14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 1, Page 1
POWER SYSTEM STABILIZER TUNING IN MULTIMACHINE POWER SYSTEM BASED ON A MINIMUM PHASE CONTROL LOOP METHOD AND GENETIC ALGORITHM Komsan Hongesombut,
Yasunori Mitani, and
Kiichiro Tsuji
Osaka University, Graduate School of Engineering, 2-1 Yamada-oka, Suita, Osaka 565-0871, JAPAN
[email protected] 1
Abstract – The incorporated use of an analytical method so called minimum phase control loop and intelligent method using a genetic algorithm (GA) has been proposed in this paper for an off-line power system stabilizers (PSS) tuning in a multimachine power system. First, the selecting PSS parameter problem is converted to an optimization problem. Then, the problem will be solved by a microgenetic algorithm (micro-GA) with the small population and reinitialization process combined with a hierarchical genetic algorithm (HGA). The concept of HGA is applied for automatically identifying the appropriate choice of PSS locations. A reasonable choice of initial solution is obtained by a minimum phase control loop method. These added features provide a considerably improvement of time efficiency and flexibility in PSS tuning of large power systems. The 16-generator and 68-bus power system has been used to validate the effectiveness of this tuning approach.
Keywords: power system stabilizer, minimum phase control loop, genetic algorithm, micro-genetic algorithm, hierarchical genetic algorithm, simultaneous tuning. 1. INTRODUCTION PPLYING power system stabilizers (PSSs) can extend power transfer stability limits by adding modulation signal through excitation control system. They have been used for several decades in utilities for the purpose of adding damping to electromechanical oscillations. Typically, PSSs act through the generator excitation system in such the way that electrical torque on the rotor is in phase, or nearly in phase, with speed variations. It will provide damping to the power system, thereby aid in the stability of the power system. This measure is the most cost-effective electromechanical damping control comparing to using other FACTS devices since the necessary part of power amplifier is embodied in the generator [1]. PSS tuning for stability system modes of oscillation has been the subject of much research since several years ago [1-4]. The basic tuning techniques such as phase compensation and root locus, have been successfully utilized with power system stabilizers application to the real power systems. However, these such methods are not easy to simultaneously tune PSSs in multimachine power systems due to the fact that PSSs may reduce the stability of some modes as well as increase the stability of the other modes. The results by the conventional techniques are thus a compromise.
A
Due to the fast development of intelligent techniques application to power systems during this decade, many researchers in the field of power systems have pay much more attention to applications of these such techniques to solve the problems in power systems. Genetic algorithm (GA) is one kind of those techniques in the field of artificial intelligent that its basic operation is conceptually simple. It has demonstrated its ability as a powerful optimization technique for solving many difficult problems. In this paper, the major part of the proposed PSS tuning method is based on GA’s. Similar works presented in [3,4] showed great advantages of using GA’s to simultaneously tune of PSS in multimachine power systems. However, the main disadvantages of those works are the computational time spent by GA’s which is still not satisfied and the PSS locations which must be chosen deterministically before starting the tuning procedure. Using participation factor to find the possible PSS locations has been extensively used, however, this method may not guarantee the effectiveness of damping performance. To overcome the above problems, the authors propose the incorporated use of an analytical method so called minimum phase control loop and GA which is a micro-genetic algorithm (micro-GA) combined with a hierarchical genetic algorithm (HGA). To start moving the eigenvalues lied on the s-plane from unstable to stable region with less computational efforts, a reasonable choice of solutions obtained by a minimum phase control loop method is inserted into the initial population of GA. Our experiences have shown that starting the search procedure from this point may reduce the overall searching time significantly. This agrees with [4] that better results will be obtained if certain initial population of GA corresponds to stable solutions. In [4], initial random generation proceeds until the population satisfy the stability criterion. Initialization by using this method was time consuming process for more complex problems. In this work, HGA is used for solving a problem of finding PSS locations [6]. Applying a micro-GA with an initialized solution by a minimum phase control loop method can improve the time efficiency significantly. The proposed method is applied to a 16generator and 68-bus power system [5]. The results demonstrate that PSSs of the study multimachine power system can be tuned to provide satisfactory damping performance over a set of predefined contingencies.
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 1, Page 2
2. BASIC IDEA OF MINIMUM PHASE CONTROL LOOP METHOD The idea presented in [1] is used due to its simplicity in concept and in computing. The criterion is based on the fact that power systems associated with generators and excitation systems exhibit the phase lag to the system. To compensate this phase lag, the lead networks will be used to provide phase lead large enough to overcome the phase lag by the system resulting to a minimum phase control loop. 2.1 Single Machine Connected to Infinite Bus Case K1 D
∆Te2 ∆Tm + -
PSS(s)
1
∆Te1
Ms
∆ω
ωb s
∆δ
K4
K5
GEP(s)
(3) select the PSS gain by using root locus method. The rules in [1] may be used for selecting the suitable gain to provide optimal PSS gain while implementing PSS in the filed. 2.2 Multimachine Power System Case The same idea can be applied for tuning PSS in multimachine power systems. Since until recently, the analytical calculation of simultaneous tuning PSS has been unknown yet, PSS tuning of remaining generators is performed one by one in a similar manner as for a single generator. Tuning PSS one by one is not entirely unrealistic since in the real power systems, putting in a whole set of PSS will only occur in small power systems. PSS is designed to damp only some specified frequencies. However, the optimal performance of PSS in damping out all oscillation modes in a global sense is not guaranteed. The method to deal with this problem and to be applied for simultaneous tuning will be explained later. 29
∆V + s + ∆Vref
61
28
G8
In summary, the tuning procedure is carried out following these steps: (1) identify how much phase lag due to GEP(S) is produced. Typically, a small-signal power system model is represented by the state-space equations. First the modified state-space matrix is formed by eliminating the angle and speed term of A, B, C, D matrix from the original state-space matrix based on the assumption that angle and speed of generator are held constant. GEP(S) is the transfer function derived from this state-space matrix where the reference voltage of AVR is the input and the electrical torque is the output. (2) adjust the time constants of PSS to match the ideal phase lead. The ideal phase lead is the negative value of the phase lag by GEP(S). If the compensation could be archived perfectly, the electrical torque would contain a component proportional to speed at all matching frequencies.
19
57 G5
17 14
15
G3
13
12 4
3
11 5
2
G2
6 7
8 1
30
9 32
47 G11
48
10 55
54
25 53
G1
G7
G4
20
16
27
23
56
18
36
34
G12
45 G10
41
44 39
35 50 52
31
G14
65 G13
62 40
37 64
33
63
66
(1)
21 24
60
Figure 1: Simplified model of single machine to infinite bus
∆Te1 = PSS ( s )GEP( s ) ∆ω
G6
G9
26
Considering the simplified model of a single machine connected to infinite bus system in Figure1, here, the plant transfer function denoted by GEP(s) represents the transfer function of the generator, excitation and power system. To provide the highest damping, a component of electrical torque signal should be in phase with speed change signal. The transfer function of damping component Te1 to generator speed change ∆ω around the PSS(s) and GEP(s)’s path shown in Figure 1 is given by
59
58
22
38
49
46
42
51
67
68 G16
G15
Figure 2: 16-generator and 68-bus test system
Figure 2 shows a model of multimachine power system consisting of 16 generators and 68 buses used in [5]. For this model, the tuning procedure for multimachine power system case is carried out following these steps: (1) the PSS placement is calculated based on the participation factor. The generators having the highest speed participation factor Gp will participate to that oscillation modes. Table 1 shows the result of eigenvalue calculation; there are 15 modes that are unstable modes or have damping ratios less than 5%. (2) one mode is pick up and the transfer function from the state-space model is formed by eliminating the angle and speed term of the generator that is being designed, then the ideal phase lead will be obtained. (3) adjust the time constants of PSS by trial and error method to match this ideal phase lead.
14th PSCC, Sevilla, 24-28 June 2002
Mode 1 2 3 4 5 6 7 8
eigenvalue -0.064±2.756i -0.032±3.590i -0.003±4.401i -0.131±5.170i 0.408±7.672i 0.240±7.722i 0.647±7.790i -0.157±8.357i
Gp 13 16 13 15 12 6 9 5
Mode 9 10 11 12 13 14 15
Session 14, Paper 1, Page 3
eigenvalue 0.290±8.462i 0.477±8.615i 0.167±8.690i -0.116±10.095i 0.092±10.188i -0.383±10.207i 0.519±12.543i
Gp 9 3 10 4 4 8 11
Table 1: Under damped and unstable modes (no PSSs) 15 14 13 12
mode number
11 10 9 8 7 6 5 4 3 2 1 -0.2
0
0.2 0.4 0.6 0.8 real part of normalized speed participation factor
1
1.2
Figure 3: Speed participation factor for generator 6
(4) choose PSS gain by using root locus method. (5) insert the PSS that was designed into the power system model and repeat step 2 until all PSSs are completely designed to damp out all 15 modes. The PSS tuned by using this method is only a compromise. Tuning one PSS at the generator having maximum participation factor in the electromechanical mode will interact to another modes. This can be shown as an example in Figure 3. If a PSS is installed at generator 6, the damping of mode 6 can be increased while the negative sign at mode 9 means it will be destabilized. The problem becomes more complex in large power system. By using this tuning method the interaction among stabilizers is not taken into consideration resulting in degradation of the overall damping performance. 3. GENETIC ALGORITHMS A genetic algorithm, in comparison with other optimization techniques, is desirable due to several reasons. Importantly, first all sort of criteria may be used for maximization or minimization problem without the problem that the fitness measure is discontinuous or non-smooth function. This is a great deal of freedom of users rather than having to interpret those goals into some complicated mathematical formulas. Secondly, in solving the problem that analytical solution has been unknown yet, GA is computationally simple and easy to implement. The result will be better within the time limit since GA always produce high quality solutions. However, the important drawbacks of GA are that there is the possibility to converge to a suboptimal solution and it is time consuming process.
In this paper, we propose a combined method of two genetic algorithm concepts; a hierarchical genetic algorithm (HGA) and a micro genetic algorithm (micro-GA). 3.1 Hierarchical Genetic Algorithm (HGA) The operation cycle of the HGA is similar to the operation cycle in the basic GA. The major difference between them is its hierarchical structure. Each hierarchical chromosome consists of a multilevel of genes, as demonstrated in Figure 4 showing the HGA chromosome representation with one-level control genes and parametric genes. In this configuration, the control genes are analogous to the PSS locations. The control gene signified as “0” in the corresponding site, is not being activated meaning that the PSS at the corresponding location will not be installed into the power system during the simulation. Parametric genes are analogous to the PSS parameters to be optimized. Using the concept of HGA, locations and PSS parameters can be simultaneously tuned. Figure 4 illustrates the interfacing of HGA chromosome and simulation package in calculating the fitness value. In many cases, other contingencies may be added arbitrarily by the users depending on the critical events in power system operations. 3.2 Micro Genetic Algorithm (micro-GA) Basically, for the basic GA, if the population size is too large, the GA tends to take longer time to converge upon a solution. Conversely, if the population size is too small, the GA is in danger to converge upon a suboptimal solution. The reason why the basic GA cannot apply the small population size is that there is not enough diversity in the population to allow the GA to escape from local optima. If the population size problem can be removed, the GA will improve speed significantly. The time problem becomes more evident when apply GA in large power systems. This becomes the motivation of using micro-GA in this work. A micro-GA is a genetic algorithm with a very small population size and a reinitialization process. The idea was suggested by some theoretical results obtained by [7]. Figure 5 is used for describing the micro-GA concept. The generations have been divided into multiple loops depending on the numbers of reinitialization process. For each loop in a micro-GA, the population size is very small (usually not more than 10). The average and maximum fitness are gradually improved for each micro-GA loop until the average and maximum fitness are different 5%. If there is no reinitialization process, the GA may trap in local minima. Only the mutation operation , which has a very small probability, can help in escaping the local minima with longer time. In a microGA, reinitialization process is used as a source of diversity where the diversity increases, the possibility in escaping the local minima increases. The aim of a microGA is to find the optimum as quickly as possible without improving any average performance.
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 1, Page 4
Hierarchical chromosome structure
G1
0
G2
1
G16
PSS 1 0 ... 1
PSS 1
...
PSS 16
PSS parameters (parametric genes)
PSS 2
...
location (control genes)
PSS 2
...
1
PSS16
loads
power system networks
contingency 1 contingency 2
...
power system simulation package
(equation 3)
damping ratios & fitness calculation
...
1
(Table 2) Fitness value
contingency n
Figure 4: Hierarchical chromosome structure and interfacing to power system simulation package in calculating fitness value
Restart
Database
Maximum
Average
Population Pool
reinitialization micro-GA loop
Method 2 Method n
Initial Population Selection
reinitialization micro-GA loop1
External Memory
...
Fitness
Random Population
Method 1
Save to database if program stops
Crossover
micro-GA loop2
generations Mutation
Figure 5: Micro-GA concept Elitism
Figure 6 shows the algorithm flowchart of the microGA used in this study. The population pool works as the source of diversity of the approach and the external memory is used to keep the competitive solutions from the database. The database is a set of PSS parameters obtained by several methods including the data selecting from user’s experiences. In this study, two of them were obtained from a minimum phase control loop method and another from the previous result of the micro-GA, which are always kept in “method 1” and the last field of the database respectively. Population pool is derived from two portions; from random population and external memory where the percentage of each can be regulated by the user. The initial population of the micro-GA at the beginning of each its cycles is taken from both portions, at a certain probability as to allow a greater diversity. During each cycle, the micro-GA undergoes conventional genetic operators; tournament selection, uniform crossover, uniform mutation and elitism. Until the nominal convergence is detected which is done by checking the percentage difference of the average and maximum fitness value whether it is 5% or not. The dominated string (best string) from current population will be chosen and be copied into external memory such that the best string so far from the previous micro-GA loop will always appear in the next micro-GA loop. The implementation of a micro-GA in this work used real encoding chromosome, a population size 5, maximum generation 120, a uniform crossover rate of 1 and a uniform mutation rate of zero. The approach also
New Population N
Nominal Convergence Y Update External Memory
Reinitialization process N
Termination Criteria Y Stop
Figure 6: Algorithm flowchart of a micro-GA
adopted an elitist strategy that copied the best string found in the current generation to the next generation. Selection was performed by using the tournament selection with tournament size of 2. 4. PROBLEM FORMULATION The input to the stabilizer used in this paper is generator shaft speed. It consists of a two-stage lead lag compensation with time constants T1i - T4i , and a gain Ki. We set the wash out time constant Twi with large enough value so that it can be considered as a constant (in this study Twi = 10s). Equation (2) shows the transfer function of each PSS. sTwi 1 + sT1i 1 + sT3i PSS ( s ) = K i (2) 1 + sT wi 1 + sT2i 1 + sT4i
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 1, Page 5
The problem of selecting the PSS parameters is converted to a simple optimization problem by a simple eigenvalue-based objective function shown in (3). The closed-loop eigenvalues are constrained to lie on the left hand side of s-plane (stable region) and to have global minimum damping ratio as maximum as possible. This guarantees the minimum damping ratio of all modes within the predefined contingencies analogous to the robustness of PSS controllers. min F = −(W stable + Wunstable )
− Re(λ i ) Re(λ i ) 2 + Im(λ i ) 2
if ∀(δ i ) j ≥ 0 Wstable
else
Wstable
if ∃(δ i ) j 〈0 else
= =
Wunstable = Wunstable =
(4)
210 min ( min δ i ) j 1≤ j ≤ m 1≤i ≤ n
0
4
Best Average 0
215 min ( min δ i ) j 1≤ j ≤ m 1≤i ≤ n
0
where i = 1, 2, 3, …, n (numbers of eigenvalues), λi is the ith closed-loop eigenvalue, j = 1, 2, 3, …, m (numbers of contingencies). 5. TEST SYSTEM The proposed tuning procedure was tested in a power system model shown in Figure 2. The detailed data of this model can be obtained from [5]. Table 2 shows 4 contingencies which are the critical events considered in the tuning procedure. Condition number 1 2 3 4
x 10
1
Configuration All lines in service Line 1-2 out of service Line 1-27 out of service Line 8-9 out of service
-1 fitness value
δi =
-2
-3
-4
-5
-6
0
20
40
60 generations
80
100
120
Figure 7: Optimization trace of a micro-GA 12 10%
δmin=-8.28%
5%
10
8 imaginary
subject to
(3)
shows that even if all the eigenvalues can be shift to the stable region, however, it does not guarantee the minimum damping ratio which at least 5% is required. When applying a minimum phase control loop method as the result shown in Figure 12, most of eigenvalues satisfy the minimum damping ratio 5%, however, one mode remains unstable and one mode is very close to damping ratio 5%. Since the transfer function zeros are close to unstable and low damped poles, a closed-loop pole may be attracted to the zero as the stabilizer gain is increased resulting to the limitation of improving overall damping that is not much improved. Therefore, additional PSSs placing in some generators may be need for moving away these zeros and making other PSSs more effective in damping oscillations. Figure 13 shows the result after installing additional PSSs at generator 1 and 2. The minimum damping ratio of this case is improved to 6.86%. The PSS setting of this case is used as in [5].
6
4
2
Table 2: Contingencies used in tuning procedure
-4
-3
-2 real
-1
0
1
Figure 8: Open-loop eigenvalues (no PSSs) δmin= 0.0392%
12
5%
10% 10
0.0392% 8 imaginary
6. TEST RESULT Figure 7 shows an objective function optimization trace of a micro-GA used in this paper. Figure 8 shows the open-loop eigenvalues of the test power system which corresponds to the calculated eigenvalues shown in Table 1. Figure 9 and 10 show the results of PSS tuning obtained by a basic GA and a micro-GA starting from random. We applied a basic GA with population size P = 30, maximum generation N = 20 and a microGA with population size P = 5, maximum generation N = 120.The simulation time for PN = 600 is about 1 hour. Obviously, a micro-GA can reach better result than a basic GA within the same fixed time. We applied a micro-GA for tuning 12 PSSs by fixing the locations based on the participation factor. Figure 11
0 -5
6
4
2
0 -5
-4
-3
-2 real
-1
0
1
Figure 9: Result by a basic GA (initialized by random)
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 1, Page 6
δmin=2.91%
12 10%
5%
δmin=12.53%
12
2.91% 10
8
8 imaginary
imaginary
12.53% 10
6
4
2
2
-4
-3
-2 real
-1
0
0 -5
1
Figure 10: Result by a micro-GA (initialized by random)
5%
6
4
0 -5
10%
-4
-3
-2 real
-1
0
1
Figure 14: With 15 PSSs (the proposed tuning approach) -3
4
x 10
δmin= 0.84%
12 10%
5%
fault on line 8-9 3
0.84%
generator speed variation (pu)
10
imaginary
8
6
4
2
0 -5
2
1
0
-1
-2
-4
-3
-2 real
-1
0
-3
1
Figure 11: Fixed locations of 12 PSSs (a micro-GA)
1
2
3
4 time (s)
5
6
7
8
Figure 15: Three-phase to ground fault on line 8-9 -3
δmin= -3.39%
12
0
4
x 10
fault on line 1-2 10%
5%
3 generator speed variation (pu)
10
imaginary
8
6
4
2
2
1
0
-1
-2 0 -5
-4
-3
-2 real
-1
0
1
-3
Figure 12: Fixed locations of 12 PSSs (a minimum phase control loop method)
0
1
2
3
4 time (s)
5
6
7
8
Figure 16: Three-phase to ground fault on line 1-2 -3
4
δmin=6.86%
12 10%
x 10
fault on line 1-27 3
6.86% 5%
generator speed variation (pu)
10
imaginary
8
6
4
2
1
0
-1
2
-2 0 -5
-4
-3
-2 real
-1
Figure 13: Fixed locations of 14 PSSs (a minimum phase control loop method)
0
1
-3
0
1
2
3
4 time (s)
5
6
7
Figure 17: Three-phase to ground fault on line 1-27
8
14th PSCC, Sevilla, 24-28 June 2002
However, comparing to the case in Figure 11 and Figure 12 which are for 12 PSSs, but different locations, the PSS locations in case of Figure 11 were at generator 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14 and in case of Figure 12 were at generator 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16. It should be noted that with the same numbers of PSSs, using variable locations of PSS based on the proposed HGA concept gave better result for the minimum damping performance. This obviously confirms that the interaction between stabilizers is existing. The participation factor may not confirm the correct numbers of PSSs. Tuning PSS by fixing the PSS locations inappropriately may result in poor overall damping performance. By using our proposed tuning approach, the interaction among stabilizers is taken into consideration resulting to the improvement of minimum damping requirement. We started our proposed method by the same set of PSS parameters in case of Figure 13. The population size P = 5, maximum generation N = 120 were used for a micro-GA. Figure 14 shows the final result which in this case, 15 PSSs is necessary and minimum damping ratio is 12.53%. The PSS parameters and locations obtained by the proposed method are given in the Appendix. We carried out the dynamic simulations as shown in Figure 15 to 17 for three-phase to ground faults on line 8-9, 1-2 and 1-27 for 6 cycles respectively to validate the effectiveness of the PSS parameters obtained by the proposed tuning approach. It is quite clear that an excellent improvement in the damping for each contingency has been achieved with one set of PSS parameters. 7. CONCLUSIONS This paper describes an off-line PSS tuning method by the incorporated use of an analytical method so called minimum phase control loop and GA. We propose using the concept of HGA for automatically identifying the PSS locations. Participation factor, which may lead to the incorrect numbers of PSSs when considering the overall damping performance, is not essential in this work. This is a benefit compared to other works that fix the PSS locations before the optimization. We have shown that our proposed tuning approach gives better result for the minimum damping requirement since the interaction among stabilizers is taken into consideration. We also propose the application of a micro-GA with the selected initial solution from the database which may be obtained from other calculation methods or user’s experiences. The minimum phase control loop method which is easy to implement was selected as a reasonable choice of initial solution. The results show that the combination of these features can speed up the GA calculation time significantly. An excellent improvement in the damping for every contingency has been achieved with one set of PSS parameters. Dynamic simulations were carried out using a 16-generator and 68-bus power system model to validate the effective of the proposed tuning approach.
Session 14, Paper 1, Page 7
REFERENCES [1] E. V. Larsen and D. A. Swann, “Applying Power System Stabilizers, Part I; General Concepts, Part II; Performance Objectives and Tuning Concepts, Part III; Practical Considerations”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100, 1981, pp.3017-3046. [2] C. Y. Chung, C. T. Tse, K. W. Wang, R. Niu, “PSS Design for Multi-Area System Based on Combined Sensitivity Concept”, Proc. of PowerCon 2000, Vol.3, Dec. 2000, pp. 1197-1202. [3] Y. L. Abdel-Magid, M. A. Abido, S. Al-Baiyat, and A. H. Mantawy, “Simultaneous Stabilization of Multimachine Power Systems Via Genetic Algorithms”, IEEE Trans. on Power Systems, Vol.14, No.4, Nov 1999, pp.1428-1439. [4] A. L. B. do Bomfim, G. N. Taranto and D. M. Falcão, “Simultaneous Tuning of Power System Damping Controllers Using Genetic Algorithms”, IEEE Trans. on Power Systems, Vol.15, No.1, Feb 2000, pp.163-169. [5] G. J. Rogers, “Power System Oscillations”, Kluwer Academic Publishers, Boston, 2000. [6] K. Hongesombut, Y. Mitani, and K. Tsuji, “An Automated Approach to Optimize Power System Damping Controllers Using Hierarchical Genetic Algorithms”, Proc. of Intelligent System Application to Power Systems, June 2001, pp.3-8. [7] D. E. Goldberg, “Sizing Populations for Serial and Parallel Genetic Algorithms. Proc. of the Third International Conference on Genetic Algorithms, 1989, pp. 70-79. APPENDIX A. PSS parameter setting obtained by the proposed tuning approach. PSS no.
K
T1
T2
T3
T4
1
10.5868
0.0731
0.0846
0.0797
0.0207
2
16.4888
0.0608
0.0095
0.0720
0.0108
3
11.2988
0.0426
0.0373
0.0974
0.0108
4
13.3828
0.1000
0.0421
0.0895
0.0261
6
17.3617
0.0917
0.0347
0.1000
0.0889
7
20.0000
0.0586
0.0146
0.0628
0.0114
8
8.0714
0.1000
0.0076
0.0612
0.0220
9
10.1642
0.0599
0.0499
0.0924
0.0187
10
17.6483
0.0894
0.0076
0.0526
0.0171
11
3.5956
0.0987
0.0386
0.0740
0.0173
12
15.3035
0.0237
0.0071
0.0890
0.0044
13
20.0000
0.0390
0.0654
0.0391
0.0197
14
19.3929
0.0598
0.0512
0.0930
0.0514
15
14.8343
0.0937
0.0397
0.0944
0.0405
16
20.0000
0.0703
0.0574
0.0759
0.0679
