Optimal reactive power dispatch by chaotic biogeography based optimization

TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 Optimal reactive power dispatch by chaotic biogeography based optimization . Truong Xuan Quy . Vo Ngoc Dieu Ho Chi Minh city University of Technology, VNU-HCM, Vietnam (Manuscript Received on July 15, 2015, Manuscript Revised August 30, 2015) ABSTRACT This paper proposes a chaotic transformers tap changer limits and biogeography based optimization (CBBO) for switchable capacitor bank limits. The BBO solving optimal reactive power dispatch has been enhanced its search ability by (ORPD) problem. Based on biogeography adding chaotic theory. Therefore, the based optimization (BBO) theory proposed proposed CBBO can obtain better solutiong by Dan Simon in 2008, a new artificial quality than BBO for optimization problems. intelligence with full models and equations The proposed method has been tested on the have been used to achieve the best solution IEEE-30 and IEEE-118 bus systems and the for objective function of ORPD such as total obtained results have been verified with other power loss, voltage deviation and voltage methods. The result comparison has stability index while satisying various indicated that the CBBO can be a promise constraints of power balance, voltage limits, method for dealing the ORPD problem Keywords: Optimal Reactive Power Dispatch, Biogeography Based Optimization, Chaos Theory, Power loss, Voltage Deviation, Voltage Stability Index 1. INTRODUCTION The main objective of optimal reactive power The problem control variables include the dispatch (ORPD) [1] in electrical power system is generator bus voltages, the transformer tap to minimize the objective function via the optimal settings, and the reactive power of shunt adjustment of the power system control variables, compensator, while the problem dependent while at the same time satisfying various equality variables include the load bus voltages, the and inequality constraints. Some objective generator reactive powers, and the power line functions in ORPD to evaluate the quality of flows. power system is real power loss, voltage deviation There are various techniques ranging were at load buses [2], voltage stability index [3]. The introduced to solve ORPD, from conventional equality constraints are the power flow balance methods to artificial intellgence based methods. equations, while the inequality constraints are the These conventional methods have been used for limits on the control variables and the operating approaching the ORPD is linear programming limits of the power system dependent variables. (LP) [4], mixed-integer programming (MIP) [5], Trang 55 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 interior point method (IPM) [6], dynamic Applied the above to the ORPD problem, x programming (DP) [7] and quadratic is the containing vector of the controlled programming (QP) [8]. The convetinal variables: the voltage and phase of load, reactive optimizations are easily to be carried out, the power of the generators and real power of slack results is acceptable but can be trapped in local bus. minima and this optimization can not act on the T x (P , ,...,  , V ,..., V , Q ,..., Q ) (2) discrete variables. Recently, meta-heuristic search G1 2 N L 1 LNL g 1 gng methods become more popular in doing with u is the containing vector of the controlling ORPD. Several methods, most of them are based variables: voltage of generators, tap-setting of on the biological model like evolutionary and transformers and the reactive power at behavior in species, were used such as compensator. evolutionary programming (EP) [9], genetic T (3) u ( V g1...VTTQQ gng , 1 ... NT , c 1 ... cNc ) algorithm (GA) [10], differential evolution (DE) The objective function is depended on the [11], ant colony optimization (ACO) [12] and target of optimization. Normally, there are three particle swarm optimization (PSO) [13]. These functions used: methods can improve the solutions for ORPD although it is more complex and slow in - The total active power loss in transmission: F PL  2 2 performance.  gk( V i V j  2 VV i j cos ij ) (4) k Nbr In this project, we discuss about an where gk is the conductance of branch k, evolutionary algorithm that was found in 2008 by Dan Simon [14], called Biogeography Based Vi is the voltage magnitude at bus i and ij is Optimization. It is based on the migration and the voltage angle different between bus i and j . mutation of species in natural and the status of - Voltage deviation at loaded buses for ecosystem in different time. By supplying the full voltage profile improvement: theory and model of CBBO, we proved the useful Nd VD  sp of this algorithm by testing on IEEE-30 bus VVi i (5) sp i1 system and IEEE-118 bus system. The results is where Vi is the standard value to evaluate the compared with the other paper to evaluate the deviation, normally set at 1 p.u. advantage or disadvantage of this method. - Voltage stability index for voltage stability 2. FORMULATION OF ORPD enhancement: The ORPD problem is built based on the F(,) x u Lmax max{Li }; i  1,..., N d (6) mathematics concepts Min F (x,u) where g(,) x u is the equality constraints, it follow the power conservation law: g(x,u)  0 P PP  (7) h(x,u)  0 G DL (1) This equation can be spread: where F(,) x u called the objective function P P  v v( g cos  B sin  )  0  gi di i jij ij ij ij whose output is the minimum value we want.  Q Q  v v( g sin  B cos  )  0 g(x , u ) is the equality constraints and h(x , u ) is  gi di i jij ij ij ij (8) the inequality constraints. Trang 56 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 where gii, Bii are the transfer conductance and 3. CHAOTIC BIOGEOGRAPHY BASED susceptance between bus i and bus j ; Pdi,Q di OPTIMIZATION are the real and reactive power outputs of 3.1 Migration [14] generating at bus i ; P ,Q are the real and gi gi Mathematical models of biogeography reactive power outputs of generating unit i . describe how species migrate from one island to This equality constraints is checked by another, how new species arise, and how species running Power Flow by Newton-Raphson method become extinct. Geographical areas that are well in Matlab. suited as residences for biological species are said h(,) x u is the inequality constraints to have a high habitat suitability index (HSI) [14]. represented as follows: The variables that characterize habitability are a) The power limitations: called suitability index variables (SIVs) [14]. min max SIVs can be considered the independent variables PPPgslack gslack  gslack  (9) of the habitat, and HSI can be considered the QQQmin  max  gi gi gi dependent variable. Habitats with a high HSI tend b) The voltage limitations: to have a large number of species, while those min max (10) with a low HSI have a small number of species. ViVV i  i Habitats with a high HSI have many species that c) Transformers tap-settings constraints: emigrate to nearby habitats, simply by virtue of min max (11) TTTi i  i the large number of species that they host. d) The compensator capacitor limitations: Habitats with a high HSI have a low species immigration rate because they are already nearly QQQmin  max (12) c c c saturated with species. Therefore, high HSI e) The power flow limitations: habitats are more static in their species max (13) SSi i distribution than low HSI habitats. By the same token, high HSI habitats have a high emigration where Si is the maximum power flow between bus i and bus j . rate; the large number of species on high HSI islands have many opportunities to emigrate to SSS max{| |,| |} (14) i ij ji neighboring habitats. To check the inequality constraint, we use the The parameters below is used for BBO Static Square method. The objective function F investigation: not only have the output value but also adding the 2 - Habitat suitability index (HSI): to evaluate penalty function k (f ( x )) with:  i the capability of the island for the creatures.  0 xminxi  x max  2 (15) - Suitability index variables (SIVs): the f() x  ()xi  xmax xi  xmax independent variables such as rainfall, ()x x 2 x  x  min i i min temperature, humidity So with the penalty function, the objective - Immigration rate:  function will be rewritten as: - Emigration rate:  NG NPQ Nl F F  k f( Q )  k f (V)  k f (S ) p gi  i  i (16) i1 i  1 i  1 Trang 57 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 With the linear curves, the value of ,  whether there are s species in habitat can be written as: E s s   s I (1  ) s n n (17) where: - E is the highest emigration rate - I is the highest immigration rate Figure 1. Linear curve of fitness – migration - n is the maximum species in habitat Considering the immigration curve. The We use the emigration and immigration rates maximum possible immigration rate to the habitat of each solution to probabilistically share is which occurs when there are zero species in the information between habitats. If a given solution habitat. As the number of species increases, the is selected to be modified, then we use its habitat becomes more crowded, fewer species are immigration rate  to probabilistically decide able to successfully survive immigration to the whether or not to modify each suitability index habitat, and the immigration rate decreases. The variable (SIV) in that solution. If a given SIV in a largest possible number of species that the habitat given solution Si is selected to be modified, then can support is at which point the immigration rate we use the emigration rates  of the other becomes zero. solutions to probabilistically decide which of the Now considering the emigration curve. If solutions should migrate a randomly selected SIV there are no species in the habitat then the to solution Si . emigration rate must be zero. As the number of species increases, the habitat becomes more The BBO migration strategy is similar to the crowded, more species are able to leave the global recombination approach of the breeder GA habitat to explore other possible residences, and and evolutionary strategies in which many parents the emigration rate increases. The maximum can contribute to a single off-spring, but it differs emigration rate is which occurs when the habitat in at least one important aspect. In evolutionary contains the largest number of species that it can strategies, global recombination is used to create support. new solutions, while BBO migration is used to change existing solutions. Global recombination The equilibrium number of species is, at in evolutionary strategy is a reproductive process, which point the immigration and emigration rates while migration in BBO is an adaptive process; it are equal. However, there may be occasional is used to modify existing islands. excursions from due to temporal effects. Positive excursions could be due to a sudden spurt of As with other population-based optimization immigration, or a sudden burst of speciation. algorithms, we typically incorporate some sort of Negative excursions from could be due to disease, elitism in order to retain the best solutions in the the introduction of an especially ravenous population. This prevents the best solutions from predator, or some other natural catastrophe. It can being corrupted by immigration. take a long time in nature for species counts to 3.2 Mutation [14] reach equilibrium after a major perturbation. Trang 58 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 Now, consider the probability Ps that the Step 7: Back to the step 3 for the next habitat contains exactly S species. By calculate iteration. the limit of the changing time of habitat, t  0 If the variables after step 5 and 6 is not , we have the probability equation: satisfied the constraints, we optimize them by set ()  PP   S  0  s s s s1 s  1 the threshold for the variables:  P  ()  PPP     1SS   1 (18) XX s s s s s1 s  1 s  1 s  1 max  XX max max   ()  PP   SS  s s s s1 s  1 max XX XX  min min (22) If a given solution S has a low probability P s 3.4 Chaos theory and application in BBO , then it is surprising that it exists as a solution. It algorithm is likely to mutate to some other solution. This can In BBO algorithm, we used the random value be implemented as a mutation rate m that is to define whether migration, mutation or not. It is inversely proportional to the solution probability: 1 P absolutely incidental process. Various researches m(s) m ()s (19) max before and my results have pointed that this Pmax where mmax is the user-defined parameters. process complied with Normal (Gaussian Distribution) [15]. The solutions complied with This mutation scheme tends to increase diversity this distribution have very high probability near among the population. Without this modification, average point, means that the solutions is the highly probable solutions will tend to be more concentrated at a specific value which is not the dominant in the population. This mutation minimum value. (see the Figure 2) approach makes low HSI solutions likely to mutate, which gives them a chance of improving. To demolish this disadvantage of BBO, chaos It also makes high HSI solutions likely to mutate, theory was used to supply the comparing value in which gives them a chance of improving even migration or mutation step. Chaos theory is used more than they already have. to research about systems that seem to be chaotic 3.3 Application BBO to ORPD problem but can be predicted. This is applied in dynamic systems that is sensitive with initial conditions Step 1: Set the initial value for the BBO and have unlimited dimensions. This is popular variables. The i-th species in BBO is a vector of applied in Soil Mechanics, Solar-system, Liquid controlling variables: convection, Geography and Economics. XVVQQTT [ ... , ... ,... ] (20) id G1 GNG C 1 CNC 1 NT A chaotic map in this paper is a reflect: The starting value of Xid is defined by: [0,1]→[0,1] by the recursive function: X Xm in  rand (X max  X min ) rand [0;1] (21) id id id id x n1  F x n with x n is the value of Step 2: Set the value of BBO algorithm. chaotic map at n-th iteration. The orbit of function Step 3: Run the Power-flow by Newton- can be easily predict by the characteristics of Raphson method and check the constraint of value and convergence. Each chaotic map has controlling variables. unique characteristics and with the different initial Step 4: Calculate the fitness value and values, we have the different displays of the graph compute ,  . of function. The chaotic maps used in this project are list below [16]. Step 5: Do the migration step  Chebyshev: Step 6: Do the mutation step Trang 59 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 2012 R2012b and the CPU: Intel core i5, 2.4 Ghz, xi1 cos(i . a cos( x ( i )) (23) 2.00 GB RAM.  Circle: 4.1 IEEE-30 bus System K xmod( x    sin(2 x ()),1) i (24) i1 i The IEEE-30 bus system is available in [17] 2 with the data in the two following tables.  Gauss: Table 2. The structure of the experimented IEEE-30 2  x() i bus system xi 1  e   (25)  Iterative: Branches Genera- Transfo- Capacitors Controlling tors rmers variables   x  41 6 4 9 19 i 1   (26) x() i   Table 3. Basic values in IEEE-30 bus test system  Logistic: P Q P Q  di  di  gi  gi MW  MVAr MW  MVAr xi1  xi(1x i )   4 (27)  Piecewise: 283.4 126.2 287.92 89.2  0.4  x i x i   0, 0.4  0.4  0.5  x i  x i  0.4,0.5  0.1 (28) xi 1   0.6  x i  x i   0.5, 0.6  0.1  1 x i  x i  0.6,1  0.4  Sine: xi1  asin( . x ( i )) (29)  Sinusoid: n x ax( i ) sin( . x ( i )) (30) i1 Figure 2 Values of Plossmin with multi running time in  Saw: random BBO  1 In this paper, the power flow solutions for the  .x i x i    (31) systems are obatined from Matpower toolbox xi1   1  x i 1/   x i  [18]. In test system, the generators are located at   buses 1, 2, 5, 8, 11, 13 and the available 4. RESULTS transformers are located on lines 6-9, 6-10, 4-12 We use the CBBO algorithm to apply in the and 27-28. The switchable capacitor banks will be IEEE-30 bus and IEEE-118 bus system to installed at buses 10, 12, 15, 17, 20, 21, 23, 24 and calculate and evaluate with the other recent 29 with the minimum and maximum values of 0 project. With a chaotic map, we run with the and 5 MVAr, respectively. The limits for controls initial value in {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, variables are given in [20], generation active 0.9}.The algorithm is simulated on MATLAB power in [21], and power flow transmisson lines Trang 60 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 in [22]. The number of population is set to 10, the Table 5. Result by CBBO methods for the IEEE-30 maximum iterations is 200 and the results were bus system with voltage deviation objective and got by 50 independent runs. The comparion comparison results were from [19]. P Voltage Voltage loss min Running Method Stability time (s) Deviation ()MW Index CBBO 0.19 6.07 0.15 18.06 PSO- 0.09 5.84 0.15 9.97 TVIW PSO- 0.12 5.38 0.15 9.88 TVAC HPSO- 0.11 5.73 0.15 9.59 TVAC PSO-CF 0.09 5.82 0.15 9.89 PGPSO 0.09 5.80 0.15 11.11 Table 6. Result by CBBO methods for the IEEE-30 Figure 3 Values of P with multi running time in lossmin bus system with voltage stability index objective and a random CBBO comparison Two following figures shows the results of 50 Voltage P loss min Voltage Running independent runs of “random” BBO and a random Method Stability Deviation time (s) CBBO in optimal total power loss, respectively, Index ()MW to clear the optimization of chaos theory to BBO. CBBO 0.13 5.28 1.32 15.47 Clearly, only 6% of solutions in BBO is far PSO- 0.12 4.91 1.94 13.42 away the average point but the CBBO have high TVIW probability (18%) of values near the minimum PSO- 0.12 4.86 1.91 13.39 TVAC value of computing. The minimum value in HPSO- 0.13 5.26 1.68 13.05 CBBO is better a lot than the BBO’s. TVAC PSO-CF 0.12 5.00 1.94 13.39 Table 4. Result by CBBO methods for the IEEE-30 PGPSO 0.12 4.81 2.04 14.57 bus system with power loss objective and comparison The results in CBBO is presented in below P Voltage loss min Voltage Running table with comparing results by three criteria: total Method Stability time (s) ()MW Deviation Index power loss, voltage deviation and voltage stability index, respectively. CBBO 4.94 0.31 0.14 30.37 PSO- 4.51 2.05 0.13 10.98 TVIW 4.2 IEEE-118 bus System PSO- 4.53 1.98 0.13 10.85 TVAC The IEEE-118 bus system is available in [17] with the data in the two following table HPSO- 4.53 1.93 0.13 10.38 TVAC PSO-CF 4.51 2.06 0.13 10.65 PGPSO 4.51 2.06 0.13 12.21 Trang 61 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 Table 7. The structure of the experimented IEEE-118 PSO- 0.39 179.80 0.07 78.70 TVAC bus system HPSO- 0.21 146.81 0.07 74.90 Branches Genera- Transfo- Capacitors Controlling TVAC tors rmers variables PSO-CF 0.18 164.97 0.07 78.13 186 54 9 14 77 Table 11. Result by CBBO methods for the IEEE-118 bus system with voltage stability index objective and Table 8. Basic values in IEEE-118 bus test system comparison Pdi Qdi Pgi Qgi Voltage P     loss min Voltage Running Method Stability time (s) MW  MVAr MW  MVAr Index ()MW Deviation 4242 1438 4357.28 650.7 CBBO 0.07 125.71 1.06 146.57 PSO- 0.06 183.87 1.38 119.66 Table 9. Result by CBBO methods for the IEEE-118 TVIW bus system with power loss objective and comparison PSO- 0.06 184.56 1.21 119.22 P loss min Voltage Voltage Runnin TVAC Metho Deviatio Stabilit g time d ()MW HPSO- 0.06 155.39 1.34 1119.16 n y Index (s) TVAC CBBO 113.93 0.53 0.07 143.45 PSO- 0.06 203.72 1.54 119.86 CF PSO- 116.65 2.07 0.06 91.72 TVIW The results in CBBO is presented in below PSO- 124.33 1.43 0.07 85.32 table with comparing results by three criteria: total TVAC power loss, voltage deviation and voltage stability HPSO- 116.20 1.86 0.07 85.25 TVAC index, respectively. PSO- 115.65 2.13 0.06 91.86 5. CONCLUSIONS CF In this paper, a new artificial intelligence The limits of variables is similar with IV.a. based method BBO has been presented with full The limits for controls variables are given in [20], overview and results. With the optimization by generation active power in [21], and power flow chaos theory, CBBO have high probability for transmisson lines in [22]. The number of searching and approach the minimum value of population is set to 30, the maximum iterations is objective function of the ORD probem better than 200 and the results were got by 50 independent BBO algorithm. For the result comparison, the runs. The comparion results were from [19]. method is shown more useful with the large Table 10. Result by CBBO methods for the IEEE-118 searching space with more variables althoungh the bus system with voltage deviation objective and CBBO is not effective in searching voltage comparison deviation and voltage stability index value. By testing on the IEEE-30 bus and IEEE-118 bus P Voltage Voltage loss min Running Method Stability systems, the proposed method has shown that it is Deviation ()MW time (s) Index more effective for large scale systems. Therefore, CBBO 0.48 130.02 0.07 74.22 the proposed CBBO is very favoable for solving PSO-TVIW 0.19 176.46 0.07 78.49 the large-scale ORPD problem. Trang 62 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 Điều độ tối ưu công suất kháng sử dụng phương pháp tối ưu hóa dựa trên địa sinh học và lý thuyết hỗn loạn . Trương Xuân Quý . Võ Ngọc Điều Trường Đại học Bách Khoa – ĐHQG-HCM, Việt Nam TÓM TẮT Bài báo đề xuất phương pháp tối ưu hóa và giới hạn công suất các tụ bù ngang. dựa trên địa sinh học hỗn loạn (CBBO) để Phương pháp BBO được tăng cường khả giải bài toán điều độ tối ưu công suất kháng năng tìm kiếm bằng cách thêm lý thuyết hỗn (ORPD). Trên cơ sở lý thuyết tối ưu dựa trên độn. Vì vậy, phương pháp CBBO có thể đạt địa sinh học (BBO) do Dan Simon đề xuất được chất lượng lời giải tốt hơn phương pháp năm 2008, một phương pháp thông minh BBO cho các bài toán tối ưu. Phương pháp nhân tạo mới với đầy đủ mô hình và các đề xuất CBBO được áp dụng tính toán cho phương trình được áp dụng để đạt được lời các hệ thống chuẩn IEEE 30 nút và IEEE 118 giải tốt nhất cho hàm mục của bài toán ORPD nút và kết quả đạt được đã được chứng với như tổng tổn thất công suất, độ lệch điện áp các phương pháp khác. Từ kết quả so sánh và chỉ số ổn định điện áp thỏa mãn các ràng cho thấy rằng CBBO là một phương pháp đầy buộc khác nhau cân bằng công suất, giới hạn hứa hẹn để giải bài toán ORDP. điện áp, giới hạn các bộ đổi nấc máy biến áp, Từ khóa: Điều độ tối ưu công suất kháng, Tồi ưu hóa dựa trên địa sinh học, Lý thuyết hỗn loạn, Tổn thất công suất, Độ lẹch điện áp, Chỉ số ổn định điện áp. REFERENCES [1]. J. Nanda, L. Hari, and M. L. Kothari, Emerding Trends in Electrical and Computer Challeging algorithm for optimal reactive Technology (ICETECT), pp. 180-185, power dispatch through classical co- (2011). ordination equations, IEE Proceedings – C, [4]. D. S. Kirchen, and H. 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