Value-Based Radial Distribution System Reliability Optimization

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY 2006

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Value-Based Radial Distribution System Reliability Optimization Jin-Man Sohn, Soon-Ryul Nam, Member, IEEE, and Jong-Keun Park, Senior Member, IEEE

Abstract—The protective devices and switches play an important role in the reliability of electrical distribution systems by minimizing the impact of interruption. In this paper, a method for identifying the type and location for protection devices and switches on the prerouted distribution system using value-based optimization is proposed. The proposed method is based on the contingency analysis of various components, e.g., the faults of line sections, switches, protective devices, and the restoration through switching actions such as upstream restoration and downstream restoration. The formulation of the optimization problem in this paper is appropriate for linear integer programming, where each nonlinear term can be translated by only two additional linear constraints. The detailed design of the protection devices and the switches are determined by minimizing the total cost of reliability that comprises apparatus investment, maintenance, and interruption cost. The efficiency and validity of the proposed method are demonstrated in case studies. Index Terms—Integer programming, power distribution, power distribution planning, power distribution protection, power distribution reliability.

NOMENCLATURE Number of Category 2 laterals. Number of Category 3 laterals. Permanent failure rate of component for section of feeder or lateral . Temporary failure rate of component for section of feeder or lateral . Repair time of line in section . Repair time of isolating switch in section . Repair time of three-phase protection device in section . Repair time of fuse in section . Time required to perform the required isolation. Time required to close the normally open switch. Product of customer damage function and average load of load point affected by component . Set of all load points of lateral adjacent to and downstream from section . Average load of load point adjacent to and downstream from section of lateral .

Manuscript received March 16, 2005; revised August 3, 2005. This work was supported by KESRI, which is funded by the Ministry of Commerce, Industry, and Energy. Paper no. TPWRS-00160-2005. J.-M. Sohn and J.-K. Park are with the School of Electrical Engineering, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). S.-R. Nam is with the Next-Generation Power Technology Center and is also with the Department of Electrical Engineering, Myongji Uiversity, Yongin 449728, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2005.860927

SCDF of load point adjacent to and downstream from section of lateral . 0/1 variable representing absence or presence of three-phase devices in the section of feeder or lateral . 0/1 variable representing absence or presence of fuses in the section of lateral . 0/1 variable representing absence or presence of isolating switches in the section of feeder . Sum of capital investment cost and annual opera. tional cost of I. INTRODUCTION

T

HE FUNCTION of an electric power distribution system is to deliver energy from bulk power supply points to individual customers with a high level of reliability. Recently, as more customers are affected by even short outages, which may cause heavy damage, there is increasing interest in the quantitative analysis of distribution system reliability worth and its applications, such as value-based reliability optimization. This optimization systematically attempts to balance both utility reliability costs, comprised of costs for capital investment, maintenance, etc., and customer interruption cost in the planning, operation, and design process. It is expected that the consideration of customer interruption cost will increase in the future, with the restructuring and privatization of utilities, regulators, and other market participants. Protective devices, such as fuses, reclosers, and isolating switches, play an important role in reliability of distribution systems; they reduce the annual failure rate and outage duration, which affect each load point considerably, thus reducing the total customer interruption cost. In addition, it is important to identify the type and location for protective devices and switches for minimization capital investment cost and maximizing customer benefits. Extensive research has been conducted on reliability optimization of distribution systems. In the early days, the network model, which translates a physical network into a reliability network based on series and parallel component connections, was widely used [1], [2]. It has a very simple structure and is easy to use, but it cannot respond to various contingencies, such as fault isolation and restoration. There are many applications using a value-based approach for various options, such as isolating switches, circuit breakers, recloser allocation, and alternative supply, which are predetermined by highly experienced engineers [3]–[8]. In [3]–[8], the optimization technique is not used, but the best option can be selected through evaluating the limited number of candidates. Recently, more attention is being

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Fig. 1. Typical distribution feeder and laterals.

paid to use the optimization techniques [9]–[16]. The contingency analysis-based optimization technique has been applied to various optimization problems; the optimal feeder routing problem is solved using genetic algorithm, simulated annealing, and network flow [9]–[11]. The optimal switching placement problem is solved using simulated annealing and genetic algorithms [12]–[14]. In [12]–[14], implicit formulation is used to find the expected annual failure rates and the expected annual interruption duration for each customer while describing various system configurations through operation of switches and protective devices; the tree search algorithm is used in [12], and the Markov model is used in [13] and [14]. A more explicit formulation will be helpful in the optimization problem. References [15] and [16] propose the explicit mathematical formulation using binary integer programming for the optimal protection devices allocation problem. The basic approach of this paper is similar to [15] and [16]. The method of [15] and [16] is, however, not appropriate for value-based optimization since it cannot deal with the outage duration and the effects between main feeder and other laterals. We have developed a new contingency analysis-based optimization method to identify the type and location of protective devices and switches in a prerouted distribution system for minimization of total reliability costs. The problem is formulated in linear binary programming with the introduction of only two additional linear constraints for each nonlinear product term. II. DISTRIBUTION RELIABILITY A. Basic Operation of Radial Distribution System The failure of a component in a radial distribution system always leads to customer interruptions. Reliable distribution systems minimize this impact by allowing faults to clear themselves, by minimizing the number of customers affected with protection device operations and by quickly restoring customers through system reconfiguration. For this purpose, the radial distribution systems basically consist of main feeders and laterals with protective equipments and are normally operated in a radial configuration for effective coordination of their protective system, as shown in Fig. 1. Moreover, supply can be restored from alternative sources (Sub2) through interconnections using switching operations. Protective equipment is normally installed at the beginning of line sections in the main feeder or lateral line section. Some of the basic protective equipment used in a radial distribution system are the following:

Fig. 2. Three categories of laterals. (a) Category 1. (b) Category 2. (c) Category 3.

1) 2) 3) 4) 5)

breakers and relays; fuses; reclosers; sectionalizers; automatic and manual isolating switches.

When a fault F1 (e.g., an overhead line fault or a fault at isolating switch) occurs in the system shown in Fig. 1, the circuit breaker B1 is tripped and leads to customer interruption of all load points—LP1, LP2, and LP3. After the fault is located, the operators or crews will open the normally closed isolating switch SW2 and close the circuit breaker B1 automatically or manually, so that the load point LP1 can be fed from the main source (Sub1). In addition, the load point LP3 can be fed from alternative sources (Sub2) by closing the normally open switch SW3. When a three-phase protection device such as a recloser is installed in line section LS2 and a fault F1 occurs, there will be customer interruption at load points LP2 and LP3 but not at load point LP1. Thus, the additional installation of three-phase protection devices or switches on the main feeder will increase the reliability of radial distribution systems. Depending on utilities’ practices, the lateral taps can be divided into three categories for identifying the types and locations of protection devices and isolating switches. A Category 1 lateral is short and has limited exposure (usually less than three spans). Due to the cost, this type of lateral is not fused. A Category 2 lateral is usually longer than three spans and is protected by fuses only. A Category 3 lateral is heavily loaded or long, and various types of protective devices can be installed at the tap or at other points [15], [16]. The highlighted region in Fig. 1 can be one of the three categories of laterals, as shown in Fig. 2. When the load point LP3 in Fig. 1 is a Category 3 lateral in Fig. 2(c), it is divided into LP31 and LP32. B. Distribution Reliability Indexes The basic indexes normally used to evaluate the reliability of a distribution system are the load point failure rate , av, and annual unavailability . The erage outage duration system indexes can be calculated using these three basic indexes, the number of customers, and the amount of load at each

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TABLE I SECTOR INTERRUPTION COST (US$/KW)

load point. The most common customer-oriented system indexes are system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), customer average interruption duration index (CAIDI), average service availability index (ASAI), average service unavailability index (ASUI), energy not supplied (ENS), and average energy not supplied (AENS) [18]. These indexes are insufficient for representing the cost of reliability. There have been extensive surveys to associate reliability with customer costs, and many indexes have been proposed [3], [14], [17], [18]. One of those reliability indexes is the expected interruption cost (ECOST) as follows [3], [18]: ECOST

$ year

Fig. 3. Possible locations of protection devices and isolating switches in radial distribution system.

linear function of many variables subject to linear constraints and variables that must be integers. A linear programming (LP)based branch-and-bound algorithm can be used to solve the following binary (zero-one) programming problem [19]: (3) such that

(1)

is the load curtailment of contingency and are where the failure rate and interruption duration of contingency , reis the unit interruption cost that is a function spectively, of the duration of interruption, and is the set of all contingencies. The survey data have been analyzed to give the sector customer damage functions (SCDFs) shown in Table I, which describes the interruption cost for the seven different customer sectors as a function of duration of interruption. Interpolation and extrapolation of the cost data are used when the duration lies between two durations given in Table I or is longer than eight hours. Since SCDF is nonlinear in nature, it is undesirable to oversimplify the failure of component in (1).

where are cost coefficients, and are parameters describing the constraints, and is the number of constraints. The cost function in this paper is expressed as a sum of linear combination over control variables and their product terms. Such nonlinearity can be eliminated by replacing any product and adding only the following two terms with a variable constraints:

C. Total Cost of Reliability (TCR)

is the set of product terms, and is the number of where , then element in (for example, given and ). The additional constraints for introducing a new variable will be proven in the Appendix.

The TCR is equal to the sum of utility cost of reliability (UCR) and the ECOST TCR

UCR

ECOST

(4)

(2)

The UCR consists of annual operational and maintenance cost and capital investment cost, e.g., the cost of the required equipment. III. PROBLEM FORMULATION A. Fundamentals The problem of identifying the types and locations of the protective devices and switches on a distribution feeder and laterals is modeled as a linear binary integer programming problem. Integer linear programming seeks to maximize or minimize a

B. Objective Function A set of possible locations where various protective devices and isolating switches can be installed on the distribution system is illustrated by the blank squares of Fig. 3. The optimization problem is to choose a subset of the locations at which to install a specific protective device and isolating switches in order to minimize TCR. The objective function is derived with the following basic assumptions. 1) All faults are exclusive; all failures are repaired before the next fault occurs. 2) All faults are first-order failures.

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3) The protective equipment isolates all permanent faults instantaneously. 4) The protective equipment is perfectly coordinated, i.e., the device closest to the fault operates first. 5) The feeder is operated as a radial feeder. 6) The main feeder can be fed through closing the normally open switch from alternative supply. 7) Every lateral is one of the three predefined categories. 8) A breaker is located at the substation, and no fuse is allowed to be installed on the main feeder. 9) Only an isolating switch may be allowed to be installed only on the main feeder. 10) An isolating switch is installed next to protective devices on the same section. 11) For Category 3 laterals, a three-phase protection device or fuse must be installed at the tap points, and only one device can be installed in each section. The UCR of protective devices and isolating switches in (2) is described as follows: UCR

(5)

, where if the device will be installed in the location and otherwise, ; and is the number of the possible locations on the main feeder or laterals where the device may be located. Since the fuses in Category 1 laterals are considered to have been already installed, the cost of those is not included in (5). ECOST in (2) consists of the part of main feeder, Category 2 laterals, and Category 3 laterals as follows: ECOST The required calculation for main feeder

(6)

TABLE II SUBSTITUTION FOR CALCULATION OF MAIN FEEDER IN (7)

Since it is assumed that there is no fuse on the main feeder, in (7). The second and third term of there exists no variable (7) represent downstream restoration from an alternative supply. Upstream restoration and the interaction between three-phase protection devices and switches are considered in the fourth and fifth term of (7). The formulation is complicated by the failure of protective devices and isolating switches themselves. For example, assume that reclosers and switches are installed in all possible locations on the main feeder in Fig. 3. If a line fault in section LS3 occurs, the recloser of section LS3 operates. However, if a fault in the recloser in the same section occurs, the recloser of section LS2 operates. As a result, the coefficients and in the product term of (8a) are different according to the type of component involved. A set of coefficients is summarized in Table II. Note that the calculation over Category 1 laterals is included in (7). A component fault in a Category 1 lateral as well as a fault in a three-phase protection device or fuse at tap points in Category 2 or 3 laterals can be treated as a component in the main feeder. In calculating a fault at a tap point of a Category 2 or 3 lateral, load points in the same lateral should not be considered; the effect of those load points affected by a protective device fault will be calculated in the formulation over Category 2 or 3 laterals. The required calculation for Category 2 laterals is

is

(9) Since the fuse is installed at lateral tap on a Category 2 lateral, . The required calculation for a Category there is no variable 3 lateral is

(7) where (10)

(8a) (8b) (8c)

where

is (11)

SOHN et al.: VALUE-BASED RADIAL DISTRIBUTION SYSTEM RELIABILITY OPTIMIZATION

TABLE III SUBSTITUTION FOR CALCULATION OF CATEGORY THREE LATERAL IN (9)

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TABLE IV DEVICE RELIABILITY DATA AND COST DATA (IN DOLLARS)

TABLE V COMPARISON OF RESULTS

Fig. 4. Radial distribution system—test system.

Note that in (11) is different from in (8). in (11) deals with a single load point adjacent to and downstream from in (11) deals with all load points of laterals section , but adjacent to and downstream from section of the main feeder. A set of coefficients for product terms is summarized in Table III. Since the substitution for temporary failure rate is the same as is omitted in Table III. permanent failure rate One of the constraints for the objective function is the coordination of three-phase protection devices. The number of three-phase devices that can be installed in a series is limited for purposes of maintaining coordination. The other constraint is that a three-phase protection device or fuse must be installed in Category 3 laterals by definition, since minimization of TCR can propose no installation, even in Category 3 laterals.

IV. CASE STUDY The test system used in this paper is the radial distribution system at Bus 5, as described in [21], which represents a typical urban distribution system consisting of residential, government and institutional, office buildings, and commercial customers, as shown in Fig. 4. The reliability data, customer data, and device cost used in this paper are obtained from previously published data [14], [20], [21]. A summary of this data is presented in Table IV. In addition, all switches are assumed to take an average time of one hour to switch, the temporary failure rate is assumed to be four times more than the permanent failure rate, and the operational cost is assumed to be 10% of the device cost. All laterals are assumed to be Category 3 laterals.

The result of the optimization is shown in Fig. 4. Since the failure rate of reclosers is greater than that of line sections in this test system, the result shows no recloser on the main feeder. Load point LP7 and LP19 consist of commercial customers and office building customers, respectively. The interruption cost over these customers is higher than other types of customers. When a fuse is installed instead of a recloser for these load points, ECOST will be increased because of a temporary failure at load points LP7 and LP19 and will more than account for the difference between the cost of a recloser and that of a fuse. Furthermore, isolating switches are also installed in all possible locations, which is the base case. TCR, UCR, and ECOST for each feeder are summarized in Table V. The optimal solution is $111 529, which is 9.2% less than the base case. The percentages of UCR over TCR are 58.1, 59.4, 38.5, and 56.8%, respectively, for feeder F1, F2, F3, and F4 in the base case and 61.2, 49.1, 41.3, and 49.3% in the optimization case. When installation of more isolating switches for reinforcement is required, it is shown that feeder F3 is preferable. To compare with other methodology, a genetic algorithm is applied to the formulation used in this paper. The simulation is performed while the number of constraints is reduced by not translating nonlinear product terms in the formulation into additional linear constraints and by reducing the number of independent 0/1 variables using equality constraints. Fig. 5 shows that the required number of objective function evaluations for genetic algorithms is approximately nine times greater than that for the proposed method in this paper. In case of genetic algorithms, the value of parameters, such as the percentage of mutation and stopping criteria, should be determined carefully since only a local optimal point can be found. The percentage of mutation in this paper is 5%, and the evolution is performed repeatedly until the optimal solution already known for the proposed method is found. The manipulation of reliability data and optimization was performed using Matlab.

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Then, it can be concluded that .

for Q.E.D.

ACKNOWLEDGMENT The authors would like to thank the Electrical Industry Research Center of the Ministry of Commerce, Industry, and Energy of Korea through the Electrical Power Reliability/Power Quality Research Center for the advice. REFERENCES

Fig. 5. Comparison of the proposed method and genetic algorithm.

V. CONCLUSION This paper has presented a method for identifying the type and locations for protection devices and the switches on prerouted distribution systems using value-based optimization. The protective devices and isolating switches are effective for improving reliability in radial distribution systems. Implementation, however, can increase the total cost of reliability. The proposed method enables evaluation of the effect of additional protective devices and isolating switches by themselves as well as the interdependencies between switching actions and protection system behavior in reliability optimization. The optimization technique used in this paper is binary linear integer programming. The objective function in this paper is expressed as a sum of linear combination over control variables and their product terms. Any product terms can be replaced with a new variable by adding only two constraints while maintaining linearity. The efficiency and validity are demonstrated in the case study. The proposed method can be an effective tool for radial distribution system planning and design. APPENDIX Let the new additional variable representing the product term be

times (A1)

(A2) Let all follows:

have the value of 1. (A2) is described as (A3)

Then, it can be concluded that . have the value of 0. (A2) is described Let a certain as follows:

(A4)

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SOHN et al.: VALUE-BASED RADIAL DISTRIBUTION SYSTEM RELIABILITY OPTIMIZATION

Jin-Man Sohn received the B.S. and M.S. degrees from Seoul National University, Seoul, Korea, in 1994 and 1996, respectively. He is currently working toward the Ph.D. degree at Seoul National University. He worked at Hyundai Engineering Company, Ltd. and Hyundai Engineering and Construction Company, Ltd. from 1996 to 2000 and was a Researcher with the Korea Electrical Engineering and Science Research Institute for three years. His research interests include reliability and power quality in distribution networks.

Jong-Keun Park (SM’97) was born in YuSeong, Korea, on October 21, 1952. He received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 1973 and the M.S. and Ph.D. degrees in electrical engineering from the University of Tokyo, Tokyo, Japan, in 1979 and 1982, respectively. He worked as a Researcher at the Toshiba Heavy Apparatus Laboratory in 1982. He was a Visiting Professor with Technology and Policy Program and Laboratory for Electromagnetic and Electronic Systems, Massachusetts Institute of Technology, Cambridge, in 1992. He is currently a Professor in the School of Electrical Engineering, Seoul National University. Prof. Park is a Senior Member of the Japan Institute of Electrical Engineers (JIEE). Also, he is presently acting as a Fellow of the Institution of Electrical Engineers (IEE), a Life Member of the Korean Institute of Electrical Engineers (KIEE), and the Korean representative of the study committee SC5 “Electricity Markets and Regulation” in CIGRE.

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Soon-Ryul Nam (S’96–M’02) received the B.S., M.S., and Ph.D. degrees from Seoul National University, Seoul, Korea, in 1996, 1998, and 2002, respectively. Currently, he is a Research Professor with Myongji University, Yongin, Korea, where he is also with the Next-Generation Power Technology Center. His research interests are the analysis, control, and protection of power systems.