- Email: [email protected]
Optimal meter placement using genetic algorithm to maintain network observability
Expert Systems with Applications 31 (2006) 193–198 www.elsevier.com/locate/eswa
Optimal meter placement using genetic algorithm to maintain network observability Amany El-Zonkoly * Department of Electrical and Computer Control Engineering, Arab Academy for Science & Technology, P.O. 1029, Miami-Alexandria, Egypt
Abstract This paper presents a genetic algorithm based method by which measurement system can be optimally determined and upgraded to maintain network observability. Accurate monitoring of power system operation has become one of the most important functions in today’s deregulated power markets. State estimators are the essential tools of choice in the implementation of this function. Determination of the best possible combination of meters for monitoring a given power system is referred to as the optimal meter placement problem. The proposed algorithm yields a measurement configuration that withstands any single branch outage and/or loss of single measurement, without losing network observability. The proposed algorithm is based on the measurement Jacobian and sparse triangular factorization in its numerical part and based on artificial intelligence in the decision making part. Details of the algorithm are presented using two case studies. q 2005 Published by Elsevier Ltd. Keywords: Artificial intelligence; Network observability; Meter placement; State estimation
1. Introduction Accurate monitoring of power system operation has become one of the most important functions in today’s deregulated power markets. State estimators are the essential tools of choice in the implementation of this function. Whether a new state estimator is put into service or an existing one is being upgraded, placing new meters for improving or maintaining reliability of measurement system; is of great concern. Determination of the best possible combination of meters for monitoring a given power system is referred to as the optimal meter placement problem. This problem has been addressed earlier in various studies. Some of them study the problem in the transmission networks (Fetzer & Anderson, 1975; Koglin, 1975; Aam, & Holten., et al, 1983; Park, Moon, Choo, & Kwon, 1988; Sarma, & Raju, et al., 1994; Clements, & Krumpholz, et al., 1983; Momticelli & Wu, 1985; Korres & Contaxis, 1994; Baran, & Zhu et al., 1995; Gouvea & SimoesCosta, 1996; Celik & Liu, 1995; Abur & Magnago, 1999; Magnago & Abur, 2000; Milosevic & Begovic, 2003) and others study the distribution systems (Wang & Schulz, 2004). However, distribution systems have
* Tel.: C20 35622586; fax: C20 35622366. E-mail address: [email protected]
0957-4174/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.eswa.2005.09.016
many features that are different from transmission systems such as (i) radial topology, (ii) three-phase unbalanced systems, (iii) high resistance to reactance ratio and (iv) very limited number of real-time measurements. In this paper, we will concentrate our research on transmission networks only. While the majority of studies are concerned about designing an observable system with minimum variance of errors in the state estimates, others consider loss of lines and/or measurements. They design measurement systems, which can keep the systems observable during such unexpected disturbances. As the power markets become more competitive, having reliable measurement systems that can withstand branch outages or loss of meters, will become more important and the costs associated with metering upgrades will be justified. Abur and Magnago (Abur & Magnago, 1999; Magnago & Abur, 2000) presented a systematic method which provide optimal measurement configuration for a given power system that incorporated meter installation costs into the problem formulation. This method made the selection not only technically but also financially sound. In Abur and Magnago (1999), the authors developed a topological method that accounts for single branch outages only and solved the optimal meter placement problem using linear programming. In Magnago and Abur (2000), the authors presented a method that generalizes the meter placement problem formulation in such a way that considerations of both types of contingencies, namely loss of a branch or a measurement, could be simultaneously taken into account. The problem was then
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solved using integer programming. In Milosevic and Begovic (2003), the authors consider the phasor measurement unit (PMU) placement problem in which they try to minimize the number of PMU’s used, so that the system is topologically observable during its normal operation and following any single-line contingency only. They tried also to maximize the measurement redundancy so that instead of a unique optimal solution, there was a set of best tradeoffs between competing objectives, the so-called Pareto-optimal solutions (POS). A specially tailored nondominated sorting genetic algorithm (NSGA) for the PMU placement to find the POS was used. However, they did not consider the meter installation costs into the problem formulation. In this paper, the optimal meter placement problem against both types of contingencies at the same time and taking into consideration the meter installation costs will be solved using an artificial intelligence method, namely the ‘Genetic Algorithm (GA)’. The proposed algorithm provides global solution of the problem that may be missed by other heuristic, sequential selection schemes. It can also be extended to solve the problem in case of loss of multi-branches and/or multi-meters. The paper is organized in such a way that a review of a direct method for network observability analysis is given first. Then, the proposed algorithm for optimal meter placement is introduced. Finally, simulation results are presented at the end of the paper to demonstrate method’s application to typical power system.
The decoupled gain matrix for the real power measurements can be formed as: G Z HTH
(2)
where, measurement error covariance matrix is assumed to be the identity matrix without loss of generality. Note that, since the slack bus is also included in the formulation, the rank of H (and G) will be at most (nK1) (n being the number of buses), even for a fully observable system. This leads to the triangular factorization of a singular and symmetric gain matrix. Consider the step where the first zero pivot is encountered during the factorization of the singular gain matrix, as illustrated below:
ð3Þ
where 1 K1 K1 KT KT KT G 0 Z LK I LiK1 .L1 GL1 .LiK1 Li
(4)
And Li’s are elementary factors given by:
ð5Þ 2. Network observability analysis using direct numerical method The meter placement algorithm presented in this paper is based on the observability analysis method introduced earlier in Gou and Abur (2000). This method will be briefly reviewed first. The static state estimation is a mathematical procedure to compute the best estimate of the node voltage magnitude and angle for each node from a given set of measurements. Network observability must be checked prior to state estimation. If there were any unobservable parts of the network, then a meter placement procedure will be followed, in order to make the entire network observable. Consider the real power versus phase angle part of the linearized and decoupled measurement equation. This is obtained by using the first order approximation of the decoupled nonlinear measurement equation around an operating point: z Z Hq C e
(1)
where z mismatch between the measured and calculated real power measurements; H decoupled Jacobian of the real power measurements versus all bus phase angles; q incremental change in the bus phase angle at all buses including the slack bus; e measurement error vector.
lTi Z ½LiC1;i ; LiC2;i ; .; Lni
(6)
Lij is the ijth entry of Li. Setting LiC1ZInxn, the triangular factorization of G’ in Eq. (3) can proceed with the (iC2)-nd column. This procedure can be repeated each time a zero pivot is detected until completion of the entire factorization. The following expression can then be written: 1 K1 K1 KT KT KT K1 KT D Z LK n LnK1 .L1 GL1 .LnK1 Ln Z L GL
(7)
where D is a singular and diagonal matrix with zeros in rows corresponding to zero pivots encountered during the factorization of G, and L is a nonsingular lower triangular matrix. If the matrix D have more than one zero on its diagonal then the system is unobservable. The above method is simple and non-iterative, which makes it computationally very efficient. Thus, it will be used to initialize the optimal meter placement algorithm developed in this paper. 3. Proposed algorithm for optimal meter placement This algorithm will be applied in two steps. In the first step, an optimal measurement scheme is determined such that
A. El-Zonkoly / Expert Systems with Applications 31 (2006) 193–198
the network will be fully observable. The optimal scheme will represent the essential measurement set. In the second step of the algorithm, the needed set of measurement that is to be added such that the network regains its observability during both types of contingencies mentioned before is optimally determined. The optimization problem in the two steps will be solved using genetic algorithm. 3.1. Solution using genetic algorithm The genetic algorithm will proceed as follows: 1. Initialization: in this step a population of possible solutions of the problem is initiated. Each individual consists of a number of bits corresponding to the number of possible meters to be added in the system. Each individual will be assigned a fitness value which equal to the reciprocal of the cost of the measurement set suggested by the individual. 2. Selection: selection of individuals with highest fitness value is performed to generate the next generation of individuals. 3. Crossover and mutation: these are the main two operators of genetic algorithm. Through them the information contained by each individual are exchanged and manipulated in order to have new individuals with better and better fitness values. 4. The previous steps are repeated until the optimal solution is found.
3.2. Step 1: selection of optimal measurement scheme The novel algorithm, which will identify all the necessary measurements to be used in order to have an observable system will proceed as follows: 1. For each set of measurement suggested by an individual the Jacobian (H) of the system is formed. 2. Calculate the gain matrix (G) and perform triangular factorization to get the diagonal matrix (D). 3. If the matrix (D) have more than one zero in its diagonal then the system is classified to be unobservable. Hence, this individual is refused and another one is generated instead and tested in the same manner. 4. Every individual in any population must yield an observable system. 5. The GA search is performed with testing all of the individuals through the generations until an optimal solution is reached.
3.3. Step 2: optimal meter placement against measurement loss and branch outage With the set of measurements determined in the first step of the solution, the proposed algorithm is to be applied
195
again. The algorithm will be applied in case of loss of any of the predetermined measurements and in case of outage of any single branch that will affect the observability of the system to find the optimal additional set of measurements to maintain the system observability. The measurement Jacobian is to be formed considering the set of measurements determined in step 1. During any of the two types contingencies under consideration, the Jacobian must be modified in accordance. 3.3.1. Loss of single measurement In case of loss of one of the measurements, the Jacobian (H), will be modified (Hmod) by removing one row that corresponds to the lost measurement. Calculating the gain matrix of the modified Jacobian and decomposing it to its factors will show that the system is now unobservable. The GA is applied to determine the additional set of measurement to be used to regain the system observability. To carry on with the observability analysis as described in Section 2, the Jacobian (Hmod) is modified again by adding rows corresponding to the suggested set of measurements. 3.3.2. Loss of a single branch It is known that network observability will be drastically affected by topology changes. In general, it is not necessary to check network observability following the outage of every single branch in the system. It is sufficient to make the measurement system robust against the outage of every single tree branch only. The tree should correspond to the chosen set of essential measurements. Thus, the loss of a co-tree branch (or link) will not have to be considered since it will have no effect on observability. Therefore, for a system of n buses, it is sufficient to check the outage of nK1 tree branches. Assuming that the tree branch kKj is outaged, the measurement Jacobian will be modified as H mod, where: Hikmod Z Hijmod Z 0, if measurement i is a line flow. Hijmod Z 0, Hikmod Z Hik C Hij , if measurement I is an injection at bus k. Hikmod Z 0, Hijmod Z Hik C Hij , if measurement I is an injection at bus j (Magnago & Abur, 2000). 3.3.3. Optimal selection The selection of optimal meter placement must satisfy the condition that the system is observable under any case of losing either a single measurement or a single branch. The optimal solution will yield the minimum number of added measurements with minimum cost that will make the system observable again. The selection mechanism will proceed as follows: 1. Form the Jacobian (H) corresponding to the set of measurements determined in step1. 2. Apply the GA search as described in Section 3.1 such that each solution is tested for observability under all possible contingencies as described in Section 2. 3. The optimal solution is determined in the end of the search.
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4. Simulation results 4.1. Case study 1 The simple 6 bus system shown in Fig. 1 is considered to illustrate the proposed method. All the branch impedances are set equal to j1 pu. The resulting Jacobian for all possible measurements is given below (Magnago & Abur, 2000): q1 Inj:1 Inj:2 Inj:3 Inj:4 Inj:5 Inj:6 Fl:1 Fl:2 Fl:3
2
q2
q3
2
q4 K1
q5
q6 K1
3
7 6 2 K1 K1 7 6 7 6 7 6 K1 2 K1 7 6 7 6 6K1 K1 3 K1 7 7 6 7 6 6 K1 2 K1 7 7 6 7 6 7 6K1 K 1 K 1 3 7 6 7 6 7 6 1 K1 7 6 6 1 K1 7 5 4 1
K1
where, Inj.k and Fl.j represent the net injection at bus k and the power flow through branch j, respectively. 4.1.1. Step 1 The genetic algorithm parameters used in the first step to run the search for the optimal set of measurements to make the system observable are set as follows: Maximum generationZ100 Population sizeZ100 Crossover probabilityZ0.8 Mutation probabilityZ0.01 The cost of the best solution through out the generations is shown in Fig. 2. Where measurements are assigned relative cost to each other, which can be replaced by true values.
Fig. 2. Cost value through the generation.
The optimal and essential set of measurements were found to be: ½Inj:1; Inj:2; Inj:4; Inj:5; Fl:4
4.1.2. Step 2 The genetic algorithm parameters used in the second step are the same as in the first one. In this step, the search is run to find the optimal set of added measurements to maintain the system observability under all possible loss of a measurement or a branch. The contingencies considered are the loss of any of the measurements determined in step 1 and the outage of branches 2–3, 2–5 and 5–6. The outage of these branches only, one at a time, will make the system unobservable. The cost of the best solution through out the generations is shown in Fig. 3. The optimal measurement to be added to maintain system observability under any single branch outage or loss of any single measurement was found to be Inj.6 (the power injection at bus 6). 4.2. Case study 2 The IEEE 30 bus system shown in Fig. 4 is considered also to illustrate the proposed method. The system data is given in Saadat (1994).
Fig. 1. 6 bus system example.
Fig. 3. Cost value through the generation.
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Fig. 6. Cost value through the generation.
½Inj:8; Inj:9; Inj:10; Inj:12; Inj:17; Inj:18; Inj:21; Inj:27; Fl:1K2; Fl:1K3; Fl:2K5; Fl:2K6; Fl:3K4; Fl:4K12; Fl:6K7; Fl:8K28; Fl:9K11; Fl:10K21; Fl:12K13; Fl:14K15; Fl:15K18; Fl:15K23; Fl:16K17; Fl:22K24; Fl:24K25; Fl:25K26; Fl:27K30; Fl:29K30 Fig. 4. IEEE 30 bus system.
4.2.1. Step 1 The genetic algorithm parameters used in the first step to run the search for the optimal set of measurements to make the system observable are set as follows: Maximum generationZ100 Population sizeZ100 Crossover probabilityZ0.8 Mutation probabilityZ0.01 The cost of the best solution through out the generations is shown in Fig. 5. Where measurements are assigned relative cost to each other which can be replaced by true values. The optimal and essential set of measurements were found to be:
4.2.2. Step 2 The genetic algorithm parameters used in the second step are the same as in the first one. In this step, the search is run to find the optimal set of added measurements to maintain the system observability under all possible loss of a measurement and/or a branch. The contingencies considered are the loss of any of the measurements determined in step 1 and the outage of any branch of the network one at a time. The cost of the best solution through out the generations is shown in Fig. 6. The optimal set of measurements to be added to maintain system observability under any single branch outage and/or loss of any single measurement was found to be as follows: ½Inj:20; Inj:28; Inj:23; Inj:25; Inj:7; Fl:5K7
5. Conclusion
Fig. 5. Cost value through the generation.
This paper presents a novel and unified algorithm to account for contingencies when designing or upgrading measurement systems for state estimation. Loss of a single branch and a single measurement are considered as the two possible contingencies, however the type and number of contingencies can be enlarged without affecting the formulation of the proposed algorithm. The developed algorithm avoids iterative addition of measurements and instead allows simultaneous placement of a minimal and optimal set of measurements that will maintain the system observability. It is based on a previously developed method for observability analysis and
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makes use of genetic algorithm in deciding on the placement of measurements that account for all possible contingencies considered. The proposed algorithm is computationally very attractive, yet simple to implement in existing state estimators as an off-line planning tool. References Aam, S., Holten, L., & Gjerde, O. (1983). Design of the measurement system for state estimation in the Norwegian high-voltage transmission network. IEEE Transactions on PAS, PAS-102, 3769–3777. Abur, A., & Magnago, F. H. (1999). Optimal meter placement for maintaining observability during single branch outages. IEEE Transactions on Power Systems, 14, 1273–1278. Baran, M. E., Zhu, J., Zhu, H., & Garren, K. E. (1995). A meter placement method for state estimation. IEEE Transactions on Power Systems, 10, 1704–1710. Celik, M. K., & Liu, W. H. E. (1995). An incremental measurement placement algorithm for state estimation. IEEE Transactions on Power Systems, 10, 1698–1703. Clements, K. A., Krumpholz, G. R., & Davis, P. W. (1983). Power system state estimation with measurement deficiency: An observability/measurement placement algorithm. IEEE Transactions on PAS, PAS-102, 2012–2020. Gou, B., & Abur, A. (2000). A direct numerical method for observability analysis. IEEE Transactions on Power Systems, 15, 625–630.
Gouvea, J. P. S., & SimoesCosta, A. J. A. (1996). Critical pseudo-measurement placement in power system state estimation. In Proceeding of the 12th power systems computation conference, Dresden, Germany. Fetzer, E. E., & Anderson, P. M. (1975). Observability in the state estimation of power systems. IEEE Transactions on PAS, 6, 1981–1988. Koglin, H. J. (1975). Optimal measuring system for state estimation. In Proceeding of PSCC, Cambridge, UK, paper no.2.3/12. Korres, G. N., & Contaxis, G. C. (1994). A tool for the evaluation and selection of state estimator measurement schemes. IEEE Transactions on Power Systems, 9, 1110–1116. Magnago, F. H., & Abur, A. (2000). A unified approach to robust meter placement against loss of measurements and branch outages. IEEE Transactions on Power Systems, 15, 945–949. Milosevic, B., & Begovic, M. (2003). Nondominated sorting genetic algorithm for optimal phasor measurement placement. IEEE Transactions on Power Systems, 18, 69–75. Momticelli, A., & Wu, F. F. (1985). Network observability: Identification of observable islands and measurement placement. IEEE Transactions on PAS, PAS-104, 1035–1041. Park, Y. M., Moon, Y. H., Choo, J. B., & Kwon, T. W. (1988). Design of reliable measurement system for state estimation. IEEE Transactions on Power Systems, 3, 830–836. Saadat, H. (1994). Power system analysis. New York: McGraw-Hill, Inc.. Sarma, N. D. R., Raju, V. V., & Prakasa Rao, K. S. (1994). Design of telemetering configuration for energy management systems. IEEE Transactions on Power Systems, 9, 770–778. Wang, H., & Schulz, N. N. (2004). A revised branch current-based distribution system state estimation algorithm and meter placement impact. IEEE Transactions on Power Systems, 19, 207–213.
Optimal meter placement using genetic algorithm to maintain network observability Amany El-Zonkoly * Department of Electrical and Computer Control Engineering, Arab Academy for Science & Technology, P.O. 1029, Miami-Alexandria, Egypt
Abstract This paper presents a genetic algorithm based method by which measurement system can be optimally determined and upgraded to maintain network observability. Accurate monitoring of power system operation has become one of the most important functions in today’s deregulated power markets. State estimators are the essential tools of choice in the implementation of this function. Determination of the best possible combination of meters for monitoring a given power system is referred to as the optimal meter placement problem. The proposed algorithm yields a measurement configuration that withstands any single branch outage and/or loss of single measurement, without losing network observability. The proposed algorithm is based on the measurement Jacobian and sparse triangular factorization in its numerical part and based on artificial intelligence in the decision making part. Details of the algorithm are presented using two case studies. q 2005 Published by Elsevier Ltd. Keywords: Artificial intelligence; Network observability; Meter placement; State estimation
1. Introduction Accurate monitoring of power system operation has become one of the most important functions in today’s deregulated power markets. State estimators are the essential tools of choice in the implementation of this function. Whether a new state estimator is put into service or an existing one is being upgraded, placing new meters for improving or maintaining reliability of measurement system; is of great concern. Determination of the best possible combination of meters for monitoring a given power system is referred to as the optimal meter placement problem. This problem has been addressed earlier in various studies. Some of them study the problem in the transmission networks (Fetzer & Anderson, 1975; Koglin, 1975; Aam, & Holten., et al, 1983; Park, Moon, Choo, & Kwon, 1988; Sarma, & Raju, et al., 1994; Clements, & Krumpholz, et al., 1983; Momticelli & Wu, 1985; Korres & Contaxis, 1994; Baran, & Zhu et al., 1995; Gouvea & SimoesCosta, 1996; Celik & Liu, 1995; Abur & Magnago, 1999; Magnago & Abur, 2000; Milosevic & Begovic, 2003) and others study the distribution systems (Wang & Schulz, 2004). However, distribution systems have
* Tel.: C20 35622586; fax: C20 35622366. E-mail address: [email protected]
0957-4174/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.eswa.2005.09.016
many features that are different from transmission systems such as (i) radial topology, (ii) three-phase unbalanced systems, (iii) high resistance to reactance ratio and (iv) very limited number of real-time measurements. In this paper, we will concentrate our research on transmission networks only. While the majority of studies are concerned about designing an observable system with minimum variance of errors in the state estimates, others consider loss of lines and/or measurements. They design measurement systems, which can keep the systems observable during such unexpected disturbances. As the power markets become more competitive, having reliable measurement systems that can withstand branch outages or loss of meters, will become more important and the costs associated with metering upgrades will be justified. Abur and Magnago (Abur & Magnago, 1999; Magnago & Abur, 2000) presented a systematic method which provide optimal measurement configuration for a given power system that incorporated meter installation costs into the problem formulation. This method made the selection not only technically but also financially sound. In Abur and Magnago (1999), the authors developed a topological method that accounts for single branch outages only and solved the optimal meter placement problem using linear programming. In Magnago and Abur (2000), the authors presented a method that generalizes the meter placement problem formulation in such a way that considerations of both types of contingencies, namely loss of a branch or a measurement, could be simultaneously taken into account. The problem was then
194
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solved using integer programming. In Milosevic and Begovic (2003), the authors consider the phasor measurement unit (PMU) placement problem in which they try to minimize the number of PMU’s used, so that the system is topologically observable during its normal operation and following any single-line contingency only. They tried also to maximize the measurement redundancy so that instead of a unique optimal solution, there was a set of best tradeoffs between competing objectives, the so-called Pareto-optimal solutions (POS). A specially tailored nondominated sorting genetic algorithm (NSGA) for the PMU placement to find the POS was used. However, they did not consider the meter installation costs into the problem formulation. In this paper, the optimal meter placement problem against both types of contingencies at the same time and taking into consideration the meter installation costs will be solved using an artificial intelligence method, namely the ‘Genetic Algorithm (GA)’. The proposed algorithm provides global solution of the problem that may be missed by other heuristic, sequential selection schemes. It can also be extended to solve the problem in case of loss of multi-branches and/or multi-meters. The paper is organized in such a way that a review of a direct method for network observability analysis is given first. Then, the proposed algorithm for optimal meter placement is introduced. Finally, simulation results are presented at the end of the paper to demonstrate method’s application to typical power system.
The decoupled gain matrix for the real power measurements can be formed as: G Z HTH
(2)
where, measurement error covariance matrix is assumed to be the identity matrix without loss of generality. Note that, since the slack bus is also included in the formulation, the rank of H (and G) will be at most (nK1) (n being the number of buses), even for a fully observable system. This leads to the triangular factorization of a singular and symmetric gain matrix. Consider the step where the first zero pivot is encountered during the factorization of the singular gain matrix, as illustrated below:
ð3Þ
where 1 K1 K1 KT KT KT G 0 Z LK I LiK1 .L1 GL1 .LiK1 Li
(4)
And Li’s are elementary factors given by:
ð5Þ 2. Network observability analysis using direct numerical method The meter placement algorithm presented in this paper is based on the observability analysis method introduced earlier in Gou and Abur (2000). This method will be briefly reviewed first. The static state estimation is a mathematical procedure to compute the best estimate of the node voltage magnitude and angle for each node from a given set of measurements. Network observability must be checked prior to state estimation. If there were any unobservable parts of the network, then a meter placement procedure will be followed, in order to make the entire network observable. Consider the real power versus phase angle part of the linearized and decoupled measurement equation. This is obtained by using the first order approximation of the decoupled nonlinear measurement equation around an operating point: z Z Hq C e
(1)
where z mismatch between the measured and calculated real power measurements; H decoupled Jacobian of the real power measurements versus all bus phase angles; q incremental change in the bus phase angle at all buses including the slack bus; e measurement error vector.
lTi Z ½LiC1;i ; LiC2;i ; .; Lni
(6)
Lij is the ijth entry of Li. Setting LiC1ZInxn, the triangular factorization of G’ in Eq. (3) can proceed with the (iC2)-nd column. This procedure can be repeated each time a zero pivot is detected until completion of the entire factorization. The following expression can then be written: 1 K1 K1 KT KT KT K1 KT D Z LK n LnK1 .L1 GL1 .LnK1 Ln Z L GL
(7)
where D is a singular and diagonal matrix with zeros in rows corresponding to zero pivots encountered during the factorization of G, and L is a nonsingular lower triangular matrix. If the matrix D have more than one zero on its diagonal then the system is unobservable. The above method is simple and non-iterative, which makes it computationally very efficient. Thus, it will be used to initialize the optimal meter placement algorithm developed in this paper. 3. Proposed algorithm for optimal meter placement This algorithm will be applied in two steps. In the first step, an optimal measurement scheme is determined such that
A. El-Zonkoly / Expert Systems with Applications 31 (2006) 193–198
the network will be fully observable. The optimal scheme will represent the essential measurement set. In the second step of the algorithm, the needed set of measurement that is to be added such that the network regains its observability during both types of contingencies mentioned before is optimally determined. The optimization problem in the two steps will be solved using genetic algorithm. 3.1. Solution using genetic algorithm The genetic algorithm will proceed as follows: 1. Initialization: in this step a population of possible solutions of the problem is initiated. Each individual consists of a number of bits corresponding to the number of possible meters to be added in the system. Each individual will be assigned a fitness value which equal to the reciprocal of the cost of the measurement set suggested by the individual. 2. Selection: selection of individuals with highest fitness value is performed to generate the next generation of individuals. 3. Crossover and mutation: these are the main two operators of genetic algorithm. Through them the information contained by each individual are exchanged and manipulated in order to have new individuals with better and better fitness values. 4. The previous steps are repeated until the optimal solution is found.
3.2. Step 1: selection of optimal measurement scheme The novel algorithm, which will identify all the necessary measurements to be used in order to have an observable system will proceed as follows: 1. For each set of measurement suggested by an individual the Jacobian (H) of the system is formed. 2. Calculate the gain matrix (G) and perform triangular factorization to get the diagonal matrix (D). 3. If the matrix (D) have more than one zero in its diagonal then the system is classified to be unobservable. Hence, this individual is refused and another one is generated instead and tested in the same manner. 4. Every individual in any population must yield an observable system. 5. The GA search is performed with testing all of the individuals through the generations until an optimal solution is reached.
3.3. Step 2: optimal meter placement against measurement loss and branch outage With the set of measurements determined in the first step of the solution, the proposed algorithm is to be applied
195
again. The algorithm will be applied in case of loss of any of the predetermined measurements and in case of outage of any single branch that will affect the observability of the system to find the optimal additional set of measurements to maintain the system observability. The measurement Jacobian is to be formed considering the set of measurements determined in step 1. During any of the two types contingencies under consideration, the Jacobian must be modified in accordance. 3.3.1. Loss of single measurement In case of loss of one of the measurements, the Jacobian (H), will be modified (Hmod) by removing one row that corresponds to the lost measurement. Calculating the gain matrix of the modified Jacobian and decomposing it to its factors will show that the system is now unobservable. The GA is applied to determine the additional set of measurement to be used to regain the system observability. To carry on with the observability analysis as described in Section 2, the Jacobian (Hmod) is modified again by adding rows corresponding to the suggested set of measurements. 3.3.2. Loss of a single branch It is known that network observability will be drastically affected by topology changes. In general, it is not necessary to check network observability following the outage of every single branch in the system. It is sufficient to make the measurement system robust against the outage of every single tree branch only. The tree should correspond to the chosen set of essential measurements. Thus, the loss of a co-tree branch (or link) will not have to be considered since it will have no effect on observability. Therefore, for a system of n buses, it is sufficient to check the outage of nK1 tree branches. Assuming that the tree branch kKj is outaged, the measurement Jacobian will be modified as H mod, where: Hikmod Z Hijmod Z 0, if measurement i is a line flow. Hijmod Z 0, Hikmod Z Hik C Hij , if measurement I is an injection at bus k. Hikmod Z 0, Hijmod Z Hik C Hij , if measurement I is an injection at bus j (Magnago & Abur, 2000). 3.3.3. Optimal selection The selection of optimal meter placement must satisfy the condition that the system is observable under any case of losing either a single measurement or a single branch. The optimal solution will yield the minimum number of added measurements with minimum cost that will make the system observable again. The selection mechanism will proceed as follows: 1. Form the Jacobian (H) corresponding to the set of measurements determined in step1. 2. Apply the GA search as described in Section 3.1 such that each solution is tested for observability under all possible contingencies as described in Section 2. 3. The optimal solution is determined in the end of the search.
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4. Simulation results 4.1. Case study 1 The simple 6 bus system shown in Fig. 1 is considered to illustrate the proposed method. All the branch impedances are set equal to j1 pu. The resulting Jacobian for all possible measurements is given below (Magnago & Abur, 2000): q1 Inj:1 Inj:2 Inj:3 Inj:4 Inj:5 Inj:6 Fl:1 Fl:2 Fl:3
2
q2
q3
2
q4 K1
q5
q6 K1
3
7 6 2 K1 K1 7 6 7 6 7 6 K1 2 K1 7 6 7 6 6K1 K1 3 K1 7 7 6 7 6 6 K1 2 K1 7 7 6 7 6 7 6K1 K 1 K 1 3 7 6 7 6 7 6 1 K1 7 6 6 1 K1 7 5 4 1
K1
where, Inj.k and Fl.j represent the net injection at bus k and the power flow through branch j, respectively. 4.1.1. Step 1 The genetic algorithm parameters used in the first step to run the search for the optimal set of measurements to make the system observable are set as follows: Maximum generationZ100 Population sizeZ100 Crossover probabilityZ0.8 Mutation probabilityZ0.01 The cost of the best solution through out the generations is shown in Fig. 2. Where measurements are assigned relative cost to each other, which can be replaced by true values.
Fig. 2. Cost value through the generation.
The optimal and essential set of measurements were found to be: ½Inj:1; Inj:2; Inj:4; Inj:5; Fl:4
4.1.2. Step 2 The genetic algorithm parameters used in the second step are the same as in the first one. In this step, the search is run to find the optimal set of added measurements to maintain the system observability under all possible loss of a measurement or a branch. The contingencies considered are the loss of any of the measurements determined in step 1 and the outage of branches 2–3, 2–5 and 5–6. The outage of these branches only, one at a time, will make the system unobservable. The cost of the best solution through out the generations is shown in Fig. 3. The optimal measurement to be added to maintain system observability under any single branch outage or loss of any single measurement was found to be Inj.6 (the power injection at bus 6). 4.2. Case study 2 The IEEE 30 bus system shown in Fig. 4 is considered also to illustrate the proposed method. The system data is given in Saadat (1994).
Fig. 1. 6 bus system example.
Fig. 3. Cost value through the generation.
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Fig. 6. Cost value through the generation.
½Inj:8; Inj:9; Inj:10; Inj:12; Inj:17; Inj:18; Inj:21; Inj:27; Fl:1K2; Fl:1K3; Fl:2K5; Fl:2K6; Fl:3K4; Fl:4K12; Fl:6K7; Fl:8K28; Fl:9K11; Fl:10K21; Fl:12K13; Fl:14K15; Fl:15K18; Fl:15K23; Fl:16K17; Fl:22K24; Fl:24K25; Fl:25K26; Fl:27K30; Fl:29K30 Fig. 4. IEEE 30 bus system.
4.2.1. Step 1 The genetic algorithm parameters used in the first step to run the search for the optimal set of measurements to make the system observable are set as follows: Maximum generationZ100 Population sizeZ100 Crossover probabilityZ0.8 Mutation probabilityZ0.01 The cost of the best solution through out the generations is shown in Fig. 5. Where measurements are assigned relative cost to each other which can be replaced by true values. The optimal and essential set of measurements were found to be:
4.2.2. Step 2 The genetic algorithm parameters used in the second step are the same as in the first one. In this step, the search is run to find the optimal set of added measurements to maintain the system observability under all possible loss of a measurement and/or a branch. The contingencies considered are the loss of any of the measurements determined in step 1 and the outage of any branch of the network one at a time. The cost of the best solution through out the generations is shown in Fig. 6. The optimal set of measurements to be added to maintain system observability under any single branch outage and/or loss of any single measurement was found to be as follows: ½Inj:20; Inj:28; Inj:23; Inj:25; Inj:7; Fl:5K7
5. Conclusion
Fig. 5. Cost value through the generation.
This paper presents a novel and unified algorithm to account for contingencies when designing or upgrading measurement systems for state estimation. Loss of a single branch and a single measurement are considered as the two possible contingencies, however the type and number of contingencies can be enlarged without affecting the formulation of the proposed algorithm. The developed algorithm avoids iterative addition of measurements and instead allows simultaneous placement of a minimal and optimal set of measurements that will maintain the system observability. It is based on a previously developed method for observability analysis and
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