www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 27th October 2007 Revised on 25th February 2008 doi: 10.1049/iet-gtd:20070446
ISSN 1751-8687
Algorithm for a class of constrained weight least square problems and applications S.-S. Lin1 S.-C. Horng2 1
Department of Electrical Engineering, St John’s University, 499, Sec. 4, Tam King Road, Tamsui, Taipei25135, Taiwan, Republic of China 2 Department of Computer Science and Information Engineering, Chaoyang University of Technology, 168 Jifong E. Rd., Wufong, Taichung 41349, Taiwan, Republic of China E-mail:
[email protected]
Abstract: An algorithm, based on ordinal optimisation (OO) and sensitive theories, is presented to solve a class of constrained weight least square problems with continuous and discrete variables. The proposed algorithm can cope with an enormous amount of computational complexity problems and has a high probability of obtaining a good enough solution according to the OO theory. This method has some advantages, such as computational efficiency, numerical stability and the superiority of the good enough solution. The proposed algorithm is explicit, compact and easy to program. Test results demonstrate that the proposed approach is more computational-efficient than other existing approaches for solving constrained-state estimation problems with continuous and discrete variables on the IEEE 30-bus and the IEEE 118-bus systems.
1
Introduction
The aim of the state estimation problem is to determine the best estimate of the current system state (voltage magnitude and phase angles of all buses). The system states are estimated from available measurements, which are functions of states, such as power transmission line flow, bus voltage magnitude and bus power injection [1 – 8]. State estimation problems with equality and/or inequality constraints, which are very important in power-system research, can be formulated as a class of constrained weight least square (WLS) problems in its exact problem formulation. Several solution schemes that have been developed [9] along with many technologies corresponding to power system are listed in [10]. State estimation problems with continuous variables are typical convex programming problems. Solution to these problems mostly originates from nonlinear programming algorithm and exploits the linear constraint structure with various approaches [11– 16]. These solutions include successive quadratic programming (SQP) [11], Lagrange –Newton (LN) [1, 2], interior point [12– 14], Lagrange relaxation (LR) 576 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
[15] and conventional Lagrange [16]. The SQP method has a good convergence rate, but results in a dense Hessian matrix. The LN method successfully adopts the sparse structure of network, but has a coupling inequality constraint. The LR method is a gradient method, and thus is simple to program but has a slow convergence rate. Several efficient methods have recently been developed to solve the state estimation problems with equality and/or inequality constraints in power systems [1– 3, 17, 18] and obtained some successful results. The above investigations focus only on continuous variables, although few attempt to solve state estimation problems with continuous and/or discrete variables. In real applications, continuous and/ or discrete variables both existed in the optimisation problems such as the optimal power flow problem with discrete variables [19 – 22]. The problems become NP-complete optimisation problems because of the huge sample space of discrete variables setting. Some problems are NP-complete, such as raveling salesman problem, job shop scheduling problem and Hamiltonian circuit problem [23, 24]. This work presents a solution technique to solve an instance of the state estimation problem
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www.ietdl.org including both continuous and discrete variables. Discrete variables are usually not free to assume any value in a state estimation problem. In conventional approaches, discrete variables can include the switching shunt capacitor banks, which are switched on and off in order to reduce active power transmission losses and transformer taps, which are adjusted step by step to ensure that a voltagecontrolled bus maintains its voltage within acceptable limits. Nonetheless, optimisation problems with both continuous and discrete variables are difficult to solve. An efficient and powerful ‘ordinal optimisation’ (OO) theory has recently been developed and explored in [25–30] to solve these NP-complete optimisation problems to obtain a good enough solution with high probability. We briefly describe the OO theory below. The typical problem formulation considered in the OO theory is of the following form minu[Q f (u )
(1)
where Q is a huge input-variable space and f () the objective function which may be an expected output or a function of the mixed with the continuous and discrete input variables. This work presents an algorithm based on OO [25–30] and sensitive theories [31] to solve a class of NP-complete optimisation problems-constrained WLS problems with continuous and discrete variables. This paper is organised as follows. Section 2 describes the mathematical formulation of the state estimation problems with continuous and discrete variables. Section 3 presents the proposed OO and sensitive theories based algorithm for solving NP-complete optimisation problems. Section 4 tests the proposed algorithm in two power systems, the IEEE 30-bus and IEEE 118-bus systems, and compares the results with those obtained by other existing approaches. Finally, conclusions and suggestions for further research are given in Section 5. The OO theory is briefly given in Section 8.
nx-dimensional vector of the continuous state variables, that is, xc ¼ (V, u ), where V denotes voltage magnitude and u denotes phase angles g g g¯ the physical limits such as the maximum active/reactive power output constraints of generators, where g and g denote the lower and upper bounds of g, respectively z measurement-value vector, which may include power injection measurements and tier-line flow measurements measured from the end bus nonlinear measurement vector function of h states corresponding to the measurement z J objective function of the constrained WLS problems with continuous and discrete variables, that is, J ¼ 1=2[z h (xc , ad )]T R1 [z h (xc , ad )] R diagonal covariance matrix of the Gaussian measurement-error vector [z h (xc , ad )] set of all the zero injection buses Izero set of all the buses connecting with bus l Bl lj lj real and reactive power flow from bus l to j p ,q P real and reactive power flow balance j[Bl lj constraints in zero injection bus l p ¼ 0, P xc
j[Bl
q lj ¼ 0 Ad j(.)j ac ac j
[ac j
a c j]
ll
2 Mathematic formulation of the state estimation problems with continuous and discrete variables We first introduce the notations for the state estimation problems with continuous and discrete variables as follows. na-dimensional vector of the discrete ad variable such as switching shunt capacitor banks and transformer taps continuous version of ad ac ac , a¯c lower and upper bounds of ac
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@p lj/@ac , @q lj/@ac h
sample space of ad cardinality of the set (.) vector of the optimal solution of the discrete variable of ad replaced by the vector of continuous variable ac jth component of the continuous variable vector ac nearest lower bound integer (near the left-hand side integer value) of the acj nearest upper bound integer (near the right-hand side integer value) of the acj l th component of the Lagrange multiplier l, lT ¼ (l1 , . . . , ljIzero j )T , vector corresponding to the real and reactive power flow balance P constraints in l zero lj injection bus, P j[Bl p (xc , ac ) ¼ 0, lj at the optimal j[Bl q (xc , ac ) ¼ 0, solution of (xc (ac ), ac ) partial derivative of plj (xc , ac ), qlj ðxc ; ac Þ w.r.t. ac , respectively positive real numbers
The state estimation problem with continuous and discrete variables can be formulated as a type of constrained WLS problem described in the following IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576– 587/ 577 doi: 10.1049/iet-gtd:20070446
www.ietdl.org min xc ,ad
1 J(xc , ad ) ¼ [z h (xc , ad )]T R1 [z h (xc , ad )] 2
subject to (2a) X j[Bl
plj (xc , ac ) ¼ 0,
X
qlj (xc , ac ) ¼ 0, 8l [ Izero (2b)
j[Bl
g g g
(2c)
3.1 Representative set N with size N ffi 1000 A crude but efficient model using the heuristic scheme to select the representative set N was constructed in two stages. The first stage regards the discrete variable ad as the continuous variable ac . Therefore (2a) – (2d) can be rewritten as (3a) – (3d) 1 J(xc , ac ) ¼ [zh (xc , ac )]T R1 [zh (xc , ac )] 2
min xc ,ac
subject to ad [ Ad
(2d)
(3a) X
The goal of the state estimation problem with continuous and discrete variables is to find an optimal continuous state variable xc and a discrete variable ad from the sample space Ad such that the objective function J(xc , ad ) is minimised, while satisfying the power flow balance constraints (2b) and the physical limits (2c).
3
Solution method
This section proposes an algorithm based on OO and sensitive theories to cope with the computational complexity of the immense discrete sample space. The OO theory indicates that the ‘OO’ performance of the sample is likely to be preserved, even when it is evaluated using a crude model. Therefore this work constructed a crude but efficient model, using a heuristic scheme to select the representative set N from the sample space. A more accurate scheme based on the sensitive theory was then used to rank the N samples and choose the top s samples to form the selected subset SS. Finally, these s discrete samples, in the SS of the constrained WLS problems with continuous and discrete variables (2a) – (2d), were solved by the exact model. The OO theory guaranteed that the top setting (i.e. k ¼ 1, the smallest objective value) selected from SS must belong to the good enough solution set GS with high probability [25 –30]. The good enough solution set represents a collection of the top 5% good enough reasonable samples among N.
Remark 1: Heuristic generation for obtaining N excellent samples may depend on how well one’s knowledge about the considered system is. For instance, in the wafer testing process problems with discrete threshold values, Lin and Horng [28] proposed an algorithm based on the ordinal optimisation (OO) theory and engineering intuition to select N excellent threshold values. 578 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
j[Bl
plj (xc , ac ) ¼ 0,
X
qlj (xc , ac ) ¼ 0, 8l [ Izero (3b)
j[Bl
g g g
(3c)
ac ac a c
(3d)
Constrained WLS problems (3a) – (3d), which are convex problems due to having only continuous variables (i.e. xc and ac), can be solved by the previous research such as the efficient dual-type method [32] to obtain an optimal solution (xc (ac ), ac ). Each jth component of the discrete ad , say adj, is set to either the nearest lower bound integer (near the left-hand side integer value) or the nearest upper bound integer (near the right-hand side integer value) of the acj (i.e. jth component of the ac ), denoted by 0 [acj 0 or 0 acj ]0, respectively. If each discrete variable has m possible discrete values, then the search sample space is reduced from mna to 2na . Clearly, the good solution should be among these 2na samples. However, 2na is still larger than the size of N, N ffi 1000, for a typical constrained WLS problem with continuous and discrete variables in an averagesize of power system. Because 2na samples cannot be applied to the OO theory, the samples space needs to be reduced further to obtain N discrete samples. The second stage uses the weighting of the minimum norm deviation, Dac j =Dac j , and the fitness of adj , to evaluate each jth component of ac , say acj, to fit each component adj . The deviation, Dacj ¼ (acj [acj ) or (acj ] acj ), results from the change from continuous value acj to the set of discrete values adj (either the 0 0 0 0 integer values [acj or acj ] ). If Dacj =acj h, where h is a predetermined positive value, then the
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www.ietdl.org discrete value adj is fixed at 0 [acj 0 or 0 acj ]0. If k 0 s adj components have been fixed by this method, then (na k)’s components of adj still have not yet been fixed. Thus, the number of search samples can be reduced further from 2na to 2na k. Therefore the value of h should satisfy 2na k ffi N, N ¼ 1000. This heuristic method is run to obtain the representative set N from the sample space.
3.2 Selected subset The N samples in the representative set N determined by the heuristic method presented in the previous section are denoted by ad (i), i ¼ 1, . . . , N. It should be noticed that the objective function in (2a) is expressed as J(xc , ad ) ¼ 1=2[z h (xc , ad )]T R1 [z h (xc , ad )]. An accurate three-phase method for obtaining the SS, based on the sensitive theory approach [31], is now described. The first phase is to estimate the deviation owing to the change of ac from the ac to the ad(i) of the optimal objective in (2a) –(2d) by following [31], that is, DJ(ad (i)) ¼ J(xc (ac ), ac ) J(xc (ad (i)), ad (i)) X X @plj @qlj (a a (i)) ll þ DJ(ad (i)) ffi d c @a @a c c ac l[I j[B zero
l
(4) where ll indicates the lth component of the Lagrange multiplier vector l, lT ¼ (l1 , . . . , ljIzero j )T , corresponding to the real and reactive power flow balance constraints in the l zero injection bus, P P lj lj j[Bl p (xc , ac ) ¼ 0, and j[Bl q (xc , ac ) ¼ 0, respectively, at the optimal solution of (xc (ac ), ac ), and Izero denotes the cardinality of the set Izero . The second phase is to rank the N samples based on the value DJ(ad(i)) . A sample ad (i) with a smaller value DJ(ad (i)) is ranked higher, since it is less sensitive to the optimal objective value. Finally, the top s samples with smaller value DJ(ad (i)) are selected to form the SS. The number of s can be obtained by [26].
3.3 Good enough solution These s samples in the SS are represented by ad (i), i ¼ 1, . . . , s. The efficient dual-type method [32] is employed again to solve the constrained WLS problems in (2a) – (2d) for every sample ad (i), i ¼ 1, . . . , s. The OO theory guarantees that the top setting [i.e. k ¼ 1, the smallest objective value of J(xc , ad )] selected from the SS with s discrete
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samples belongs to the good enough solution set with probability 0.95 [25 – 30].
3.4 Complete algorithm for solving the state estimation problems with continuous and discrete variables The proposed method for solving state estimation problems with continuous and discrete variables in (2a) – (2d) is briefly described below. First, a crude model is constructed, and the representative set N is selected using a heuristic scheme. The proposed efficient dual-type method [32] is applied to obtain an optimal solution (xc (ac ), ac ), with the discrete variable ad in (2a) – (2d) being adopted as the continuous variable ac in (3a) – (3d). The minimum norm deviation approach is then applied to evaluate each jth component of ac , say acj, to fit the optimal discrete variables adj . The sensitive theory based method in (4) is then used to determine the SS. Finally, the s discrete samples in the SS of the constrained WLS problems (2a) –(2d) are solved to obtain the good enough solution set. The full algorithm for solving the state estimation problem with continuous and discrete variables is stated below. Step 0. Specify the value of s and set the parameter h. Step 1. Use the efficient dual-type method to solve the constrained WLS problems (3a) – (3d) and obtain the (xc (ac ), ac ). Step 2. Set each of the adj to be the near bound integer value of the acj , that is, adj ¼ [acj or acj ], and form the 2na discrete samples of adj . Step 3. Calculate the deviation of each acj and find the k0 s samples with smaller value of the weighting of minimum norm deviation, Dacj =acj h, such that 2na k ffi 1000 (the representative set N ), then fix these k0 s samples adj to be the nearest bound integer value of acj , that is, adj ¼ [acj or acj ]. Step 4. Compute DJ(ad (i)) by (4) and calculate DJ(ad (i)), 8i, i ¼ 1, . . . , N. Step 5. Rank all N samples based on the value of DJ(ad (i)) and select the top s samples with smaller value to form the SS. Step 6. Use the efficient dual-type method [32] to solve the constrained WLS problems in (2a)-(2d) for these samples ad (i), i ¼ 1, . . . , s, the one with the smallest objective value is the good enough solution. Remark 2: Significantly, if the numbers of these 2na discrete samples of adj obtained in Step 2 are not IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576– 587/ 579 doi: 10.1049/iet-gtd:20070446
www.ietdl.org larger than 1000, that is, N samples are smaller than 1000, then the value of s in the SS can still be calculated using a Monte Carlo study [26].
3.5 Comments on the advantage of the proposed algorithm The proposed algorithm has the following two merits. The first is the reduction of the immense searching space. Given na discrete variables, each with m possible discrete values, sample space Ad contains mna samples of ad . For example, if na ¼ 10 and m ¼ 12, then Ad contains 1210. Because the sample space is huge, solving for an optimal solution of constrained WLS problems shown in (2a) – (2d) using traditional global searching will be computationally intractable. The second merit is that the dual-type method is used to solve constrained WLS problems efficiently. The conventional Lagrange duality method [33] utilises Lagrange multiplier to relax the inequality constraints and the Kuhn –Tucker condition to solve the problem. Adding or dropping the active inequality constraints from the working set [29] usually causes unexpected computational loading during their solution processes. The proposed efficient dual-type method [32] differs from the conventional Lagrange method by regarding the inequality constraints as the domain of primal variables in the dual function, and employing the projection theory [34] to manage the projection problem induced by the inequality constraints.
4
Simulation and test results
The constrained state estimation problems are classified as constrained WLS problems [16– 18]. The proposed algorithm was tested in two different power systems, one small (IEEE 30-bus system) and one large (IEEE 118-bus system), as shown in Figs. 1 and 2, respectively. A system with n buses has 2n 2 1 state variables. Therefore the IEEE 30-bus system has 59 continuous variables, comprising 30 voltage
Figure 1 Diagram of the IEEE 30-bus system 580 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
magnitudes and 29 phase angle variables, and the IEEE 118-bus system has 235 continuous variables, comprising 118 voltage magnitudes and 117 phase angle variables. Additionally, both the IEEE 30-bus and the IEEE 118-bus systems have some discrete variables in the power flow balance constraints of zero injection buses. Notably, no-generating and no-loading buses are zero injection buses. For example, bus 2 in Fig. 1 is a generating bus, bus 7 is a loading bus, but bus 5 is neither a generating nor a loading bus. Fig. 2 uses the same symbols as Fig. 1. The discrete variables, which include the capacitor bank and transform tap ratio settings in the zero injection buses, satisfy the power flow balance constraints. Each capacitor has four capacitor banks, and each transform tap has six discrete steps. Moreover, the physical limits (e.g. the reactive power output constraints) in some generator buses are also addressed. The measurements considered in the constrained state estimation problem with continuous and discrete variables include the real power injection measurements, voltage magnitude measurements and real power line flow measurements [18]. The objective function is J(xc , ad ) ¼ 1=2[z h (xc , ad )]T R1 [z h (xc , ad )] in all cases. To compare the efficiency of the different methods, four cases [Cases (a)– (d) and Cases (A)– (D)] on the IEEE 30-bus and the IEEE 118-bus systems were tested, including four different numbers of discrete variables considered in the power flow balance constraints in zero-injection buses. The numbers of discrete variables were increased from Case (a) to Case (d), as well as from Case (A) to Case (D). The physical limits considered in this paper are the reactive power output constraints in some power generating buses and are the same in all test cases. The detailed descriptions for both the IEEE 30-bus and the IEEE 118-bus systems are stated below. In the IEEE 30-bus system (Fig. 1), the power flow balance constraints considered in the zero injection buses are buses 5, 6, 9, 10 and 22 for Case (a); buses 5, 6, 9, 10, 22 and 25 for Case (b); buses 5, 6, 9, 10, 22, 25 and 27 for Case (c) and buses 5, 6, 9, 10, 22, 25, 27 and 28 for Case (d). Because each capacitor has four capacitor banks and each transform tap has six discrete steps, the corresponding sizes of the discrete sample of ad in Ad are (4 6)5, (4 6)6, (4 6)7 and (4 6)8 in Cases (a)–(d), respectively. The reactive power output constraints considered in some power generating buses are the same in all cases with setting in buses 1, 2, 8, 11 and 13. In the IEEE 118-bus system (Fig. 2), the power flow balance constraints considered in the zero injection buses are buses 60, 63, 64, 37, 39, 41, 17, 28, 115, 2, 11, 12, 117, 48 and 22 for Case (A); buses 50, 51, 60, 63, 64, 37, 39, 41, 17, 28, 29, 114, 115, 2, 11, 12,
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Figure 2 Diagram of the IEEE 118-bus system
117, 48, 67, 21 and 22 for Case (B); buses 50, 51, 52, 53, 60, 63, 64, 37, 39, 41, 17, 28, 29, 114, 115, 2, 7, 9, 11, 12, 117, 48, 67, 21, 22, 71 and 118 for Case (C) and buses 50, 51, 52, 53, 57, 58, 60, 63, 64, 37, 39, 41, 43, 44, 17, 28, 29, 114, 115, 2, 7, 9, 11, 12, 117, 48, 67, 79, 81, 21, 22, 71 and 118 for Case (D). Since each capacitor has four capacitor banks and each transform tap has six discrete steps, the corresponding sizes of the discrete sample of ad in Ad are (4 6)15, (4 6)21, (4 6)27 and (4 6)33 in Cases (A) – (D), respectively. The reactive power output constraints considered in some power generating buses are the same in all cases with setting in buses 54, 55, 56, 59, 34, 36, 40, 42, 46, 15, 18, 19, 27, 31, 32, 1, 4, 6, 8, 10, 62, 65, 66, 78, 24, 25, 26, 72, 73, 103, 104, 104, 111, 112, 85, 87, 89, 90, 91 and 92.
Remark 3: This work considers the cases corresponding to discrete variables in some transit buses. The discrete device (including discrete variable) usually need not be located in transit buses (buses with no generation and no demand). The discrete device can be located in some buses. For example, discrete devices such as switching shunt capacitor banks are switched on and off in order to reduce active power transmission losses, and transformer taps are adjusted step by step to ensure that a voltage-controlled bus maintains its voltage within acceptable limits. Remark 4: The measurements set-up in the IEEE 30-bus system (Fig. 1) is as follows: 62 real and reactive power flow measurements on transmission lines in some buses,
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22 real and reactive power injection measurements in some buses, 10 voltage magnitude measurements in some buses and a redundancy of about 1.59. The measurements set-up in the IEEE 118-bus system (Fig. 2) is as follows: 306 real and reactive power flow measurements on transmission lines in some buses, 84 real and reactive power injection measurements in some buses, 27 voltage magnitude measurements in some buses and a redundancy of about 1.77. The measured values were assumed to be the values obtained from the power load flow running with 100 MV and 100 MVAR base for real and reactive power [16], respectively. The value of s to form the SS was determined from the universal alignment probability [26]. To determine the value of s used in the OO theory conservatively, this work considered about three types of noise j(). Medium noise, which has a noise range equal to the range of the ordered performance curve (OPC), was generated from the uniform distribution random variable U[ 20.5, 0.5]. Large noise, which has a noise range twice as wide as the OPC range, was generated from the uniform distribution random variable U[21.0, 1.0]. The very large noise was generated from the uniform distribution random variable U [22.5, 2.5], which indicates the noise range between large noise and infinite noise. Therefore depending on the three types of noise and some parameters, three different sizes of SS, s ¼ 12 s ¼ 13 and s ¼ 28 were obtained by the direct numerical calculation form [26]. The detailed descriptions are given in Section 8. IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576– 587/ 581 doi: 10.1049/iet-gtd:20070446
www.ietdl.org s ¼ 28, respectively. Cases (A) – (D) in Tables 4 –6 represent the test cases on the IEEE 118-bus system and the sizes of SS are s ¼ 12, s ¼ 13 and s ¼ 28, respectively.
A single PC was used as the experimental computer; the CPU processor speed was 3200 MHZ and the RAM size was 512 MB. The algorithm was tested on the state estimation problems with continuous and discrete variables for two test systems, four cases of power flow balance constraints and three different sizes of SS, based on the conservative consideration of the noise type as described above. The reactive power output constraints considered in some generating buses were the same in each test case in the same test system. The proposed algorithm converged in all cases, as expected. Tables 1 –4 show the average CPU time (time) in seconds (s), and the resulting objective values (obj.).
To show the efficiency, the proposed algorithm was compared with the conventional approach incorporating the LN method [33], abbreviated as the Conv. þ Lag. Newton method henceforth. First, the LN method [33] was used to solve the constrained WLS problems (3a) – (3c) for Cases (a) – (d) in the IEEE 30-bus system and Cases (A) – (D) in the IEEE 118-bus system. The resulting optimal continuous value of the discrete variable, ac , was then rounded to the nearest discrete value. After setting the value of the discrete variable at ad , the LN method [33] was applied again to solve the constrained WLS problems (2a) – (2d) in all cases for this discrete sample ad only.
A summary is described below. Cases (a) – (d) in Tables 1 – 3 represent the test cases on the IEEE 30bus system and the sizes of SS are s ¼ 12, s ¼ 13 and
Table 1 Comparison of our algorithm (s ¼ 12) with the Conv. þ Lag. Newton and competing MFT methods on IEEE 30-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(a)
28.025
2.878
116.768
3.311
3.166
0.150
1.822
3.363
(b)
28.104
2.883
118.236
3.379
3.207
0.172
1.826
3.384
(c)
28.419
2.893
123.321
3.458
3.339
0.195
1.834
3.471
(d)
28.844
2.905
125.558
3.476
3.353
0.196
1.845
3.482
Table 2 Comparison of our algorithm (s ¼ 13) with the Conv. þ Lag. Newton and competing MFT methods on IEEE 30-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(a)
30.360
2.878
116.768
3.311
2.846
0.150
1.822
3.363
(b)
30.445
2.883
118.236
3.379
2.883
0.172
1.826
3.384
(c)
30.788
2.893
123.321
3.458
3.009
0.195
1.834
3.471
(d)
31.247
2.905
125.558
3.476
3.018
0.196
1.845
3.482
Table 3 Comparison of our algorithm (s ¼ 28) with the Conv. þ Lag. Newton and competing MFT methods on IEEE 30-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(a)
65.391
2.878
116.768
3.311
0.785
0.150
1.822
3.363
(b)
65.574
2.883
118.236
3.379
0.803
0.172
1.826
3.384
(c)
66.309
2.885
123.321
3.458
0.859
0.198
1.834
3.471
(d)
67.301
2.898
125.558
3.476
0.865
0.199
1.845
3.482
582 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
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www.ietdl.org Table 4 Comparison of our algorithm (s ¼ 12) with the Conv. þ Lag Newton and competing MFT methods on IEEE 118-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(A)
45.401
7.811
425.589
9.214
8.374
0.179
4.965
9.451
(B)
45.441
7.814
434.527
9.401
8.562
0.203
5.237
9.459
(C)
45.545
7.821
449.391
9.574
8.867
0.224
5.431
1
(D)
45.735
7.828
465.322
9.598
9.174
0.226
5.439
1
Table 5 Comparison of our algorithm (s ¼ 13) with the Conv. þ Lag. Newton and competing MFT methods on IEEE 118-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(A)
49.183
7.811
425.589
9.214
7.653
0.179
4.965
9.451
(B)
49.249
7.814
434.527
9.401
7.823
0.203
5.237
9.459
(C)
49.341
7.821
449.391
9.574
8.107
0.224
5.431
1
(D)
49.546
7.828
465.322
9.598
8.391
0.226
5.439
1
Table 6 Comparison of our algorithm (s ¼ 28) with the Conv. þ Lag. Newton and competing MFT methods on IEEE 118-bus system Case
Our algorithm
MFT method
Efficiency
Conv. þ Lag. Newton
Time, s
Obj.
Time, s
Obj.
TSF
OSF
Time, s
Obj.
(A)
105.934
7.811
425.589
9.214
3.017
0.179
4.965
9.451
(B)
106.029
7.814
434.527
9.401
3.098
0.203
5.237
9.459
(C)
106.272
7.815
449.391
9.574
3.228
0.225
5.431
1
(D)
106.715
7.822
465.322
9.598
3.360
0.227
5.439
1
All the reactive power output constraints considered in some generating buses were the same in each case in the same test system as described above. Tables 1– 6 also show the average CPU time (time) in seconds (s), and the resulting objective values (obj.) consumed by the Conv. þ Lag. Newton method. The proposed algorithm was also compared with the competing mean field theory (MFT) method [19] to demonstrate its ability to solve constrained WLS problems with continuous and discrete variables. The MFT method adopts the Potts spin model [35] to derive the mean field equations and the Newton updating scheme of the mean field equations to improve the optimisation efficiency. The competing MFT method was run to solve the same constrained WLS problems (2a) – (2d) for all cases in both the IEEE 30-bus and the IEEE 118-bus systems. Tables 1–6 introduce two evaluation factors, the time saving
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factor (TSF) and the objective value saving factor (OSF), to evaluate the performance of the proposed algorithm and the competing MFT method. The Conv. þ Lag. Newton method had a shorter CPU time consumed than our algorithm in all cases in Tables 1 – 6, because it solved only one discrete sample ad of the constrained WLS problem (2a) – (2d), whereas the proposed algorithm considered the s discrete samples ad . However, the conventional approach may be infeasible for Cases (C) and (D) in Tables 4–6, which involve infinite objective values. Since the proposed algorithm can obtain a good enough solution in all cases, TSF and OSF comparisons between the proposed algorithm and the Conv. þ Lag. Newton method were not performed. Simulation results demonstrate that the proposed algorithm had smaller objective values than both Conv. þ Lag. Newton method and competing MFT IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576– 587/ 583 doi: 10.1049/iet-gtd:20070446
www.ietdl.org method. Among all of discrete variables, the weighting of the minimum norm deviation, Dacj =acj , corresponding to the transformer tap ratio changes was less than that corresponding to the capacitor bank changes. Hence, the transformer tap ratios were fixed as described in the minimum norm deviation approach of our algorithm. This result is consistent with the study in [36]. The test results of two conservative cases for s ¼ 28 in Tables 3 and 6 are described in detail below. These results reveal that the values of TSF were slightly increasing for Cases (a)– (d) in Table 3 when the discrete samples were slightly increasing. The average value for TSF was about 0.828. Conversely, the competing MFT method exhibited a moderate rise in CPU time when solving the same problems for Cases (a) –(d), whereas the numbers of discrete sample slightly increased. Meanwhile, the proposed algorithm produced smaller objective values than the competing MFT method for all cases in Table 3 and had an average value OSF of about 0.179. Thus, the proposed approach is efficient for solving the state estimation problems with continuous and discrete variables. According to Tables 1 and 2, the values of TSF achieved by our algorithm were much higher than those obtained by the competing MFT method with respect to Table 3 at low values of SS. The objective values resulting from our algorithm were also smaller than those from the competing MFT method in every case in Tables 1 and 2. The objective values obtained by the proposed algorithm were the same in each corresponding case, because the top setting [i.e. k ¼ 1, the smallest objective value of J(xc , ad )] selected from the SS in Tables 1 (s ¼ 12) and 2 (s ¼ 13) was identical. Likewise, the objective values resulting from our algorithm in Cases (a) and (b) in Table 3 (s ¼ 28) were also the same as those Cases (a) and (b) in Tables 1 and 2. Cases (c) and (d) in Table 3 yielded slightly better discrete samples owing to the conservative consideration of the noise type, and slightly higher OSF values, than Cases (c) and (d) in Tables 1 and 2. However, the size of s ¼ 12 (the medium noise) and s ¼ 13 (the large noise) in Tables 1 and 2 already provided the proposed algorithm to obtain a good enough solution more effectively than competing MFT method, as indicated by the values of OSF and the average values of TSF listed in Tables 1 and 2. To make a comparison of the efficiency within two different sizes of test systems, the proposed algorithm was tested in a large system (the IEEE 118-bus system). The values of TSF were highly rising for Cases (A) –(D) in Table 4 when the discrete samples 584 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
were moderately increasing. The average value of TSF was about 3.175. Nevertheless, the competing MFT method exhibited a strong rise in CPU time for solving the same problems for Cases (A) – (D) when the numbers of discrete samples moderately increased. The objective values obtained by our algorithm were also smaller than those from the competing MFT method in each case in Table 6, and the average value of the OSF was about 0.208. Thus, the proposed method is more suitable for solving the state estimation problems with continuous and discrete variables in large test systems. The conventional approach produced some infeasible solutions, as indicated in Cases (C) and (D) in Tables 4 – 6, which produced infinite objective values. However, the proposed algorithm still obtained good enough solutions for all cases, revealing that the probability of our algorithm being unable to yield any feasible solution is extremely low. These tests results demonstrate that the performance of our algorithm is significant for solving state estimation problems with continuous and discrete variables. Remark 5: The tests results indicates that the size of s ¼ 12 according to the universal alignment probability [26] can already obtain a good enough solution in solving the state estimation problems with continuous and discrete variables in both IEEE 30-bus and IEEE 118-bus systems.
5 Conclusions and further research This paper presents an efficient algorithm, based on the OO and sensitive theories, to solve a class of NPcomplete optimisation problems-constrained WLS problems with continuous and discrete variables. The proposed algorithm can yield a good enough solution with smaller objective values than the conventional approach. Simulation test results were performed to compare the efficiency of the proposed algorithm and the competing MFT in solving the constrained state estimation problems with continuous and discrete variables on the IEEE 30-bus and the IEEE 118-bus systems. The results were measured according to two evaluation factors, TSF and OSF. Further research will investigate combining the proposed algorithm with the decentralise step-size rule [37] and the active set strategy based dual-type method [38] in order to solve distributed constrained WLS problems with continuous and discrete variables in real asynchronous distributed computing environments [18].
6
Acknowledgments
The authors deeply appreciate the Associate Editor and the anonymous reviewers for their useful suggestions
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www.ietdl.org and comments to improve the quality of this paper. This research work is supported in part by the National Science Council in Taiwan under Grant NSC94-2622E129-008-CC3.
7
power flows’, IEEE Trans. Power Syst., 2001, 16, (2), pp. 222– 228 [13] WU Y.: ‘Fuzzy second correction on complementary condition for optimal power flows’, IEEE Trans. Power Syst., 2001, 16, pp. 360 – 366
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Appendix
‘OO’ theory has been developed and explored in [25–30] for solving NP-complete optimisation problems to obtain a good enough solution with high probability. There are two basic principles of the OO theory. The first one is that order against value in decision-making. It is clear that determining whether 586 /IET Gener. Transm. Distrib., 2008, Vol. 2, No. 4, pp. 576 – 587 doi: 10.1049/iet-gtd:20070446
f (u1 ) , f (u2 ) is much more easier than calculating the value of f (u2 ) f (u1 ) ¼ ? In other words, considering the intuitive example of determining which of the two melons in two hands is heavier against identifying how heavier one is than the other. The second principle is the goal softening. Instead of asking the best for sure in optimisation, it settles for the good enough with high probability. A word drawn form the OO theory is the following. Suppose the designers simultaneously evaluate a large set of alternatives very approximately and order them according to the approximate evaluation. Then there is high probability that the designers can find the actual good alternatives if the designers limit themselves to the top n% of the observed good choices. Therefore the goal softening policy based on OO theory is to decrease the searching space gradually, which can be carried out in the following three steps: [25] (a) Uniformly select the representative set N,N ffi 1000, settings from Q. (b) Estimate and sort the N settings via a rough model of the considered problem, then select the top s [the settings samples to form the SS and the exact numbers s can be obtained by [26] and we also had a brief description in Section 8], which is the candidate samples of the good enough subset GSN. (c) Estimate and sort all the s settings in SS via the accurate form, then select the top k, k 1 settings (k is called the numbers of alignments in OO theory [26]). The OO theory guarantees that for N ffi 1000 in (a) and a rough form with moderate noise in (b), the top setting (i.e. k ¼ 1) selected from (c) within s samples must belong to the good enough subset GSN with probability around 0.95 where good enough subset GSN represents a collection of the top 5% actually good enough settings among N. This represents that the actual top setting in SS selected from (c) is among the real top 5% of the N settings with probability 0.95. Nonetheless, the good enough solution of problem (1) that the designers are searching for should be a good enough setting in Q instead of the N settings unless the set u is quite smaller as well as N [26, 27]. An OO theory-based two-level algorithm to solve for a vector of good enough threshold values in the wafer testing process was presented in [28]. Recently, a more conservative theory corresponding to the universal alignment probability, the Technical Note of the Universal Alignment Probability revisited was presented in [29]; the Note addressed that less than a moderate modelling noise, the top 3.5% of the uniformly selected N settings, say GSrN (the reduced good enough subset of N ), will be among the top 5% settings of a huge Q, say GSQ (the truly good enough subset of the whole sample space Q), with an extremely high probability 0.991 (i.e. Prob{GSrN # GSQ } ¼ 0:991), and the best case can be among the top 3.5% settings of Q if there is no modeling error existed [29]; in other words, if the
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www.ietdl.org designers choose a more restricted definition of good enough subset GSN, the top 5% actually good enough settings among N defined in (c), instead of the reduced good enough subset GSrN , the top 3.5% actually good enough settings among N, the designers can use the corresponding technology of universal alignment probability curves [26] to predict the alignment probability within the intersection of SS and GSQ under the alignments k (i.e. Prob {jSS > GSQ j k}) [29]. The OO approach is a noble technology designed to handle the tough problems such as problems with enormous sample space that grows exponentially with respect to the dimension and will obtain a good enough solution with high probability. Furthermore, OO theory provides a mathematic formula for calculation the SS size value of s as the value of six-parameter function z(k, g, N, C, j(), PA ) [26]. The notation of these parameters is described as follows: k represents the desired number of alignments between SS and GSN, g the size of GSN, C a category of OPC. The OPC category chosen for the N sample strongly depends on the designers’ knowledge of the target system. For example, if one is familiar with the structure of the system, the designers may use a better heuristic method than uniform selection to determine the representative set N from Ad . j() represents the noise characteristics of the modelling error v. If a uniform noise density U[ W , W ] is assumed, the magnitude W strongly depends on the crude model J(xc , ad ) þ v chosen for determining SS. PA is the alignment probability and is taken to be 0.95 in the majority of the
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applications. Straightforward terms for the function z(k, g, N, C, j(), PA ) based on a Monte Carlo study over numerous OPCs distributed uniformly among the five broadly generic classed and the detailed description can be found in [26]. The formula allows the designers to determine s, the size value of SS, by a simple and direct calculation. Once the value of s is obtained, the designers need just to solve the constrained WLS problems with continuous and discrete variables in (2a) – (2d) for each sample ad (i), i ¼ 1, . . . , s. The resulting k top samples ranked by the objective values of the constrained WLS problems with continuous and discrete variables in SS will be the candidates of GSN with probability PA ¼ 0.95. Since what we care is a good enough solution, the corresponding smallest number of alignments between SS and GSN, k, is set to l. Hence, the value of s, s ¼ Z(k, g, N, C, j(), PA ), can be calculated by the mathematic formula in [26]. Taking the reduced good enough subset GSrN instead of the good enough subset GSN [29], the parameters are set as follows: g ¼ 35, k ¼ 1, N ¼ 1000, PA ¼ 0:95, and C is chosen as the worst case that is type (v) OPC (lots of bad designs). In this paper, more conservative consideration for determining the SS size is taken. Thus, we measured three types of uniform noise distribution: U[20.5, 0.5], U[21.0, 1.0] and U[22.5, 2.5] for the noise characteristics j(). Therefore depending on the three types of noise and some parameters, three different sizes of SS, s ¼ 12, s ¼ 13 and s ¼ 28 were obtained by the mathematical formula. Notably, the values obtained s according to the above setup are conservative due to the selection of C and the three different types of uniform noise j().
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