and currents (direction of the powers and currents) in Fig. 3 [2,4]. 2.3. Further examples In [14] composite system reliability is evaluated using a DC Ward equivalent of the EA. In order to simulate branch outages in the EA, the statistical mean of the positive and negative injections and the admittance values of the boundary branches are used in the EA Ward equivalent. Shunt branches are not considered in the process of AC Ward equivalent extraction for the EA in [15]. Electrical distance between buses, based on mutual admittance, is considered as a guide in partitioning a system into a SA and an EA. As contingencies do not result in considerable admittance changes in the EA, it is suggested in [15] that the statistical mean of the admittances to be used in the equivalent network. As shown in Fig. 4, papers [16,17] separate a power system into three regions of equipment outage area, optimization area and external area with the intention of increasing the calculation performance in a composite system reliability evaluation. In the equipment outage area, the probabilistic behavior of generators and transmission lines are fully considered. In the optimization area it is assumed that equipments are fully reliable and participate in optimization problems associated with remedial actions. Using the assumption of fixed generators and loads, the external area is substituted by its DC Ward equivalent. Capacity limitations of the transmission lines in the external area are neglected.
In [4], reliability evaluation for two interconnected systems is accomplished by determining one generation and one load REI node for each system. This paper creates a REI network based on DC loadflow equations. Ref. [13] applies an AC REI network of the EA for analysis of power transactions between an EA and a SA and system security. Two REI nodes are introduced for the generators and loads in the EA. Transaction patterns are simulated through power injection variations in the REI nodes by the user. In [18,19], a DC Ward equivalent is determined for passive EA network in which all generators and loads of EA are deactivated. Then, as shown in Fig. 5, the virtual slack bus and branches are added to the remaining system. Generation and load balance in EA is assigned to the introduced slack node. Reactances of the fictitious branches among the slack node and boundary buses are calculated such that power entries in the boundary buses are the same as the base condition. It is assumed that the EA and SA do not assist each other during emergency conditions. The method mentioned in [18,19] is very sensitive to the selection of the EA and SA and direction of power flow between them. Reliability evaluation of the entire power network in Iran is difficult and in [20], the Iran network is divided into five areas. Separation of areas is performed such that a minimum number of tie lines remain between them. As shown in Fig. 6, each tie line is cut at its mid point and virtual loads and generators are added at
Fig. 4. System partitioning into three areas of equipment outage, optimization and external.
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Fig. 5. Determination of an equivalent for active and passive elements of EA in boundary buses with SA.
Fig. 7. Sample system to illustrate finding an equivalent for the EA. Table 1 Reliability data for the generators and transmission lines in Fig. 7. Component
Failure rate (occ./year)
Repair rate (occ./year)
Rating (MW)
Generating units Transmission lines
1 2
99 998
30 35
3. Adequacy equivalents in the form of a capacity-probability table 3.1. Vertical integrated environment
Fig. 6. Tie lines bisection and addition of virtual generators.
this point in order to simulate real and reactive power flows under normal conditions. Generators with zero real generation, as shown in Fig. 6, are added to increase the accuracy of the contingency analysis.
Refs. [21–26] present a method in which an adequacy equivalent of an EA is obtained in the form of equivalent generation and load (available capacity) for a desired contingency level in the EA. Fig. 7 shows a sample system to explain this approach. The associated data is given in Table 1. Table 2 shows the available capacities or powers at the boundary bus 2 of the EA with their probabilities and frequencies for contingencies up to the second level. It has been assumed that all the generators and lines are the same. During the process of tabulating, the tie line between buses 2 and 3 is considered to be in an open state. Once the contingencies or different system failure states are analyzed, states with the same effect (same available capacity) are combined [21–24,27]. The equivalent generator column in Table 2 represents the total available generation in bus 2 if all the loads in the EA are neglected. Subtraction of the equivalent generator and available capacity columns produces the equivalent load column in Table 2. Consequently, as shown in Fig. 8, the EA can be modeled for each outage state j, as an equivalent generator and load at boundary bus 2. Introducing an equivalent load facilitates modeling of load shedding philosophies [24].
Table 2 Adequacy equivalent for the EA at boundary bus 2 in Fig. 7. State no.
Contingency level
Equivalent generator at bus 2 (MW)
Available capacity at bus 2 (MW)
Equivalent load at bus 2 (MW)
Probability
Frequency (occ./year)
1 2 3 4 5 6
0 G L 2G G&L 2L
60 30 35 0 30 0
35 5 15 −25 5 −20
25 25 20 25 25 20
0.976184 0.019721 0.003913 0.000010 0.000079 0.000004
5.857101 2.050971 3.920384 0.020119 0.086946 0.007833
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Fig. 8. Replacement of the EA with an equivalent generator and load for each outage state j.
Refs. [21–24] apply the network flow method (as described in [28]) for determining the maximum flow from a source node to a sink node. The network flow method solves a transportation problem with only constraint of branch capacity limitation. This method has the advantage of simplicity and low calculation burden [25,26]. In general, determination of available capacity leads to an optimization problem subject to generation and transmission constraints. 3.2. Restructured or deregulated environment Adequacy equivalents at each bulk load point (BLP) have been determined in [29–35] in order to evaluate the reliability in a restructured power system. A distribution entity with customers is considered as a BLP [32]. An adequacy equivalent is in the form of a table containing different service states (available capacities) with their probabilities and frequencies from a BLP view. This model is designated as an equivalent multi-state service provider (EMSP). Fig. 9 depicts the concept of EMSP at a BLP [31]. In subsequent works, the EMSP has been divided into two equivalents representing a generation provider and transmission provider according to a model of the power market [30,32,33]. Equivalents are determined at each BLP for generation and transmission service providers considering the transactions agreed between producers and consumers (market model). These equivalents are designated as an equivalent multi-state generation provider (EMGP) and an equivalent multi-state transmission provider (EMTP). The EMGP is a table containing outage states with their probabilities and frequencies and the real power that producers can deliver to the BLP in each outage state [30,32]. In the power pool operation mode of the market, one EMGP is created for all Gencos, as shown in Fig. 10. In the bilateral transaction market operation mode, each Genco has an EMGP as displayed in Fig. 11 [30,32,33].
Fig. 9. Determination of a composite system equivalent for a bulk load point in the form of an equivalent multi-state service provider (EMSP).
Fig. 10. Replacement of producers with equivalent multi-state generation provider (EMGP) in power pool operation mode.
In addition to the type of agreement between producer and load entity (the parties), the load point reliability also depends on transmission configuration between these parties. Refs. [32,33] propose a multi-state EMTP model for the transmission network between the EMGP and the related BLP. During the determination of the EMTP model, power transfer possibilities between the EMGP and related load points are evaluated for each outage state [30]. Figs. 12 and 13 illustrate modeling the transmission network between the producers and the load points [30,32–34]. After finding the adequacy equivalents for the generation and transmission service providers, the load point reliability indices are calculated in three stages. In the first stage, indices corresponding to convolution of the generation and load models are obtained. Combination of the transmission and load models provides reliability indices for the second stage. The calculated indices in the previous stages are aggregated in the third stage. This summation gives an over-estimation of the reliability indices [30,32,36]. 3.3. Further examples In the first step of the procedure proposed in [37], maximum power injections from boundary buses of EA to SA are determined for selected contingencies. In the next step, power injection states of the EA are categorized in different groups based on their available power levels. Finally, the states in each group are merged and therefore the rank of the transition-rate matrix, or M, is reduced.
Fig. 11. Replacement of producers with equivalent multi-state generation provider (EMGP) for each Genco in bilateral transaction operation mode.
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4. Probabilistic load flow The determined table in the adequacy equivalent models, described in the previous section, is a variant form of the probability density function of the available power at the boundary buses. This reasoning could lead to the conclusion that it is possible to apply probabilistic load flow to determine the adequacy equivalent. Although probabilistic load flow obtains probability density functions for voltages and power flows, it is not suitable for finding the adequacy equivalent as remedial actions are not included in probabilistic load-flow assessments [39,40]. 5. Methods based on graph theory 5.1. Binary and capacitated-flow graphs
Fig. 12. Equivalent for the transmission network (EMTP) between multi-state generation provider (EMGP) and bulk load points (BLPs) in the power pool operation mode.
If the number of EA injection states and the power categories are respectively n and c, then the rank of the matrix M is reduced from n × n to c × c. Ref. [38] uses the fact that outage of the farther elements in the EA has less impact on the risk indices in the SA. As shown in Fig. 14, the system is partitioned into several areas. The first area is the SA. The second area includes the SA and all the elements with one transmission line farther to SA and so on. The last area includes the whole system. The effect of elements in each area on the SA risk indices, defines up to which area, the outage of elements should be considered. No noticeable changes in the SA indices were observed in [38] beyond area 2.
If connectivity of supply and demand nodes is the principle factor in a reliability evaluation, then the graph under consideration is called a binary network [41]. Reliability evaluation methods using binary networks are mostly applicable to reliability calculations in the networks where branch ratings or limitations are not of direct concern. A more comprehensive criterion in a reliability evaluation is the probability of service continuity in a graph liable to element failures and route traffic or congestion [42]. This topic is related to an important type of graph with branch current limitations or to capacitated-flow graphs. Graphs with current limitations are more consistent with the operating conditions in composite power systems. Cut-sets and tie-sets are two conventional tools for reliability evaluation in graphs [43–46,47]. The time required to determine the cut-sets and tie-sets increases rapidly with graph size and hence it is normal to apply graph simplifying methods. In this regard, reduction and factoring are the two major simplification methods [42,43,45,48,49]. Simplifying transformations are used in the reduction method and omission of graph vertices and edges are applied in the factoring method. 5.2. Further examples A comprehensive discussion on graph reduction is provided in [42,50] with the purpose of application in k-terminal reliability evaluation in communication networks. Equations for parallel, series, delta–star and star–delta transformations in capacitated-flow graphs are presented in [51]. These equations have questionable ability to cope with large systems. In order to facilitate reliability evaluation in composite power systems, Ref. [52] applies graph reduction techniques for EA simplification. Two methods are proposed to extract important generation and load nodes in the EA which are not candidates for graph reduction. In the first method, the generation nodes in the EA with more generation and less distance (impedance) from SA consumers are considered as important nodes. In the same manner, load nodes with higher consumption and less distance to the SA generators are essential nodes. Bus risk index errors in this method are noticeable. In the second method important generation and load nodes in the EA are those with higher impact on the risk indices of all the SA buses. In spite of the increased precision, this method suffers from immense variations in sensitivity analysis between buses. 6. Intelligent-based methods
Fig. 13. Equivalent for transmission network among each multi-state generation provider (EMGP) and related bulk load points (BLPs) in bilateral transaction operation mode.
Since intelligent-based methods have been used in variety of power system problems, it seems necessary to give an answer in this section for the question that whether there are instances
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Fig. 14. Network areas surrounding the SA in the IEEE-RTS.
for applicability of these methods for determination of adequacy equivalent? Based on the reviewed references, it can be argued that intelligent-based methods have not been applied for this purpose in the relevant literatures. For instance, Refs. [53–58] have used intelligent-based methods to categorize system states into success and failure states (or state filtering). Filtering system states decreases the necessary execution duration of the load curtailment optimization problem and hence reduces the computation time for risk index calculation. It is evident that this kind of system filtering is not suitable for obtaining adequacy equivalent.
7. Limitations of the reviewed methods 7.1. Equivalent networks in load-flow studies • The models are highly dependent on the base condition or load point. • Contrary to PQ buses, the elimination of PV buses has a noticeable negative effect on voltage and reactive power flow precision [12]. • Neglecting shunt branches during the determination of an equivalent network introduces no considerable error in real (active) power flows [11,12]. • Inserting a buffer zone between the SA and the EA increases the result accuracy. High capacity PV buses and transmission lines can be retained in the buffer zone [6,12,59]. • In a Ward equivalent network, it is difficult to designate PV or PQ behavior to the remaining buses which have power injections and the user cannot conveniently simulate generation or load changes in the EA [3,5,7,9,10,12,59–61].
• Due to the existence of at least one generation and one load node in the REI network, the simulation of generation and load variation in the EA is more straightforward [2–4,6,8,9,13]. • Admittances of fictitious branches in a REI network may be more unusual in comparison to these admittances in a Ward equivalent [2,4]. 7.2. Adequacy equivalents in the form of a capacity-probability table The main feature of the adequacy equivalent tabular method is the combination of states with similar available capacity at the boundary buses and not real network reduction. For example, in the system of Fig. 7 the total number of events in the EA up to the N-2 level is:
4 0
+
4 1
+
4 2
= 11
Due to the merging of states with similar capacity at boundary bus 2, the number of events is decreased to 6, as shown in Table 2. The adequacy equivalent tabular method loses its efficiency in large systems, where the number of combinable states at boundary buses is not considerable or the number of boundary buses is high. 7.3. Methods based on graph theory Methods based on graph theory and related simplifying transformations are often suitable for distribution and small-scale systems. These methods are less useful in composite (generation
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and transmission) systems where several power sources connect to the transmission network with branch limitations. In composite systems, branch flows considerably affects the reliability evaluation results while, graph-based simplifications have difficulty to cope with branch ratings [51]. In addition, the following limitations can be stated for the methods based on graph theory [6,47,51]: • Reduction formulas for binary graphs are not valid in capacitatedflow graphs. • Applying the transformation method in capacitated-flow graphs leads to numerous flow states with different probabilities. • Nearly all reliability evaluation methods in capacitated-flow graphs depend on cut-sets and tie-sets, which have questionable performance in large graphs. 8. Conclusions A review of the methods, which are directly or indirectly applicable to the determination of composite system adequacy equivalents is presented in this paper. Classification of the methods with a brief discussion about their specifications and limitations is provided. The following conclusions relating to adequacy equivalents can be drawn based on previous sections of this paper: • Heuristic methods for finding adequacy equivalents in large systems could be more practical than circuit-based or mathematicalbased methods [2,59]. • Partitioning of EA buses into essential and nonessential sets have been applied in some publications. Essential buses (for instance high capacity PV or large PQ buses in an EA) have considerable impact on the operating conditions at the boundary buses. It is preferred to retain these buses and elements that are connected to them in the equivalent network of an EA [3,4,6,9]. • There is no general rule for partitioning areas and it is usually done based on engineering judgment [17]. The definition of areas depends on the power transfer agreements or assistance between the areas and the capacity or the number of tie lines [18,19]. • From time or calculation saving view point, it is less efficient to determine an AC equivalent network and hence most of the proposed methods are based on DC load-flow equations [11,21,38]. • It is an advantage in developing a probabilistic equivalent model, to observe the following points [3–6,9–13,18,22,24,37,39,62]: - Keeping the identity or effects of important or determinant elements. - Possibility of including generation and load changes without wide variation in the model parameters. - Efficiency in simulating outage events. - Less dependency on the size of the EA. - Practicability and no need for a very large number of input data or modeling parameters. - Reduction of the time and computation burden compared to more conventional methods. References [1] R. Billinton, W. Li, Reliability Assessment of Electric Power Systems Using Monte Carlo Method, Plenum Press, 1994. [2] W.F. Tinney, W.L. Powell, The REI approach to power network equivalents, in: PICA Conference, 1977, pp. 314–320. [3] T.E. Dy Liacco, S.C. Savulescu, K.A. Ramarao, An on-line topological equivalent of a power system, IEEE Trans. Power Syst. 97 (5) (1978) 1550–1563. [4] A.D. Patton, S.K. Sung, A transmission network model for multi-area reliability studies, IEEE Trans. Power Syst. 8 (2) (1993) 459–466. [5] K.I. Geisler, A. Bose, State estimation based external network solution for online security analysis, IEEE Trans. Power Syst. 102 (8) (1983) 2447–2454. [6] S.C. Savulescu, Solving open access transmission and security analysis problems with the short-circuit currents method, Paper Presented at the Latin America Power Conference, Controlling and Automating Energy Session, Mexico, 2002.
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