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Acceptance No – 14 Summit Paper -137 Michael Faraday IET India Summit, Kolkata, India, November 25, 2012 Organized by Young Professional Section, IET Kolkata Local Network 172 Cost-Optimal Design of a 3-Phase Core Type 3-Winding Transformer Raju Basak Arabinda Das Abstract— 3-phase core type transformers are extensively used as power transformers. Some of these transformers are equipped with a third or tertiary winding for supplying station auxiliaries and for other reasons. Their cost is a sizable proportion of the total system cost. A number of methodologies have already been developed for reaching cost-optimal design. However, in this paper an attempt has been made to develop a simplified method for optimizing design, in presence of constraints specified by the customer and the regulatory authorities, through exhaustive search using nested loops. The method appears to be best at present considering the enormous speed of the present-day computer. A case study has been made based on this method and has been given at the end of the paper. Index Terms— Design variable, Exhaustive search, Objective function, Optimal design, 3-winding transformer Symbols used Rating, VA S V1 ,V2 I1 , I 2 Bm Primary and secondary phase voltage, V δ Current density, A/sq. mm Stacking factor, window space factor Ks , Kw H w ,Ww Rw Et T1 , T2 a1 , a2 CT VR TR Io Primary and secondary phase current, A Maximum value of flux-density, Tesla Window height, window width Window height/width ratio Volts/turn No. of turns/phase, the primary and the secondary C.S. of the primary and secondary, sq.mm. Total cost of transformer inclusive of overheads,Rs. Efficiency, % Voltage regulation, % Temperature rise, o C No load current, % Raju Basak is a research scholar in the Department of Electrical Engineering, Jadavpur University, Kolkata - 700 032, India (e-mail: [email protected] Arabinda Das is an Associate Professor in the Department of Electrical Engineering, Jadavpur University; Kolkata - 700 032, India (e-mail: [email protected]) Amar Nath Sanyal is a Professor of Electrical Engineering at the Academy of Technology, Hooghly, West Bengal, India (e-mail: [email protected]) N st A Amar Nath Sanyal No. of steps of the core I. INTRODUCTION N empirical basis based on experience of the designer was previously adopted for engineering design. Later on, with the development of analytical tools and mathematical modelling the engineers began to apply them to design problems. This approach to design involves many variables and constraints and complex calculations to reach a feasible solution [2], [6]. It is highly time-consuming to solve such a problem by long-hand calculations, using a calculator, particularly because a set of calculations have to be performed many times to reach an optimal solution. So, recourse to computer is made for solving design problems. There are four different approaches to solve a design problem. These are enumerated below [5]: Analytical design: In this method the values of the design variables are chosen by the designer from his accumulated experience. Based on the chosen values of design variables, the dimensions of the equipment are found out step by step and finally the performance variables are calculated. If a feasible or acceptable solution is not obtained by this procedure, the designer suitably modifies the values of design variables and restarts the process. In this method, there is no feedback from output to modify the input, which is an inherent disadvantage, but the method is simple and straight-forward. It does not require skillful programming. Synthetic design: The performance variables are prescribed in this method. The program itself adjusts the design variables to reach the target as closely as possible. This method requires more skillful programming. Optimal design: This is a method in which a judiciously chosen objective function is optimized (minimized or maximized) in presence of a set of equality and non-equality constraints. Through appropriate loops in the program, the optimal solution is reached without violating any of the constraints. A skillful programmer having high degree of mathematical knowledge can only adopt this method [1], [7], [9]. Acceptance No – 14 Summit Paper -137 173 Michael Faraday IET India Summit, Kolkata, India, November 25, 2012 Organized by Young Professional Section, IET Kolkata Local Network Standard design: bitumen-filled. In any case, the maximum allowable This method is used to get best possible economy in mass temperature rise must not be exceeded. The design variables production. The stampings, frame size etc. are standardized should be chosen with a look to these points. for a series of standard mass production items. The manufacturing unit may itself standardize and produce the III. 3-WINDING 3-PHASE CORE TYPE TRANSFORMER stampings or frames or it may purchase them straight-way from the market whichever is economically more The distribution transformers [3] are two-winding advantageous [1]. transformers, generally with tap-settings for adjustment of the The synthetic design is obviously better than the analytic design. The optimal design is, undoubtedly, the best. The optimal design may either be static or dynamic, constrained or unconstrained. The mathematical techniques [3], [4] , [9]] adopted to reach the optimal solution are Non-linear programming (NLP), Linear programming (LP), Quadratic programming (QP), Geometric programming (GP), Integer programming (IP) and Method of exhaustive search. Generally the design problems of electrical equipment are non-linear constrained optimization problems. The solutions may be obtained directly or indirectly. The heuristic search, constraint approximation, feasible direction and gradient projection methods are direct methods. The transform method for eliminating the constraints, the interior and exterior penalty function methods are indirect methods. In this paper, a case-study has been taken up aiming at optimality of production cost, in presence of specified constraints. The transformer is a power transformer of rating 25/20/5 MVA, 66/11/3.3 kV. The minimality has been sought and found out by the method of exhaustive search using nested loops against key design variables. II. CHOICE OF DESIGN VARIABLES A 3-phase transformer may be core type or shell type. In a shell type transformer, the coils are surrounded by the core. It requires more iron and less copper (or, aluminium). In the core construction, the coils surround the core. It requires less iron and more copper. The choice depends on relative price of copper and iron in a country. For example, in India the core construction for 3-phase transformers generally proves itself to be more economic [5], [6]. The 3-phase core type transformer may be either a power or a distribution transformer. A power transformer runs at a load near to its rating for most of the times whereas the average load of a distribution transformer is much less compared to its rating. As such, a power transformer should be designed in a manner so as to have its maximum efficiency at or near its full load but the maximum efficiency of a distribution transformer should occur at a percent load of 40 to 60 depending upon the load factor at the point of use. The efficiency should not fall below a limit specified by the board / utility. The voltage regulation should be kept within a maximum limit for distribution transformers and the shortcircuit current should be kept within a maximum limit for the power transformer. The allowable temperature rise depends on the type of transformer whether oil-cooled or air-cooled or output voltage. They do not have any third or tertiary winding. Some special purpose small transformers may have a tertiary winding e.g. in one used for controlled rectification using thyristors. But in large power transformers a tertiary winding is added for various reasons [3]. Some of the examples are given below: 1) In a power station, the third winding is added for supplying power to the station auxiliaries like boiler feedpump, FD and ID fan-motors etc. 2) For supplying two transmission lines from a point at two different voltage levels. 3) Some of the H.V. transformers use star-connection for both primary and secondary winding for economic advantages. This practice gives rise to neutral shift during unbalanced operation. To obviate this difficulty, a stabilizing tertiary connected in delta is used. It suppresses the zero sequence current as well as the triple harmonics. These are, in essence, 3-winding transformers with unloaded tertiary. IV. THE DESIGN VARIABLES AND CONSTRAINTS The key variables [5] to be chosen, to optimize a design problem, depends on the objective function [4]. While it is optimized to get minimum possible cost of production subject to usual design constraints e.g. the efficiency is not less than 98%, no load current not more than 1%, voltage regulation is not more than 5% (for a distribution transformer) or the short circuit MVA is not more than 8 p.u. (for a power transformer) etc., the iron loss and copper loss are kept at their maximum possible values to reduce the cost of production but their ratio is to be adjusted to match the load factor. Accordingly, the flux density and the current density are kept at their maximum possible values without violating the design constraints. In such a case, the following design variables affect the cost: a) The e.m.f. constant K in equation Et = K.S Here Et = e.m.f. per turn, S = KVA rating b) The ratio of window height to width : Rw = Hw/Ww c) The choice of core material -- costlier CRGOS may prove itself to be more economic than cheaper HRS considering overall cost including that of copper. d) The choice of conductor materials -- sometimes costlier copper may have to be used considering overall performance and cost. e) The ratio of iron loss to copper loss : (P i/Pc) Acceptance No – 14 Summit Paper -137 174 Michael Faraday IET India Summit, Kolkata, India, November 25, 2012 Organized by Young Professional Section, IET Kolkata Local Network But if the customer’s interest is to be secured then the Step 14: Go to transformer design subroutine with values of K running cost towards lost energy units must also be included in and Rw for which the cost has been found to be the objective function. Therefore, the flux density and the minimum. current density are also to be chosen as design variables to find Step 15: Print out results the minimality conditions for the chosen objective function. In Step 16: Stop this paper, only techniques for minimizing cost of production Step 17: End subject to given design constraints have been considered. V. THE DESIGN PROCEDURE From designer’s experience the following values of design variables have been suggested for a 3-phase core type transformer [1], [5]: E.M.F. constant, K: 0.45 for distribution transformer and 0.6-0.7 for power transformer; Window height/width, Rw : 2.03.0 for power transformer and 3.0-4.0 for distribution transformer The following choice of materials is recommended: Core material: CRNOS for smaller ratings and CRGOS for larger ratings Conductor materials: Aluminium for smaller ratings and Copper for larger ratings After choosing the conductor and the core material judiciously, our task is to choose such values of K and Rw which will give rise to minimum cost of production without violating the design constraints. The value of flux density B m and current density are initially chosen so as to keep the efficiency and no load currents near to their limiting values prescribed by the customer. The selling cost is the objective function. Its minima is sought against two variables only. Therefore the method of exhaustive search has been adopted as the computer run-time has been found to be negligibly small. Considering these the following algorithm for the computer program [8] has been developed. The algorithm Step 1: Input specifications of the transformer Step 2: Input user-specified data for design variables Bm, , Nst , etc. Step 3: Choose copper as conductor material, CRS as cost material Step 4: For K = 0.5 to 0.7 in steps of 0.1 do Step 5: For Rw = 2.0 to 3.0 in steps of 0.1 do Step 6: Go to transformer design sub-routine and find the performance variables: efficiency, voltage regulation, and no load current Step 7: If efficiency 98% go to step 12 Step 8: If percent Z 10% go to step 12 Step 9: If no load current 1.0% step 12 Step 10: Find the overall cost Step 11: If the current cost is less than previous minimum then set minimum cost = current cost. Preserve the corresponding values of K and Rw Step 12: end for Step 13: end for VI. A CASE STUDY The optimal design of an oil-filled core-type 3-phase 3winding power transformer has been taken up. The transformer is in the primary substation. It steps down a part of the power to 11 kV and the other part to 3.3 kV, for feeders. The cost minimality for the transformer is obtained for the following values of design variables: EMF constant = 0.42, Window height/width = 4.2 and the minimum cost = Rs. 3964018/The design details of the optimal machine are given below: MVA-rating of the primary, secondary and tertiary: 25; 20; 5 Nominal power factor (assumed same for secondary and tertiary) = 0.85 lagging Rated line voltage of primary, secondary and tertiary: 66; 11; 3.3 kV Nominal frequency in Hz = 50 Primary, secondary and tertiary connection: Y /Y / Conductor material: Copper No. of taps = 9; % turns between taps = 1.25 The EMF- constant for cost-optimality = 0.42 No. of nominal turns of the primary = 572 No. of additional turns of the primary for tapping = 28 Total no. of turns of the primary = 600 No. of nominal turns of the secondary = 95 No. of nominal turns of the tertiary = 99 Current in primary/secondary/tertiary:218.69;1049.7; 252.53A Chosen current density = 3 A/mm2 Cross section of Primary / Secondary / Tertiary (mm2): 72.898; 349.91; 84.175 Net area of core iron (mm2) = 0.17596 Stacking factor = 0.92 Gross area of core iron (mm2) = 0.19126 3-stepped core has been used. Diameter of the core circle = 0.01288 m Length of the core sides, m: 0.487; 0.381; 0.228 Area of the window = 1.4602 mm2 Window height/width, m : 2.4765; 0.58964 Distance between core centers = 1.0106 m Width/height of yoke, m : 0.487 ; 0.39274 Total length of core = 2.7429 m Total height of core = 3.2619 m Iron loss = 40715 W; % Iron loss = 0.16286 Mean length of turn (m) of Primary/Secondary/Tertiary : 2.0617; 2.9879; 2.4322 Resistance of Primary/Secondary/Tertiary, Ω : 0.33972 ; 0.01704; 0.06007 Acceptance No – 14 Summit Paper -137 175 Michael Faraday IET India Summit, Kolkata, India, November 25, 2012 Organized by Young Professional Section, IET Kolkata Local Network Copper loss = 116550 W ; % Copper loss = 0.4662 successfully. In this process the chance of getting locked at Total % loss = 0.6291 local minima has been eliminated. The computer run-time has Efficiency at full load and 0.85 lagging p.f = 0.9927 been found to be negligible. Maximum efficiency of 0.9936 occurs at 59.11 % load. The magnetizing current in % = 0.3451 REFERENCES The core loss current in % = 0.1629 The no load current in %= 0.3816 [1] O.W. Anderson, “Optimum design of electrical machines”, IEEE Trans. % leakage reactance between primary and secondary = 4.09 (PAS), Vol-86, pp. 707-11, 1967. The % voltage regulation at rated power and p.f between [2] A.E. Dymkov, “Transformer design”, MIR publications, 1975. [3] M.G. Say, “The Performance and design of alternating current primary and secondary = 2.551 machines”, Third Edition, CBS Publishers and Distributors, Delhi, The % leakage reactance between primary and tertiary = 3.183 1983. The % voltage regulation at rated power and p.f between [4] M. Ramamoorty, “Computer-aided design of electrical equipment”, primary and tertiary = 2.073 Affiliated East-West Press Pvt. Ltd. New Delhi, ISBN 81-85095-57-4 [5] A.K. Sawhney. “A course in electrical machine design”, Dhanpat Rai & Dimension of the tank (m) length, width, height: Sons, 2003, Delhi 1.2906 x 3.4266 x 3.6119 [6] H.M. Rai, “Principles of electrical machine design”, Satya Prakashan, The no. of tubes (50 mm dia.) required = 991 New Delhi,1985 The weight of tank = 6405 Kg [7] S.S. Rao, “Engineering optimization –theory and practice, Third Edition, New Age International (P) Ltd., 1996” Cost of tank sheet/Kg = Rs. 45/[8] N.S. Kambo, “Mathematical programming techniques” Revised The cost of tank = Rs. 288222 /Edition,1991,1984, Affiliated East-West Press Pvt. Ltd. New Delhi, The volume of oil = 15974 liters ISBN 81-85336-47-4. Cost of oil/liter = Rs. 32/[9] K. Deb, “Optimization for engineering design”, PHI Pvt. Ltd., 1998. The cost of oil: Rs. 511154 /Volume of iron = 2.2726 m3 Weight of iron = 17385 Kg Cost of iron/Kg = Rs. 120/-; Cost of iron = Rs. 2086232 /Volume of copper = 0.37139 m3 Weight of copper = 3305 Kg; cost of copper/Kg = Rs. 380/Cost of copper = Rs. 1256046 /Direct cost allowing 15% labour charge = Rs. 4762903 /Selling cost allowing 25% overhead = Rs. 5953629 /VII CONCLUSION The cost of transformers in a power system is an appreciable fraction of the total cost. So they must be designed costeffectively. The design methodology suggested in refs. [1], [4] or [5] is inadequate in the sense that it fails to give a costoptimal solution and may even fail to give a feasible solution provided the design variables are freely chosen. Also there is no feedback from the values of performance variables in the process. There is as such no check against their unsatisfactory values. Authors like Dymkov, Ramamoorty etc. have dealt with computer-aided design and its optimization. They have discussed several methods for reaching the optimal solution in presence of design constraints. Such methods e.g. gradient search technique etc. are complex. Also, it is difficult to choose the starting point and the step length for the gradientsearch methods and in many cases the global minima is not reached. To obviate these difficulties, the method of exhaustive searching through nested loops with decision flyoffs has been used. Due to tremendous increase in clock speed, the computer run-time has drastically reduced. So nesting could be made without any appreciable increase in the computer run-time. Case study has been made for cost-optimality of 3-phase 3winding power transformers using copper conductors and CRGOS core with the program developed for the purpose very