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Optimal design of a furnace transformer by intelligent evolutionary methods
Electrical Power and Energy Systems 43 (2012) 1056–1062
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Optimal design of a furnace transformer by intelligent evolutionary methods K.S. Rama Rao ⇑, Mohd Noh Karsiti Department of Electrical and Electronic Engineering, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Malaysia
a r t i c l e
i n f o
Article history: Received 25 September 2010 Received in revised form 27 May 2012 Accepted 2 June 2012 Available online 15 July 2012 Keywords: Furnace power transformers Genetic Algorithm Nonlinear programming Optimal design Particle Swarm Optimization Scatter Search
a b s t r a c t This paper presents three intelligent evolutionary optimization techniques to investigate the optimal design parameters of a 3-phase furnace transformer. The transformer rating is derived from the operating conditions of a medium size direct arc furnace. Scatter Search (SS), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) techniques are employed on the developed nonlinear mathematical model of the transformer for constrained optimization minimizing the cost. The design and analysis programs of the furnace transformer are developed using codes written in C++/C language. The optimal design data results validated by an example show the efficacy of the three intelligent techniques. Among the three methods, the optimal results obtained by GA and PSO techniques show the potential for implementing as efficient search techniques for design optimization of furnace transformers. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An electric arc furnace used in steel industry essentially needs large amounts of power at a low voltage level for production of steel. In a direct-arc furnace the heat is generated by a powerful electric arc struck between carbon or graphite electrodes and the metal charge. Important characteristic features of a furnace transformer connected between the power system and the furnace load are low secondary voltages and high secondary currents. Accordingly, a transformer for an arc furnace is to be designed to supply a very large current by the secondary winding at a very low voltage dictated by the arc resistance. Besides the arc voltage must be large enough to supply the desired power and permit operation of the furnace at higher than normal current with maximum voltage [1]. The regulation of a furnace voltage over a wide range is usually achieved by tap settings of the transformer extending from maximum secondary voltage down to 50–25%. Furnace arc power and arc resistance, and secondary circuit impedance of the power system are major factors for fixing the rating of a furnace transformer. The secondary circuit consists of secondary bus, flexible cables, copper bus tubes and the electrodes. For a specified effective arc voltage and current, a wide range of circuit impedances can be implemented that result in the same average arc resistance [2]. This paper aims to focus on optimal design of a furnace transformer based on manufacturer’s reports and publications [3–8] which provided possible types of construction, electrical characteristics ⇑ Corresponding author. E-mail addresses: [email protected] (K.S. Rama Rao), nohka@petronas. com.my (M.N. Karsiti). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.06.019
and winding connections of arc furnace transformers. The design procedure of a transformer employed for furnace loads is different from a conventional power transformer. The traditional optimization techniques such as the sequential unconstrained minimization technique by Powel-Botm [9] together with Zangwill’s exterior penalty function method [10] has been applied for conventional and special purpose transformers [11]. In the recent years, a number of researchers paid attention on evolutionary procedures which are conceived to be independent of a memory based optimization method. An improved GA for single and double objective optimum design problems of power transformers is proposed [12] as a search method. The GA based parameter estimation method is compared with conventional methods [13] for the equivalent circuit parameter estimation of a three-winding transformer. A heuristic method for transformer design minimizing the cost is proposed in [14] based on the given specifications and available materials. An integrated three dimensional finite element method for the analysis and design of power transformers is developed in [15]. A survey on transformer design optimization methods as published in [16] reveals the continued interest in applications of advanced techniques. Bacterial foraging technique is applied [17] for design optimization of a conventional transformer and the results are compared with those obtained by SA and PSO. A comparative study on design optimization of high frequency power transformer by using geometric programming and by applying GA and SA is reported in [18]. However, very few publications are reported on design optimization of special purpose transformers such as furnace transformers [19]. In this paper three evolutionary methods SS, PSO and GA are selected for cost optimization of a furnace transformer, and the
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Nomenclature a, b Ai Bc, By ci, cc c1, c2 D12 f FF () Gc, Gi Gl, Gy I k1, k2
Kz Mt nA, nR Pc, Pi R RA RFT RS XFT XS Vi, Vc Xsc
dimensions of strap conductor, mm net core cross section (m2) flux density in the core and yoke (T) cost of stampings and copper (units) loss capitalization factors (units) mean diameter between the windings (m) frequency (Hz) polynomial to represent core loss (W/kg) weight of the copper and stampings (kg) weight of the limbs and yoke (kg) current per phase (A) constants
design parameters are investigated. To provide an overview of proposed optimization methods this paper is organized as follows: Section 2 describes the development of the furnace transformer rating and specifications for a case study. The problem statement using objective and constraint functions is presented in Section 3. A brief description of the evolutionary optimization procedures in general and the three proposed methods for optimal design are presented in Section 4. Then the design analysis programs for simulation are described in Section 5. Section 6 is used to analyze the results of optimization and the resulting effect of variation of control parameters of GA, SS and PSO algorithms on the objective function. Section 7 concludes the paper validating the simulation results and optimal design parameters. 2. Rating of the furnace transformer Fig. 1 shows the power system, transformer and furnace load. The transformer rating is derived from the operating conditions of the arc furnace such as arc resistance and arc current. In this paper, a 3-phase star-delta connected, core-type transformer is considered to supply a medium size direct-arc furnace. As shown in Fig. 2, the average arc resistance varies nonlinearly as the arc current increases when the transformer is operating at the highest tap [3]. Assuming a typical arc current, Ic of 60 kA and a per phase secondary circuit impedance, Zs of (0.5 + j 2.7) mX, the transformer rating is derived as follows [11]:
distance between the limb centers (m) perimeter of the tank (m) number of axial and radial straps full-load copper and iron losses (kW) resistance per phase (X) arc resistance (X) resistance of furnace transformer (X) resistance of secondary circuit (X) reactance of furnace transformer (X) reactance of secondary circuit (X) volume of stampings and copper (m3) % short-circuit reactance
Average arc resistance per phase, RA = 4.8 mX. The total circuit impedance, Zc = RA + Zs = 5.3 + j 2.7 mX. p Circuit voltage on load, Vc = 3IcZc = 620 V. p Transformer rating, S = 3IcVc = 64,500 kV A = 64.5 MV A. Normally to manage very high secondary currents over a wide secondary voltage range, an on-load tap changer, connected directly on the HV winding or placed in an intermediate circuit of a two-core design (booster regulation) within the transformer tank [8] is provided. As furnace transformers are generally operated at a medium voltage level, a primary voltage of 33 kV, and a secondary voltage range of 384–712 V, with nine tap positions on HV winding are assumed. With this arrangement, it is possible to feed the stipulated power into the furnace at the highest tap and maintain a stable arc for holding the melt on the lowest tap [3]. Table 1 presents the derived rating and other specifications of the transformer which are used to investigate the optimal design parameters. 3. Objective and constraint functions In this paper, a nonlinear mathematical model defined in terms of a set of design variables with cost as objective function and important constraint functions based on furnace transformer configuration, is developed. The design variables and constraints are
Table 1 Specifications of transformer. Infinite bus
Furnace
Transformer
Secondary circuit
X FT
RFT
Arc resistance
XS
RS
64.5 MV A, 33 kV/384 – 712 V, 50 Hz, star/delta 3-phase core type with forced oil cooling
RA
Fig. 1. Furnace transformer and the arc furnace.
Table 2 Design variables and constraints. Limits
Arc resistance, mohms
Arc resistance 20 15 10 5 0 20
30
40
50
60
70
80
Arc current, kA Fig. 2. Arc resistance for highest voltage tap.
90
Design variables x1 Flux density in the core (T) x2 Current density in HV winding (A/mm2) x3 Current density in LV winding (A/mm2) x4 height of the windings (m) x5 Voltage per turn (V) x6 Distance between core centers (m)
1.55–1.7 3.00–3.65 2.50–3.25 2.75–3.5 100–120 0.80–1.20
Constraints 1. Temperature rise of windings (°C) 2. Temperature rise of oil (°C) 3. % No-load current (I0) 4. % Short circuit impedance, Zs) 5. % Efficiency, g 6. Clearance between different phase windings (m)
<58 <50 <2.5 <12 >99 <0.1
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selected as in Table 2, considering their effect on core geometry, losses in core and windings, short-circuit reactance, temperature rise, weight, volume and finally on the cost of the transformer. The main design analysis program together with function programs coded in C language is simulated with each of the optimization techniques. The optimal design parameters are derived minimizing the cost as the objective function subjected to the constraints based on geometry and performance characteristics of the transformer. The cost objective function is the sum of initial cost and the capitalized cost of transformer losses. The initial cost is the sum of active materials cost of stampings and copper windings which are assumed to be in the ratio of 1:2 units. The capitalized cost of losses are based on (i) maximum demand charge as an additional demand caused by losses and (ii) cost of electrical energy wasted as transformer losses [11]. Thus the cost objective function is expressed as in the following equation:
F c ðXÞ ¼ ci Gi þ cc Gc þ c1 Pc þ c2 Pi
ð1Þ
where X = (x1, x2, . . . , x6), vector of design variables. The initial cost and capitalized cost of losses are expressed as functions of selected design variables x1 to x6 from Table 2. Similarly the constraints as listed in Table 2 are expressed in terms of the design variables [11]. In general any nonlinear optimization problem is stated as: Find X, such that the nonlinear objective function F(X) is a minimum, subject to the nonlinear constraints, gj(X){6=}0, j = 1, 2, . . . , m with X P 0 being a non-negative solution. Using exterior penalty function method [10] an augmented objective function, P is formulated as in the following equation:
PðX; rÞ ¼ FðXÞ þ r
m X ½g j ðXÞ2 ; r P 0
ð2Þ
j¼1
where gj(X) is defined as max [gj(X, 0)] and r is the penalty factor, which imposes penalty on the deviations of constraints. 4. Evolutionary optimization procedures Optimal design problems involve a search for the global minimum of the fitness function or objective function. To achieve global optimization, there are a variety of approaches. One way is to run any of the traditional nonlinear optimization methods from numerous random starting points such as in Powell’s method [9]. But this practice is inefficient as the traditional methods require large computational time to arrive at a global optimum value. A number of evolutionary computation methods are applied recently to engineering combinatorial optimization problems. Intelligent methods such as GA, SS and PSO techniques are a class of evolutionary methods based on random generation of initial population and natural behavior of artificial life [20,21]. Each of these methods computes a fitness value which is directly related on the distance to the optimum value. The reproduction of the population is based on the fitness value. To demonstrate the effectiveness of the proposed optimization techniques, the performance of the objective function with the variation of control parameters of the three algorithms is tested. All the three evolutionary methods proposed in this paper for design optimization of furnace transformer offer a convenient way of handling constraints and single or multi objective functions. A brief description and algorithms of the three evolutionary procedures are reported here. 4.1. Scatter Search technique Scatter Search (SS) [22,23] is an evolutionary population-based stochastic optimization technique that has been formulated combining decision rules and problem constraints. It uses small
population called a reference set, and is found to be successful for combinatorial optimization problems. SS selects two or more elements of the reference set in a systematic way with the purpose of creating new solutions using a set of control parameters and meets the desired characteristics of minimizing nonlinear optimization problems. The fundamental procedure combines the following steps: i. A Diversification Generation Method to generate a set of diverse trail solutions from an arbitrary initial solution. ii. An Improvement Method to transform the diverse solutions into one or more improved solutions. iii. A Reference Set Update Method to develop and maintain a representative set of good solutions. Here two or more elements are selected in a systematic way and a new solution is created strategically. Typically a reference set has 20 solutions or less. This method updates the reference set following intensification and diversification strategies. iv. A Subset Generation Method to operate on the reference set, selecting several subsets of solutions, for creating combined solutions. v. A Solution Combination Method to combine the solutions in each subset taking into account their good features. The usual notation of the control parameters used in SS is as follows: i. Psize = the size of the set of diverse solutions generated by the Diversification Generation Method. ii. b = the size of the reference set. iii. b1 = the size of the high quality subset. iv. b2 = the size of the diverse subset. v. MaxIter = maximum number of iterations. 4.2. Particle Swarm Optimization technique PSO is also a population based optimization technique based on the behavior of bird flocking and fish schooling. Besides PSO is one of the swarm intelligent techniques and is not affected by the size and nonlinearity of the problem [24,25]. The population of agents or particles tries to simulate its social behavior in the problem space and arrives at an optimal value. In this algorithm, the particles, the candidate solutions, use the information related to the most successful particle in the population in order to improve their position. The particles or agents in the population swarm through the design space with specified velocities. These agents try to simulate their social behavior in the problem space and arrive at an optimum value of fitness function. Each agent is represented by its current position and corresponding velocity. During simulation, the modified position of each agent is a random combination of its previous velocity and current position. In every step each agent’s position is updated by the best fitness value called as pbest. Also each agent knows the best value in the group called as gbest among the pbests. Over a number of iterations the agents searching points gradually reach the global optimal point using pbest and gbest. The basic steps of PSO are described as follows: i. Set the initial parameters of the PSO. ii. Create a population of particles. Initialize the particles with random vector of variables for position, X and velocity, V. iii. Evaluate the fitness function, F for all particles. iv. Compare the fitness value at the current position to the best fitness value at any time. Select the individual best particle, pbest. v. Identify the best particle in the group, gbest and its location corresponding to the minimum fitness value.
K.S. Rama Rao, M.N. Karsiti / Electrical Power and Energy Systems 43 (2012) 1056–1062
vi. Set gbest = pbest. vii. Update the new particle velocity and position as in (a) and (b).
V kþ1 ¼ W V ki þ C 1 rand1 ðpbest i X ki Þ þ C 2 rand2 ðgbesti X ki Þ i ðaÞ X kþ1 i
¼
X ki
þ
V kþ1 i
ðbÞ
where V ki is the velocity (current velocity) of particle i at kth iteration; V kþ1 is the velocity (modified velocity) of particle i at (k + 1)th i iteration; W is the inertia weight factor; rand1 and rand2 are the rank dom numbers between 0 and 1; pbesti is the best value found by k particle i until iteration k; gbesti is the best particle found in the group until iteration k; X ki is the current position (current searching point) of the particle i at kth iteration; X kþ1 is the current position i (modified searching point) of the particle i at (k + 1)th iteration. viii. If stopping criterion is satisfied then go to next step, else go to step ii. (The stopping criterion is good fitness value or a specified maximum number of iterations or no further improvement in fitness value). ix. Print the optimal parameters of X and the minimum fitness value. 4.3. Genetic Algorithm technique GA is a stochastic search technique derived from the principles of natural evolution and is based on genetic parameters such as inheritance, mutation, crossover and selection. Besides, GA rapidly locates the global minimum in difficult search spaces. It is a robust method and far less sensitive to parameter values [26,27]. In contrast to more traditional numerical techniques, the parallel nature of the stochastic search done by GA often makes it very effective in locating the global optimum. GA is less susceptible to getting stuck at a local minimum than gradient search methods. Also, GA is much less sensitive to initial conditions and is widely used in various optimization problems [21]. It starts with a set of random solutions called population. In GA approach each design variable is represented as a binary encoded string (chromosome) of fixed length. The chromosomes are evaluated by using a fitness function. GA provides solutions by generating a set of chromosomes referred to as a generation. A new generation is created from a pair of chromosomes (parents) by reproduction operators, mutation and crossover. If the search is to continue for an optimal solution, the GA creates a new generation from the old one until a decision is made on the convergence. In this paper, one type of selection strategy, the tournament selection, is applied to select the fittest individuals from the population, and the better one is duplicated in the next generation. The process is repeated until the individuals reach a specified population size. The basic steps of GA are listed as follows: i. Set the initial GA parameters. ii. Create a population of random solutions consisting of binary strings. iii. Evaluate the fitness of each chromosome in the population using tournament selection process. iv. Create a new population repeating the following sub-steps. a. Select two parent chromosomes having higher fitness. b. Perform crossover on the parents to form new offspring with a crossover predictably. c. Mutate the individual bits of the new offspring with a mutation probability. d. Retain the best chromosomes from the previous generation and replace the remaining with the new offspring in a new population.
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v. Compute the fitness value for the new population. vi. If the stopping condition is satisfied, return the best solution in current population. 5. Design analysis The main design analysis program together with important function programs, viz., design of HV continuous disc winding with tap settings, design of LV helical winding, estimation of total losses of the transformer, temperature rise of windings and oil etc. are coded in C language. 5.1. Core and windings As design and production techniques favor core type construction for furnace transformers, a three-legged core is considered. A high voltage continuous disc winding near the core for the primary and a low voltage helical winding for secondary to carry a large circuit current are selected to suit the specifications. A continuous disc type winding for HV is more reliable from the point of mechanical strength and helical winding for LV is generally preferred when the number of turns is small but the current is very large. The low voltage winding consists of a number of multistrand conductor coils connected in parallel and arranged one above the other. The LV winding conductor dimensions are estimated from the current and current density. Tap settings provided on the HV winding are arranged to give nine different voltages according to the specifications. An integral number of turns on each tap step were designed without exceeding the maximum permissible ratio error of ±0.5%. Standard strap conductor dimensions for the design of HV winding are selected from the conductor tables which are stored in the program. 5.2. Losses in the transformer The key contributors to the losses that affect the efficiency of the transformer are core laminations, winding copper and stray magnetic flux. Assuming 0.35 mm thick CRGO steel laminations at 50 Hz and using a standard core loss curve, the core losses are computed for core and joints. In addition to the normal I2R losses, the HV and LV conductor eddy current losses are assumed to vary with the square of the rms current and square of the frequency [28]. The eddy current losses are expressed as in the following equation: 4
Pe ¼ k1 b n2R f 2 a2 n2A ð3I2 RÞ=x4
ð3Þ
Also, the leakage fluxes produce eddy-currents in the core, clamps, tank and other metallic parts causing stray losses which increase the temperature rise of oil and windings [28]. The stray losses are expressed as in the following equation:
Ps ¼
k2 X 2sc B2c A2i x34 f Mt ½x4 þ 2ðK z 0:5D12 Þ2
ð4Þ
The data of core losses, magnetization power curves for core and joints of CRGO steel laminations are stored through polynomial expressions in the design analysis program. 5.3. Thermal modeling Based on the total losses, physical dimensions of transformer windings and core, the winding and core temperature rises above oil are determined. The temperature rises also depends on the type of cooling used. Assuming forced oil cooling, the two constraints on temperature rise of winding and top oil are expressed [28] as in (5) and (6).
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Temperature rise of winding above ambient:
Table 3 PSO parameters.
hwa ¼ hw þ ho C
ð5Þ
C hw ¼ k3 q0:7 w
Temperature rise of top oil above ambient:
hoa ¼ 1:2ho þ k4 C
ð6Þ
where hw is the temperature difference between winding surface and oil, ho is permissible average oil temperature above ambient, qw specific thermal load of winding, W/m2, k3 and k4 are constants based on winding and oil respectively. It is to be noted that both the constraints (5) and (6) are functions of design variables.
The design analysis program of the transformer together with one of the optimization techniques is simulated for estimating the optimal design parameters. The design variables are bounded by lower and upper limits as listed in Table 2. It is observed that during simulations each intelligent technique locates a global minimum out of many local minimums. 6.1. Results of optimization with GA In this paper each design variable is coded as a 16 bit binary string in GA. Optimal design parameters and performance data minimizing the objective function are derived for a set of GA parameters and a penalty factor. Extensive simulation study on the design analysis program together with GA technique is conducted with population sizes of 25, 50, 75 and 100, and for different probabilities of mutation and crossover. Experimenting with different combinations of control parameters and after performing a number of simulations, a global minimum for cost is observed with a population size as 100, number of generations as 100, probability of mutation as 0.044 and probability of cross-over as 0.8867. In order to analyze the performance of optimal design variables, the number of generations is changed from 100 to 600 for fixed values of penalty factor and other GA parameters. It is observed that an
3529000 3528500 300
400
100 5–35 0.9–0.4 C1 = C2 = 2 0–1
PSO - Optimal cost
500
600
Number of generations Fig. 3. Cost optimization by GA.
3.5600 3.5400 3.5200 3.5000 5
10
15
20
25
30
35
Swarm size Fig. 5. Cost optimization by PSO.
increase in number of generations often improves the minimization of the objective function. Fig. 3 shows the effect of variation of number of generations on the augmented cost objective function for the defined set of GA parameters and a penalty factor of 5. It can be seen from Fig. 3 that convergence of the objective function to an optimum value is around 400 generations. As the active materials cost of stampings and winding copper is specified in the ratio of 1:2 units, the cost function is represented in units. The effect of penalty parameter on the objective function is also analyzed with the same GA parameters defined previously. The simulation results yield similar optimal solutions for a variation of penalty factor in steps from 5 up to 100. But when the penalty factor is increased further, the objective function is observed to yield a local optimum or cause the solution to diverge. The difference in optimal cost with small and large penalty factors at 600 generations is only about 0.011%.
cost
SS - Optimal cost x 10 6 4.400 4.300 4.200 4.100 4.000 3.900 5
cost
The simulation results with SS technique are obtained for different values of subsets, b1 and b2, where the size of the reference set, b = b1 + b2. The combination of b1 and b2 is varied between 5 and 35, while keeping the size of the set of diverse solutions PSize at a constant value [22,23]. The maximum number of iterations, MaxIter is fixed as 10. With PSize = 20 and 30, the SS simulations were experimented with different combinations of control parameters, and optimum results are obtained when b1 = 10 and b2 = 10. Fig. 4 shows the change of optimal cost for variation of reference set size.
3529500
Cost, units
Cost, units
3530000
200
Number of iterations Swarm size, S Inertia weight factor, W Weighting factors, C1 and C2 Random numbers, rand1 and rand2
6.2. Results of optimization with SS
GA -Optimum cost
100
Value
3.5800
Cost, Units
6. Results of optimization and comparison
Parameters
10
15
20
Size of reference set Fig. 4. Cost optimization by SS.
25
30
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Table 4 Optimal design variables, constraints and objective function. Powell
SS
PSO
GA
Design variables x1 flux density in the core (T) x2 Current density in HV winding (A/mm2) x3 Current density in LV winding (A/mm2) x4 Height of the windings (m) x5 Voltage per turn (V) x6 Distance between core centers (m)
1.678 3.497 2.396 3.369 111.16 0.898
1.667 3.492 3.104 2.974 111.26 0.801
1.65 3.438 2.500 3.336 111.11 1.2
1.68 3.439 2.50 3.336 111.11 1.2
Constraints 1. Temperature rise of windings (°C) 2. Temperature rise of oil (°C) 3. % No-load current, I0 4. % Short circuit impedance, Zsc 5. % Efficiency, g 6. Clearance between different phase windings (m) 7. Cost, units 8. Execution time (s)
55.0 44.29 1.192 9.268 99.43 0.063 5.650 106 170
50.0 46.562 1.334 11.192 99.4 0.075 4.0483 106 31
57.926 47.602 1.018 10.021 99.5 0.098 3.5484 106 25
55.429 42.636 2.090 11.42 99.05 0.083 3.5289 106 1
6.3. Results of optimization with PSO Extensive simulation study is conducted on proper selection of PSO parameters. Finally the design analysis program together with PSO technique is simulated by selecting the parameters as in Table 3 to yield global optimum cost. The behavior of the objective function is as shown in Fig. 5 for varying swarm size. The PSO technique yield optimal cost function for each swarm size and fast convergence. It can be seen from Fig. 5 that convergence of the cost to an optimum value is around a swarm size of 20. 6.4. Comparison of results from SS, PSO and GA techniques Table 4 presents the optimal design variables, constraints and minimum cost of the 64.5 MV A furnace transformer obtained by the three evolutionary methods and by the conventional Powell unconstrained minimization technique [11]. A sensitivity analysis performed on design variables and constraints with variation of number of generations in GA, reference set size in SS, and swarm size in PSO show minimum variation in magnitudes at optimum cost. For comparison of optimal results a reference set size of 20 for SS, a swarm size of 20 for PSO, and 400 generations for GA are selected. However, in the case of conventional Powell method twelve minimization cycles are required to arrive at the optimal value. As the number of generations is increased in GA, it is obTable 5 Optimal results of 3-phase, 64.5 MV A, 33 kV/384–712 V, star/delta connected furnace transformer. HV continuous disc winding Turns per phase Number of coils Width of the winding (mm) Normal conductor (mm mm)
198 60 22.95 12.5 3.28
LV helical winding Turns per phase Number of coils Number of parallel conductors Width of the winding (mm) Conductor (mm mm)
4 10 4 19.95 85.55 4.05
Other design parameters Core circle diameter (m) Width of the core (m) Length of the core (m) Dimensions of the tank (m) Secondary voltages (V)
0.70 3.02 3.55 1.68 4.08 5.22 384, 407, 435, 464, 501, 540, 586, 646, 712
served that the distribution of solutions is much closer as compared to SS and PSO. It is also possible that SS and PSO may discard good solutions than GA in the problem search space. Subsequently SS and PSO have difficulty to reach global optimal point, although they move quickly towards the best general area in the solution space. Besides, GA creates new solutions by combining old ones. Thus the effectiveness of GA in the search space leads to arrive at global optimal solutions taking less execution time. It can be seen from Table 4, that minimum cost obtained during simulation with GA has taken less execution time as compared with the other two optimization techniques. It is also observed during simulations that the execution time varies from 0.01 to 1 s for a variation of number of generations from 100 to 600 in GA. In case of PSO, with the variation of swarm size from 5 to 35, the execution time varies from 23 to 25 s. The SS optimization technique required a longer execution time of 3–73 s for a change in reference set size from 5 to 30. At the end of each optimization simulation process the design analysis program returns the design data such as optimal core and winding dimensions, and other transformer details. Table 5 shows the optimal parameters and main design data of the specified transformer using GA with 400 generations and minimizing the cost. Similar design parameters are derived when SS and PSO techniques were employed. It can be seen that the design data of the furnace transformer presented in Tables 4 and 5 is satisfactory. Also it is observed that GA and PSO are proved to be efficient and robust methods for solving the nonlinear constrained design optimization of a furnace transformer. 7. Conclusions A procedure for optimum design of a furnace transformer using three intelligent evolutionary methods has been presented. As a first step in the design procedure the rating of the transformer is developed from the manufacturer’s reports. A numerical simulation tested by varying the control parameters of SS, PSO and GA techniques shows how the proposed algorithms search and improve the optimal solution. It was shown that the optimal design data of the furnace transformer found by GA and PSO techniques were better than by SA technique for minimum cost. For the nonlinear design analysis program the proposed three evolutionary methods proved to be reliable and provided global optimum solutions. A highly promising outcome of results of design data of furnace transformer proved that the proposed evolutionary methods are efficient tools for constrained optimization of other electrical equipment.
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Acknowledgment The authors gratefully acknowledge the support provided by Universiti Teknologi PETRONAS, Malaysia. References [1] Ciotti JA, Robinson CG. 400 Ton arc furnace. J Met AIME 1973:17. [2] Hatch CR. Arc furnace transformers – modern design developments. Elect Rev 1960;166:52–6. [3] Borrebach EJ. Maximum power operation of arc furnaces. Iron Steel Eng 1969:74–83. [4] Transformers with off-load tap changing for direct arc melting furnaces. AEI Publication, 1563-1. UK: GEC Ltd. [5] Transformers with on-load tap changing for direct arc melting furnaces. AEI Publication, 1563-1a, UK: GEC Ltd. [6] Bonis P, Coppadora F. Transformers for arc furnace. Brown Boveri Rev 1973;60:456–7. [7] Gorga WH. Design considerations for the installation of modern high power electric arc furnaces. Iron Steel Eng 1972;49:35–41. [8] Grundmark B. Large furnace transformers. ASEA J 1972;6:151–6. [9] Powell MJD. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput J 1964;7:155–62. [10] Zangwill WI. Nonlinear programming via penalty functions. Manage Sci 1967;13:344–58. [11] Rama Rao KS. Optimal design of electromagnetic devices. Ph.D. thesis. Indian Institute of Technology, Kanpur, India; July 1978. [12] Hui L, Li H, Bei H, Shunchang Y. Application research based on improved genetic algorithm for optimum design of power transformers. In: ICEMS 2001, proceedings of 5th international conference, vol. 1; 2001. p. 242–5. [13] Thilaigar SH, Sridhara Rao G. Parameter estimation of three-winding transformers using genetic algorithm. J Eng Appl Artif Intell 2002;15(5): 429–37.
[14] Georgilakis PS, Tsili MA, Souflaris AT. A heuristic solution to the transformer manufacturing cost optimization problem. J Mater Process Technol, Sciencedirect 2007;181(1–3):260–6. [15] Tsili MA, Kladis AG, Georgilakis PS. Computer-aided analysis and design of power transformer. J Comput Ind, Sciencedirect 2008:338–50. [16] Amoriralis EI, Tasli MA, Kladis AG. Transformer design and optimization: a literature survey. IEEE Trans Power Deliv 2009;24(4):1999–2024. [17] Subramanian S, Padma S. Optimization of transformer design using bacterial foraging algorithm. Int J Comput Appl 2011;3(Article 8):52–7. [18] Yadav AK, Rahi OP, Malik H, Azeem A. Design optimization of high frequency power transformer by GA and SA. Int J Elect Comput Eng 2011;2(2):102–9. [19] Rama Rao KS, Ramalinga Raju M. Optimal design of a furnace transformer by genetic algorithm. In: PECon 2009, proceedings IEEE int’l conf on power and energy, Kuala Lumpur, Malaysia. [20] Pham DT, Karaboga D. Intelligent optimization techniques. Great Britain: Springer; 1998. [21] Dumitrescu D, Lazzerini B, Jain LC, Dumitrescu A. Evolutionary computation. LLC, Florida: CRC Press; 2000. [22] Laguna M, Marti R. Scatter search: methodology and implementations in C. Boston: Kluwer Academic Publishers; 2002. [23] Marti R. Scatter search – wellsprings and challenges. Eur J Oper Res 2006;169:351–8. [24] Valle YD, Venayagamoorthy GK, Mohegheghi S, Hermandz JC, Harley RG. Particle swarm optimization: basic concepts, variants and applications in power systems. IEEE Trans Evol Comput 2008;12(2):171–95. [25] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings IEEE int’l conf on neural networks, Perth, Australia; 1995. [26] Reeves CR. Using genetic algorithms with small populations. In: Proceedings of the 5th international conference on genetic algorithms, Morgan Kaufmann, San Mateo, CA; 1993. [27] Thompson DR, Bilbro Gl. Comparison of a genetic algorithm with a simulated annealing algorithm for the design of an ATM network. IEEE Commun Lett 2000;4(8). [28] Vasutinsky SB. Principles, operation and design of power transformers. India: P.S.G. College of Technology; 1962.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Optimal design of a furnace transformer by intelligent evolutionary methods K.S. Rama Rao ⇑, Mohd Noh Karsiti Department of Electrical and Electronic Engineering, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Malaysia
a r t i c l e
i n f o
Article history: Received 25 September 2010 Received in revised form 27 May 2012 Accepted 2 June 2012 Available online 15 July 2012 Keywords: Furnace power transformers Genetic Algorithm Nonlinear programming Optimal design Particle Swarm Optimization Scatter Search
a b s t r a c t This paper presents three intelligent evolutionary optimization techniques to investigate the optimal design parameters of a 3-phase furnace transformer. The transformer rating is derived from the operating conditions of a medium size direct arc furnace. Scatter Search (SS), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) techniques are employed on the developed nonlinear mathematical model of the transformer for constrained optimization minimizing the cost. The design and analysis programs of the furnace transformer are developed using codes written in C++/C language. The optimal design data results validated by an example show the efficacy of the three intelligent techniques. Among the three methods, the optimal results obtained by GA and PSO techniques show the potential for implementing as efficient search techniques for design optimization of furnace transformers. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An electric arc furnace used in steel industry essentially needs large amounts of power at a low voltage level for production of steel. In a direct-arc furnace the heat is generated by a powerful electric arc struck between carbon or graphite electrodes and the metal charge. Important characteristic features of a furnace transformer connected between the power system and the furnace load are low secondary voltages and high secondary currents. Accordingly, a transformer for an arc furnace is to be designed to supply a very large current by the secondary winding at a very low voltage dictated by the arc resistance. Besides the arc voltage must be large enough to supply the desired power and permit operation of the furnace at higher than normal current with maximum voltage [1]. The regulation of a furnace voltage over a wide range is usually achieved by tap settings of the transformer extending from maximum secondary voltage down to 50–25%. Furnace arc power and arc resistance, and secondary circuit impedance of the power system are major factors for fixing the rating of a furnace transformer. The secondary circuit consists of secondary bus, flexible cables, copper bus tubes and the electrodes. For a specified effective arc voltage and current, a wide range of circuit impedances can be implemented that result in the same average arc resistance [2]. This paper aims to focus on optimal design of a furnace transformer based on manufacturer’s reports and publications [3–8] which provided possible types of construction, electrical characteristics ⇑ Corresponding author. E-mail addresses: [email protected] (K.S. Rama Rao), nohka@petronas. com.my (M.N. Karsiti). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.06.019
and winding connections of arc furnace transformers. The design procedure of a transformer employed for furnace loads is different from a conventional power transformer. The traditional optimization techniques such as the sequential unconstrained minimization technique by Powel-Botm [9] together with Zangwill’s exterior penalty function method [10] has been applied for conventional and special purpose transformers [11]. In the recent years, a number of researchers paid attention on evolutionary procedures which are conceived to be independent of a memory based optimization method. An improved GA for single and double objective optimum design problems of power transformers is proposed [12] as a search method. The GA based parameter estimation method is compared with conventional methods [13] for the equivalent circuit parameter estimation of a three-winding transformer. A heuristic method for transformer design minimizing the cost is proposed in [14] based on the given specifications and available materials. An integrated three dimensional finite element method for the analysis and design of power transformers is developed in [15]. A survey on transformer design optimization methods as published in [16] reveals the continued interest in applications of advanced techniques. Bacterial foraging technique is applied [17] for design optimization of a conventional transformer and the results are compared with those obtained by SA and PSO. A comparative study on design optimization of high frequency power transformer by using geometric programming and by applying GA and SA is reported in [18]. However, very few publications are reported on design optimization of special purpose transformers such as furnace transformers [19]. In this paper three evolutionary methods SS, PSO and GA are selected for cost optimization of a furnace transformer, and the
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Nomenclature a, b Ai Bc, By ci, cc c1, c2 D12 f FF () Gc, Gi Gl, Gy I k1, k2
Kz Mt nA, nR Pc, Pi R RA RFT RS XFT XS Vi, Vc Xsc
dimensions of strap conductor, mm net core cross section (m2) flux density in the core and yoke (T) cost of stampings and copper (units) loss capitalization factors (units) mean diameter between the windings (m) frequency (Hz) polynomial to represent core loss (W/kg) weight of the copper and stampings (kg) weight of the limbs and yoke (kg) current per phase (A) constants
design parameters are investigated. To provide an overview of proposed optimization methods this paper is organized as follows: Section 2 describes the development of the furnace transformer rating and specifications for a case study. The problem statement using objective and constraint functions is presented in Section 3. A brief description of the evolutionary optimization procedures in general and the three proposed methods for optimal design are presented in Section 4. Then the design analysis programs for simulation are described in Section 5. Section 6 is used to analyze the results of optimization and the resulting effect of variation of control parameters of GA, SS and PSO algorithms on the objective function. Section 7 concludes the paper validating the simulation results and optimal design parameters. 2. Rating of the furnace transformer Fig. 1 shows the power system, transformer and furnace load. The transformer rating is derived from the operating conditions of the arc furnace such as arc resistance and arc current. In this paper, a 3-phase star-delta connected, core-type transformer is considered to supply a medium size direct-arc furnace. As shown in Fig. 2, the average arc resistance varies nonlinearly as the arc current increases when the transformer is operating at the highest tap [3]. Assuming a typical arc current, Ic of 60 kA and a per phase secondary circuit impedance, Zs of (0.5 + j 2.7) mX, the transformer rating is derived as follows [11]:
distance between the limb centers (m) perimeter of the tank (m) number of axial and radial straps full-load copper and iron losses (kW) resistance per phase (X) arc resistance (X) resistance of furnace transformer (X) resistance of secondary circuit (X) reactance of furnace transformer (X) reactance of secondary circuit (X) volume of stampings and copper (m3) % short-circuit reactance
Average arc resistance per phase, RA = 4.8 mX. The total circuit impedance, Zc = RA + Zs = 5.3 + j 2.7 mX. p Circuit voltage on load, Vc = 3IcZc = 620 V. p Transformer rating, S = 3IcVc = 64,500 kV A = 64.5 MV A. Normally to manage very high secondary currents over a wide secondary voltage range, an on-load tap changer, connected directly on the HV winding or placed in an intermediate circuit of a two-core design (booster regulation) within the transformer tank [8] is provided. As furnace transformers are generally operated at a medium voltage level, a primary voltage of 33 kV, and a secondary voltage range of 384–712 V, with nine tap positions on HV winding are assumed. With this arrangement, it is possible to feed the stipulated power into the furnace at the highest tap and maintain a stable arc for holding the melt on the lowest tap [3]. Table 1 presents the derived rating and other specifications of the transformer which are used to investigate the optimal design parameters. 3. Objective and constraint functions In this paper, a nonlinear mathematical model defined in terms of a set of design variables with cost as objective function and important constraint functions based on furnace transformer configuration, is developed. The design variables and constraints are
Table 1 Specifications of transformer. Infinite bus
Furnace
Transformer
Secondary circuit
X FT
RFT
Arc resistance
XS
RS
64.5 MV A, 33 kV/384 – 712 V, 50 Hz, star/delta 3-phase core type with forced oil cooling
RA
Fig. 1. Furnace transformer and the arc furnace.
Table 2 Design variables and constraints. Limits
Arc resistance, mohms
Arc resistance 20 15 10 5 0 20
30
40
50
60
70
80
Arc current, kA Fig. 2. Arc resistance for highest voltage tap.
90
Design variables x1 Flux density in the core (T) x2 Current density in HV winding (A/mm2) x3 Current density in LV winding (A/mm2) x4 height of the windings (m) x5 Voltage per turn (V) x6 Distance between core centers (m)
1.55–1.7 3.00–3.65 2.50–3.25 2.75–3.5 100–120 0.80–1.20
Constraints 1. Temperature rise of windings (°C) 2. Temperature rise of oil (°C) 3. % No-load current (I0) 4. % Short circuit impedance, Zs) 5. % Efficiency, g 6. Clearance between different phase windings (m)
<58 <50 <2.5 <12 >99 <0.1
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selected as in Table 2, considering their effect on core geometry, losses in core and windings, short-circuit reactance, temperature rise, weight, volume and finally on the cost of the transformer. The main design analysis program together with function programs coded in C language is simulated with each of the optimization techniques. The optimal design parameters are derived minimizing the cost as the objective function subjected to the constraints based on geometry and performance characteristics of the transformer. The cost objective function is the sum of initial cost and the capitalized cost of transformer losses. The initial cost is the sum of active materials cost of stampings and copper windings which are assumed to be in the ratio of 1:2 units. The capitalized cost of losses are based on (i) maximum demand charge as an additional demand caused by losses and (ii) cost of electrical energy wasted as transformer losses [11]. Thus the cost objective function is expressed as in the following equation:
F c ðXÞ ¼ ci Gi þ cc Gc þ c1 Pc þ c2 Pi
ð1Þ
where X = (x1, x2, . . . , x6), vector of design variables. The initial cost and capitalized cost of losses are expressed as functions of selected design variables x1 to x6 from Table 2. Similarly the constraints as listed in Table 2 are expressed in terms of the design variables [11]. In general any nonlinear optimization problem is stated as: Find X, such that the nonlinear objective function F(X) is a minimum, subject to the nonlinear constraints, gj(X){6=}0, j = 1, 2, . . . , m with X P 0 being a non-negative solution. Using exterior penalty function method [10] an augmented objective function, P is formulated as in the following equation:
PðX; rÞ ¼ FðXÞ þ r
m X ½g j ðXÞ2 ; r P 0
ð2Þ
j¼1
where gj(X) is defined as max [gj(X, 0)] and r is the penalty factor, which imposes penalty on the deviations of constraints. 4. Evolutionary optimization procedures Optimal design problems involve a search for the global minimum of the fitness function or objective function. To achieve global optimization, there are a variety of approaches. One way is to run any of the traditional nonlinear optimization methods from numerous random starting points such as in Powell’s method [9]. But this practice is inefficient as the traditional methods require large computational time to arrive at a global optimum value. A number of evolutionary computation methods are applied recently to engineering combinatorial optimization problems. Intelligent methods such as GA, SS and PSO techniques are a class of evolutionary methods based on random generation of initial population and natural behavior of artificial life [20,21]. Each of these methods computes a fitness value which is directly related on the distance to the optimum value. The reproduction of the population is based on the fitness value. To demonstrate the effectiveness of the proposed optimization techniques, the performance of the objective function with the variation of control parameters of the three algorithms is tested. All the three evolutionary methods proposed in this paper for design optimization of furnace transformer offer a convenient way of handling constraints and single or multi objective functions. A brief description and algorithms of the three evolutionary procedures are reported here. 4.1. Scatter Search technique Scatter Search (SS) [22,23] is an evolutionary population-based stochastic optimization technique that has been formulated combining decision rules and problem constraints. It uses small
population called a reference set, and is found to be successful for combinatorial optimization problems. SS selects two or more elements of the reference set in a systematic way with the purpose of creating new solutions using a set of control parameters and meets the desired characteristics of minimizing nonlinear optimization problems. The fundamental procedure combines the following steps: i. A Diversification Generation Method to generate a set of diverse trail solutions from an arbitrary initial solution. ii. An Improvement Method to transform the diverse solutions into one or more improved solutions. iii. A Reference Set Update Method to develop and maintain a representative set of good solutions. Here two or more elements are selected in a systematic way and a new solution is created strategically. Typically a reference set has 20 solutions or less. This method updates the reference set following intensification and diversification strategies. iv. A Subset Generation Method to operate on the reference set, selecting several subsets of solutions, for creating combined solutions. v. A Solution Combination Method to combine the solutions in each subset taking into account their good features. The usual notation of the control parameters used in SS is as follows: i. Psize = the size of the set of diverse solutions generated by the Diversification Generation Method. ii. b = the size of the reference set. iii. b1 = the size of the high quality subset. iv. b2 = the size of the diverse subset. v. MaxIter = maximum number of iterations. 4.2. Particle Swarm Optimization technique PSO is also a population based optimization technique based on the behavior of bird flocking and fish schooling. Besides PSO is one of the swarm intelligent techniques and is not affected by the size and nonlinearity of the problem [24,25]. The population of agents or particles tries to simulate its social behavior in the problem space and arrives at an optimal value. In this algorithm, the particles, the candidate solutions, use the information related to the most successful particle in the population in order to improve their position. The particles or agents in the population swarm through the design space with specified velocities. These agents try to simulate their social behavior in the problem space and arrive at an optimum value of fitness function. Each agent is represented by its current position and corresponding velocity. During simulation, the modified position of each agent is a random combination of its previous velocity and current position. In every step each agent’s position is updated by the best fitness value called as pbest. Also each agent knows the best value in the group called as gbest among the pbests. Over a number of iterations the agents searching points gradually reach the global optimal point using pbest and gbest. The basic steps of PSO are described as follows: i. Set the initial parameters of the PSO. ii. Create a population of particles. Initialize the particles with random vector of variables for position, X and velocity, V. iii. Evaluate the fitness function, F for all particles. iv. Compare the fitness value at the current position to the best fitness value at any time. Select the individual best particle, pbest. v. Identify the best particle in the group, gbest and its location corresponding to the minimum fitness value.
K.S. Rama Rao, M.N. Karsiti / Electrical Power and Energy Systems 43 (2012) 1056–1062
vi. Set gbest = pbest. vii. Update the new particle velocity and position as in (a) and (b).
V kþ1 ¼ W V ki þ C 1 rand1 ðpbest i X ki Þ þ C 2 rand2 ðgbesti X ki Þ i ðaÞ X kþ1 i
¼
X ki
þ
V kþ1 i
ðbÞ
where V ki is the velocity (current velocity) of particle i at kth iteration; V kþ1 is the velocity (modified velocity) of particle i at (k + 1)th i iteration; W is the inertia weight factor; rand1 and rand2 are the rank dom numbers between 0 and 1; pbesti is the best value found by k particle i until iteration k; gbesti is the best particle found in the group until iteration k; X ki is the current position (current searching point) of the particle i at kth iteration; X kþ1 is the current position i (modified searching point) of the particle i at (k + 1)th iteration. viii. If stopping criterion is satisfied then go to next step, else go to step ii. (The stopping criterion is good fitness value or a specified maximum number of iterations or no further improvement in fitness value). ix. Print the optimal parameters of X and the minimum fitness value. 4.3. Genetic Algorithm technique GA is a stochastic search technique derived from the principles of natural evolution and is based on genetic parameters such as inheritance, mutation, crossover and selection. Besides, GA rapidly locates the global minimum in difficult search spaces. It is a robust method and far less sensitive to parameter values [26,27]. In contrast to more traditional numerical techniques, the parallel nature of the stochastic search done by GA often makes it very effective in locating the global optimum. GA is less susceptible to getting stuck at a local minimum than gradient search methods. Also, GA is much less sensitive to initial conditions and is widely used in various optimization problems [21]. It starts with a set of random solutions called population. In GA approach each design variable is represented as a binary encoded string (chromosome) of fixed length. The chromosomes are evaluated by using a fitness function. GA provides solutions by generating a set of chromosomes referred to as a generation. A new generation is created from a pair of chromosomes (parents) by reproduction operators, mutation and crossover. If the search is to continue for an optimal solution, the GA creates a new generation from the old one until a decision is made on the convergence. In this paper, one type of selection strategy, the tournament selection, is applied to select the fittest individuals from the population, and the better one is duplicated in the next generation. The process is repeated until the individuals reach a specified population size. The basic steps of GA are listed as follows: i. Set the initial GA parameters. ii. Create a population of random solutions consisting of binary strings. iii. Evaluate the fitness of each chromosome in the population using tournament selection process. iv. Create a new population repeating the following sub-steps. a. Select two parent chromosomes having higher fitness. b. Perform crossover on the parents to form new offspring with a crossover predictably. c. Mutate the individual bits of the new offspring with a mutation probability. d. Retain the best chromosomes from the previous generation and replace the remaining with the new offspring in a new population.
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v. Compute the fitness value for the new population. vi. If the stopping condition is satisfied, return the best solution in current population. 5. Design analysis The main design analysis program together with important function programs, viz., design of HV continuous disc winding with tap settings, design of LV helical winding, estimation of total losses of the transformer, temperature rise of windings and oil etc. are coded in C language. 5.1. Core and windings As design and production techniques favor core type construction for furnace transformers, a three-legged core is considered. A high voltage continuous disc winding near the core for the primary and a low voltage helical winding for secondary to carry a large circuit current are selected to suit the specifications. A continuous disc type winding for HV is more reliable from the point of mechanical strength and helical winding for LV is generally preferred when the number of turns is small but the current is very large. The low voltage winding consists of a number of multistrand conductor coils connected in parallel and arranged one above the other. The LV winding conductor dimensions are estimated from the current and current density. Tap settings provided on the HV winding are arranged to give nine different voltages according to the specifications. An integral number of turns on each tap step were designed without exceeding the maximum permissible ratio error of ±0.5%. Standard strap conductor dimensions for the design of HV winding are selected from the conductor tables which are stored in the program. 5.2. Losses in the transformer The key contributors to the losses that affect the efficiency of the transformer are core laminations, winding copper and stray magnetic flux. Assuming 0.35 mm thick CRGO steel laminations at 50 Hz and using a standard core loss curve, the core losses are computed for core and joints. In addition to the normal I2R losses, the HV and LV conductor eddy current losses are assumed to vary with the square of the rms current and square of the frequency [28]. The eddy current losses are expressed as in the following equation: 4
Pe ¼ k1 b n2R f 2 a2 n2A ð3I2 RÞ=x4
ð3Þ
Also, the leakage fluxes produce eddy-currents in the core, clamps, tank and other metallic parts causing stray losses which increase the temperature rise of oil and windings [28]. The stray losses are expressed as in the following equation:
Ps ¼
k2 X 2sc B2c A2i x34 f Mt ½x4 þ 2ðK z 0:5D12 Þ2
ð4Þ
The data of core losses, magnetization power curves for core and joints of CRGO steel laminations are stored through polynomial expressions in the design analysis program. 5.3. Thermal modeling Based on the total losses, physical dimensions of transformer windings and core, the winding and core temperature rises above oil are determined. The temperature rises also depends on the type of cooling used. Assuming forced oil cooling, the two constraints on temperature rise of winding and top oil are expressed [28] as in (5) and (6).
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Temperature rise of winding above ambient:
Table 3 PSO parameters.
hwa ¼ hw þ ho C
ð5Þ
C hw ¼ k3 q0:7 w
Temperature rise of top oil above ambient:
hoa ¼ 1:2ho þ k4 C
ð6Þ
where hw is the temperature difference between winding surface and oil, ho is permissible average oil temperature above ambient, qw specific thermal load of winding, W/m2, k3 and k4 are constants based on winding and oil respectively. It is to be noted that both the constraints (5) and (6) are functions of design variables.
The design analysis program of the transformer together with one of the optimization techniques is simulated for estimating the optimal design parameters. The design variables are bounded by lower and upper limits as listed in Table 2. It is observed that during simulations each intelligent technique locates a global minimum out of many local minimums. 6.1. Results of optimization with GA In this paper each design variable is coded as a 16 bit binary string in GA. Optimal design parameters and performance data minimizing the objective function are derived for a set of GA parameters and a penalty factor. Extensive simulation study on the design analysis program together with GA technique is conducted with population sizes of 25, 50, 75 and 100, and for different probabilities of mutation and crossover. Experimenting with different combinations of control parameters and after performing a number of simulations, a global minimum for cost is observed with a population size as 100, number of generations as 100, probability of mutation as 0.044 and probability of cross-over as 0.8867. In order to analyze the performance of optimal design variables, the number of generations is changed from 100 to 600 for fixed values of penalty factor and other GA parameters. It is observed that an
3529000 3528500 300
400
100 5–35 0.9–0.4 C1 = C2 = 2 0–1
PSO - Optimal cost
500
600
Number of generations Fig. 3. Cost optimization by GA.
3.5600 3.5400 3.5200 3.5000 5
10
15
20
25
30
35
Swarm size Fig. 5. Cost optimization by PSO.
increase in number of generations often improves the minimization of the objective function. Fig. 3 shows the effect of variation of number of generations on the augmented cost objective function for the defined set of GA parameters and a penalty factor of 5. It can be seen from Fig. 3 that convergence of the objective function to an optimum value is around 400 generations. As the active materials cost of stampings and winding copper is specified in the ratio of 1:2 units, the cost function is represented in units. The effect of penalty parameter on the objective function is also analyzed with the same GA parameters defined previously. The simulation results yield similar optimal solutions for a variation of penalty factor in steps from 5 up to 100. But when the penalty factor is increased further, the objective function is observed to yield a local optimum or cause the solution to diverge. The difference in optimal cost with small and large penalty factors at 600 generations is only about 0.011%.
cost
SS - Optimal cost x 10 6 4.400 4.300 4.200 4.100 4.000 3.900 5
cost
The simulation results with SS technique are obtained for different values of subsets, b1 and b2, where the size of the reference set, b = b1 + b2. The combination of b1 and b2 is varied between 5 and 35, while keeping the size of the set of diverse solutions PSize at a constant value [22,23]. The maximum number of iterations, MaxIter is fixed as 10. With PSize = 20 and 30, the SS simulations were experimented with different combinations of control parameters, and optimum results are obtained when b1 = 10 and b2 = 10. Fig. 4 shows the change of optimal cost for variation of reference set size.
3529500
Cost, units
Cost, units
3530000
200
Number of iterations Swarm size, S Inertia weight factor, W Weighting factors, C1 and C2 Random numbers, rand1 and rand2
6.2. Results of optimization with SS
GA -Optimum cost
100
Value
3.5800
Cost, Units
6. Results of optimization and comparison
Parameters
10
15
20
Size of reference set Fig. 4. Cost optimization by SS.
25
30
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Table 4 Optimal design variables, constraints and objective function. Powell
SS
PSO
GA
Design variables x1 flux density in the core (T) x2 Current density in HV winding (A/mm2) x3 Current density in LV winding (A/mm2) x4 Height of the windings (m) x5 Voltage per turn (V) x6 Distance between core centers (m)
1.678 3.497 2.396 3.369 111.16 0.898
1.667 3.492 3.104 2.974 111.26 0.801
1.65 3.438 2.500 3.336 111.11 1.2
1.68 3.439 2.50 3.336 111.11 1.2
Constraints 1. Temperature rise of windings (°C) 2. Temperature rise of oil (°C) 3. % No-load current, I0 4. % Short circuit impedance, Zsc 5. % Efficiency, g 6. Clearance between different phase windings (m) 7. Cost, units 8. Execution time (s)
55.0 44.29 1.192 9.268 99.43 0.063 5.650 106 170
50.0 46.562 1.334 11.192 99.4 0.075 4.0483 106 31
57.926 47.602 1.018 10.021 99.5 0.098 3.5484 106 25
55.429 42.636 2.090 11.42 99.05 0.083 3.5289 106 1
6.3. Results of optimization with PSO Extensive simulation study is conducted on proper selection of PSO parameters. Finally the design analysis program together with PSO technique is simulated by selecting the parameters as in Table 3 to yield global optimum cost. The behavior of the objective function is as shown in Fig. 5 for varying swarm size. The PSO technique yield optimal cost function for each swarm size and fast convergence. It can be seen from Fig. 5 that convergence of the cost to an optimum value is around a swarm size of 20. 6.4. Comparison of results from SS, PSO and GA techniques Table 4 presents the optimal design variables, constraints and minimum cost of the 64.5 MV A furnace transformer obtained by the three evolutionary methods and by the conventional Powell unconstrained minimization technique [11]. A sensitivity analysis performed on design variables and constraints with variation of number of generations in GA, reference set size in SS, and swarm size in PSO show minimum variation in magnitudes at optimum cost. For comparison of optimal results a reference set size of 20 for SS, a swarm size of 20 for PSO, and 400 generations for GA are selected. However, in the case of conventional Powell method twelve minimization cycles are required to arrive at the optimal value. As the number of generations is increased in GA, it is obTable 5 Optimal results of 3-phase, 64.5 MV A, 33 kV/384–712 V, star/delta connected furnace transformer. HV continuous disc winding Turns per phase Number of coils Width of the winding (mm) Normal conductor (mm mm)
198 60 22.95 12.5 3.28
LV helical winding Turns per phase Number of coils Number of parallel conductors Width of the winding (mm) Conductor (mm mm)
4 10 4 19.95 85.55 4.05
Other design parameters Core circle diameter (m) Width of the core (m) Length of the core (m) Dimensions of the tank (m) Secondary voltages (V)
0.70 3.02 3.55 1.68 4.08 5.22 384, 407, 435, 464, 501, 540, 586, 646, 712
served that the distribution of solutions is much closer as compared to SS and PSO. It is also possible that SS and PSO may discard good solutions than GA in the problem search space. Subsequently SS and PSO have difficulty to reach global optimal point, although they move quickly towards the best general area in the solution space. Besides, GA creates new solutions by combining old ones. Thus the effectiveness of GA in the search space leads to arrive at global optimal solutions taking less execution time. It can be seen from Table 4, that minimum cost obtained during simulation with GA has taken less execution time as compared with the other two optimization techniques. It is also observed during simulations that the execution time varies from 0.01 to 1 s for a variation of number of generations from 100 to 600 in GA. In case of PSO, with the variation of swarm size from 5 to 35, the execution time varies from 23 to 25 s. The SS optimization technique required a longer execution time of 3–73 s for a change in reference set size from 5 to 30. At the end of each optimization simulation process the design analysis program returns the design data such as optimal core and winding dimensions, and other transformer details. Table 5 shows the optimal parameters and main design data of the specified transformer using GA with 400 generations and minimizing the cost. Similar design parameters are derived when SS and PSO techniques were employed. It can be seen that the design data of the furnace transformer presented in Tables 4 and 5 is satisfactory. Also it is observed that GA and PSO are proved to be efficient and robust methods for solving the nonlinear constrained design optimization of a furnace transformer. 7. Conclusions A procedure for optimum design of a furnace transformer using three intelligent evolutionary methods has been presented. As a first step in the design procedure the rating of the transformer is developed from the manufacturer’s reports. A numerical simulation tested by varying the control parameters of SS, PSO and GA techniques shows how the proposed algorithms search and improve the optimal solution. It was shown that the optimal design data of the furnace transformer found by GA and PSO techniques were better than by SA technique for minimum cost. For the nonlinear design analysis program the proposed three evolutionary methods proved to be reliable and provided global optimum solutions. A highly promising outcome of results of design data of furnace transformer proved that the proposed evolutionary methods are efficient tools for constrained optimization of other electrical equipment.
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