Equivalent circuit modelling of a three-phase to seven-phase transformer using PSO and GA

Abstract

Impedance mismatching between different phases of a multiphase transformer is generally observed e.g., in a three-phase to seven-phase transformer, due to an unequal number of turns in different coils. This mismatching introduces error in the study of per phase equivalent circuit diagrams as well as induces an imbalance in output voltages and currents. Therefore, it is a challenging task to develop a per-phase equivalent circuit for the secondary and primary sides (In some cases) too. This paper proposes an artificial intelligence optimization technique like PSO based modeling of the per-phase equivalent circuit of the secondary side. This paper deals with the modeling and simulation of a three-phase to seven-phase power transformer using Artificial Intelligence technique like particle swarm optimization (PSO) and Genetic Algorithm (GA). The proposed model is optimized through PSO and GA algorithms and tested for minimum voltage error in each phase. The proposed model is designed and the objective function is optimized by PSO & GA in MATLAB environment. It is found that the optimized model can be effectively implemented as a per-phase equivalent circuit for the secondary side.

Keywords

Genetic algorithm multiphase particle swarm optimization transformer seven-phase

1 Introduction

The literature of the Multiphase system was found since 1960 s and flourished after 1960 [12]. Multiphase systems are known for reliability, fault tolerance, and high power carrying capabilities. Literatures are found for its application in transmission and utilization [3 , 19]. The various benefits offered by the multiphase system have pulled researchers’ interests are higher machine torque density, lower torque ripples, lower harmonics, better transient and steady-state performances and robust control of multiphase electrical drives [3 , 13]. Multiphase AC-DC rectifiers produce lower ripples and higher ripple frequency. Therefore Multiphase AC-DC rectifiers produce DC of higher quality.

Multiphase electric power from a three-phase power is generated by three techniques. One is by converting three-phase AC to a DC power and then converting it back to multiphase AC (i.e. AC-DC-AC); the second is through multiphase power converting transformers; the third one is by employing matrix converter (AC-AC). The second method utilizes transformer winding in a special connection and turns ratios while the first and third method needs rectifiers and inverters [1–4 , 22–24].

The literature of 6- and 12-phase transformers based multi-pulse rectifier system is found in [14, 16]. Recently, 24-phase and 36-phase transformer systems have been designed for supplying a multi-pulse rectifier system [2 , 26]. The reason of choice for a 6-, 12-, or 24-phase system is that these numbers are multiples of three, and designing this type of system is simple and straight forward. The explanation of the decision for a 6-, 12-, or 24-phase transformer is that these numbers are products of three and the design is straightforward [19].

As far as Multiphase transformer of odd numbers of phases is concerned, limited pieces of literature are found for the odd number of phases viz. 5-, 7- and 11-phase [1 , 19–21]. Prototypes were designed and tested for a three to five-phase and three to seven-phase power converting transformer and reported in the literature. One of which is Scott based three to five-phase converting power transformer design is reported in [1]. The approach of [1] is extended and reported in [24] for a three-phase to seven-phase transformer. The work reported in [24] requires further research. This paper deals with the mathematical modeling of the transformer reported in [24].

Per-phase equivalent circuit is vital for the analysis of a multiphase transformer. Although three-phase to seven-phase conversion techniques has been already reported [18, 24] but the research that addresses the secondary winding mismatching, per-phase equivalent circuit modeling is not available yet. This work is the first of its kind that addresses the mismatching issue and developed a per-phase equivalent circuit of the secondary side using intelligent techniques like GA and PSO.

Section 2 deals with the transformer design. Section 3 is problem formulation while section 4 is optimization results. The conclusion is being put in section 5.

2 Materials and Methods

2.1 General Theory of two-phase to n-phase transformation

A more simplified and graphical two-phase to n phase theory of transformation is given in [20]. It is shown in [20] that a three-phase AC can be converted to any number of phases by using a special transformer design given and turn ratios of primary and secondary can be calculated from the generalized equations.

Refer Fig. 1 taken from [20]; let’s assume voltage V_α and V_β be the output of three-phase to two-phase conversion out of primary Scott connection. Vα and Vβ are voltage induced in winding “RO” and “YB” and have equal RMS value as that of the supply phase but they are 90 degrees phase apart as both constitute a two-phase AC power.

Fig. 1

Phasor diagram for 2 phase to n phase conversion.

Let’s assume a phase conversion of two-phase to n balanced phase (refer Fig. 3). Let V_i be the i^th phasor of a balanced n phases balanced system at an angle of iδ and have a magnitude of V where $δ = \frac{2 π}{n}$ is phase difference between each phase. The projection of V_i along axis α and β would be V_i_α and V_i_β for i^th phase. For phasor of i^th phase, $V_{i α} = V_{i} cos (i δ)$ (1) $\frac{V_{i α}}{V_{i}} = cos (i δ)$ (2) $a_{α} = cos (i δ)$ (3)

Similarly, $V_{i β} = V_{i} sin (i δ)$ (4) $\frac{V_{i β}}{V_{i}} = sin (i δ)$ (5) $a_{β} = sin (i δ)$ (6)

Where i may be any phase of balanced n-phase system i.e., i = 0,1,2 ... ..(n-1),

V_i_α,-Projection of i^th phasor along x-axis/

V_i_β-Projection of i^th phasor along y-axis/

a_α- Turns ratio for i^th phase along x-axis or with respect to winding wound on Core 1.

a_β-Turns ratio for i^th phase along y-axis or with respect to winding wound on Core 2 which produces a flux 90 degrees out of phase w.r.t core 1.

Hence an n phase transformation from a two-phase balanced supply could be achieved by employing two core transformer with a pulsating flux of 90 degrees apart in phase. For n-number of phases, we can get turn ratios of individual windings by varying i from 0 to n-1 and putting $δ = \frac{2 π}{n}$ . Hence Voltage equations and turns ratios can be written in the matrix form as shown by (7) and (8).

In matrix form, the voltage relationship can be generalized as: $[\begin{matrix} V 1 \\ V 2 \\ . \\ . \\ Vi \\ . \\ Vn \end{matrix}] = [\begin{matrix} \begin{matrix} 1 \\ cos (δ) \\ . \\ . \\ cos (i - 1) δ \\ . \\ cos (n - 1) δ \end{matrix} & \begin{matrix} 0 \\ sin (δ) \\ . \\ . \\ sin (i - 1) δ \\ . \\ sin (n - 1) δ \end{matrix} \end{matrix}] [\begin{matrix} V α \\ V β \end{matrix}]$ (7)

Whereas Turn ratios can be generalized as below $[\begin{matrix} \begin{matrix} 1 \\ cos (δ) \\ . \\ . \\ cos (i - 1) δ \\ . \\ cos (n - 1) δ \end{matrix} & \begin{matrix} 0 \\ sin (δ) \\ . \\ . \\ sin (i - 1) δ \\ . \\ sin (n - 1) δ \end{matrix} \end{matrix}]$ (8)

2.2 Three-phase to seven-phase transformer

A generalized theory of three-phase to multiphase transformation is developed using the phasor diagram approach in [24]. Using that approach a three-phase to seven-phase transformer can be designed for 1:1 turns rations if windings are connected according to the schematic diagram shown in Fig. 2. Required Turns ratios are shown in Table 1. Proper connections of the multiple windings of the requisite number of turns are the key to generate a seven-phase current from a three-current. The input and output currents are shown in Figs. 3, 4.

Fig. 2

Winding arrangement of the transformer.

Table 1

Turns ratios of each windings

α	Turn ratio	β	Turn ratio	Total no.
Axis	w.r.t	Axis	w.r.t	of Turns
	Winding RO	Winding RO	w.r.t RO(N)
a₁a₂	1.00			1.000
b₁b₂	0.6234	b₃b₄	0.7818	1.4052
c₁c₂	0.2225	c₃c₄	0.9749	1.1974
d₁d₂	0.9009	d₃d₄	0.4338	1.3347
e₁e₂	0.9009	e₃e₄	0.4338	1.3347
f₁f₂	0.2225	f₃f₄	0.9749	1.1974
g₁g₂	0.6234	g₃g₄	0.7818	1.4052

Fig. 3

Three-phase input current.

Fig. 4

Seven-phase output current.

2.3 Impedance mismatch calculation

Per phase equivalent circuit model or any polyphase machine is required for transient as well as steady-state analysis of the machine. It is very clear from Table 1 taken from [24] that each phase is composed of two winding except phase 1. It is clear from Fig. 1 that the windings of two limbs are connected in series with proper polarity and turns ratios so resistance and inductance of each phase shall include both the windings. Resistance and inductance of a coil is calculated by the following expression. $\begin{matrix} R = \frac{ρ l}{A} \\ L = \frac{N^{2} μ A}{l} \end{matrix}$ Where L = Inductance of the coil

N = Number of turns of the coil

μ=Permeability of the core

A = Area of the coil

l = Average length of the coil

Here A is fixed to the core cross-sectional are for all the windings but l is proportional to the number of turns. So it can be said that $\begin{matrix} R α l α N \\ X α l α N \end{matrix}$

If Resistances and self-inductance of the reference phase is assumed to be R_2a and X_2a then resistances and inductances of the rest of the phases can be expressed in terms of the reference phase as below. $\begin{matrix} R_{b} = 1.4052 R_{a}, X_{b} = 1.4052 X_{a} \\ R_{c} = 1.1974 R_{a}, X_{c} = 1.1974 X_{a} \\ R_{d} = 1.3347 R_{a}, X_{d} = 1 . . 3347 X_{a} \\ R_{e} = 1.3347 R_{a}, X_{e} = 1.3347 X_{a} \\ R_{f} = 1.1974 R_{a}, X_{f} = 1.1974 X_{a} \\ R_{g} = 1.4052 R_{a}, X_{g} = 1.4052 X_{a} \end{matrix}$

3 Equivalent circuit and parameters determination

Per phase equivalent circuit of secondary winding poses some issues as each phase has a different number of windings. An intelligent system is applied to estimate the approximate value of secondary winding parameters for an acceptable error.

The per-phase equivalent circuit model of the secondary side of the multiphase transformer is shown in Fig. 5. Let’s assume that EMF induced in all seven secondary windings is E₂, Load current of I_L flowing through each circuit. The terminal voltage of each secondary windings will be different as each winding has a different value of impedance.

Fig. 5

Exact equivalent circuit of secondary windings.

The aim is to formulate a per-phase equivalent circuit of primary side and secondary side as it has been done for three-phase transformers. The process would be simple and straightforward if all the seven-phases has the same parameters. Unfortunately due to the very nature of transformer design, all the seven-phase secondary as well primary three-phase too has different parameters. This paper is indeed an extension of [24] and aimed to develop a per-phase equivalent circuit modeling.

3.1 Primary side per phase equivalent circuit

It is well known that in Scott connection the three-phase primary windings are connected in such a way to produce a two-phase flux separated by 90 degrees in the time domain. That is windings are in the ration of 1:0.57:0.57 and connected as shown in Fig. 1 primary section.. That too may impose imbalance but can be approximated as a two out of three-phase has equal parameters.

3.2 Secondary side per phase circuit

Let’s assume that Per phase equivalent circuit is shown in Fig. 5 while exact seven-phase circuit is shown in Fig. 6, where k is the optimum value of the factor which will decide the parameters of per phase equivalent circuit. It will deduced by suitable optimization technique.

Fig. 6

Per-phase equivalent circuit of secondary windings.

Output voltages of all seven-phases under steady-state condition can be deduced from the circuit diagram shown in Fig. 5; it is as shown by Equation 9. $\begin{matrix} V_{oa} = E_{2} - I_{L} (R_{2 a} + {jX}_{2 a}) \\ V_{ob} = E_{2} - I_{L} (R_{2 b} + {jX}_{2 b}) \\ V_{oc} = E_{2} - I_{L} (R_{2 c} + {jX}_{2 c}) \\ V_{od} = E_{2} - I_{L} (R_{2 d} + {jX}_{2 d}) \\ V_{oe} = E_{2} - I_{L} (R_{2 e} + {jX}_{2 e}) \\ V_{of} = E_{2} - I_{L} (R_{2 f} + {jX}_{2 f}) \\ V_{og} = E_{2} - I_{L} (R_{2 g} + {jX}_{2 g}) \end{matrix}$ (9)

The output voltage of per-phase equivalent secondary circuit is $V_{o} = E_{2} - I_{L} ({kR}_{2 a} + {jkX}_{2 a})$

Let us formulate the error voltage (difference of secondary voltages) as objective function,

$\begin{matrix} f_{1} (k) = V_{o} - V_{oa} \\ = E_{2} - I_{L} ({kR}_{2 a} + {jkX}_{2 a}) - (E_{2} - I_{L} (R_{2 a} + {jX}_{2 a})) \\ = I_{L} ((k - 1) R_{2 a} + j (k - 1) X_{2 a}) \end{matrix}$

Similarly $\begin{matrix} f_{2} (k) = I_{L} ((k - 1.4052) R_{2 a} + j (k - 1.4052) X_{2 a})) \\ f_{3} (k) = I_{L} ((k - 1.1974) R_{2 a} + j (k - 1.1974) X_{2 a})) \\ f_{4} (k) = I_{L} ((k - 1.3428) R_{2 a} + j (k - 1.1.328) X_{2 a})) \\ f_{5} (k) = I_{L} ((k - 1.3428) R_{2 a} + j (k - 1.1.328) X_{2 a})) \\ f_{6} (k) = I_{L} ((k - 1.1974) R_{2 a} + j (k - 1.1974) X_{2 a})) \\ f_{7} (k) = I_{L} ((k - 1.4052) R_{2 a} + j (k - 1.4052) X_{2 a})) \end{matrix}$ (10)

It is clear from Equation 10 that f₁ (k), f₂ (k), f₃ (k), and f₄ (k) are the only objective functions to be optimized for this multi-objective optimization problem. This problem is solved by making a single objective function and then using particle swarm optimization (PSO) algorithm to find the best value of variable, k. The whole process is discussed in the section 3.3.

3.3 Formulation of the optimization problem

The aim of the optimization problem is to get the best value of k for which the functions (given in Equation 10) simultaneously have the least values. For this, a common objective function (or fitness function) is defined and is written as:

$\begin{matrix} ofun (k) = \frac{1}{(\sqrt{R_{2 a}^{2} + X_{2 a}^{2}}) I_{L}^{2}} \\ [{| f_{1} (k) |}^{2} + {| f_{2} (k) |}^{2} + {| f_{3} (k) |}^{2} + {| f_{4} (k) |}^{2}] \end{matrix}$ (11)

The terms for functions f₅(k), f₆(k) & f₇(k) will have the same effect as that due to functionsf₄(k), f₃(k) & f₁(k) respectively, so these terms have not been used in the objective function.

The optimization problem is a minimization problem such that ofun has the minimum value for a particular value of k.

There are various artificial intelligence (AI) techniques like Artificial Neural Network (ANN), adaptive neuro-fuzzy inference system (ANFIS), Genetic Algorithm (GA), Particle swarm optimization (PSO), etc [6]. Particle swarm optimization (PSO) is one of the best evolutionary techniques that is being used rigorously used by researchers throughout the world [8 , 21]. Unlike GA, PSO does not use genetic operators like crossover and mutation. Particles get updated using internal velocity; and have a memory vital to the algorithm. In addition, PSO is a single direction information sharing method, as here only the ‘best’ particle gives out the information to others, and thus the process looks for the best solution. Also as compared to GA, PSO is easy to implement with few parameters to be adjusted.

In order to get the best value of k, PSO algorithm is used in this work. Particle swarm optimization (PSO) is an artificial intelligence (AI) technique modeled as per the performance of bird flocks [14, 27]. It is useful to find the keys to exceptionally tough optimization problems. In PSO algorithm, a swarm of particles is maintained, and each particle’s position represents a solution [10]. The particles change their position by getting influenced by the neighboring best particle (P_best) as well as the overall best particle (G_best). The position ( $x_{i}^{j + 1}$ ) of i^th particle in the (j + 1)^th iteration gets attuned by using the following equation: $x_{i}^{j + 1} = x_{i}^{j} + ϕ_{i}^{j + 1}$ (12)

Where, ϕ_i, is the step size or the velocity of the particle. Its value is obtained by

$ϕ_{i}^{j + 1} = ω ϕ_{i}^{j} + c_{1} r_{1} (P_{besti} - x_{i}^{j}) + c_{2} r_{2} (G_{best} - x_{i}^{j})$ (13)

$ω = ω_{max} - j (ω_{max} - ω_{min}) / (max ite)$ (14)

Where, ω represents inertia weight, acceleration coefficients are c₁ and c₂_; r₁ and r₂ are the random numbers between 0 & 1; ω_max, ω_min are the maximum and minimum values of inertia weight, and, maxite is the predefined maximum number of iterations. The typical movement of particles during the optimization process is shown in Fig. 3.

In the problem discussed in this paper, the value of k, itself represents the position of a particle and thus Equation 4 can be rewritten as $k_{i}^{j + 1} = k_{i}^{j} + ϕ_{i}^{j + 1}$ (15)

In the case of GA, which was formerly presented by Holland [28] as an EC method, the optimization process depends on three important operators (namely the selection, crossover, and mutation) on every generation. Here, a number of chromosomes (the candidate solutions) are assessed firstly. Then a fitness function is used to evaluate each chromosome. Higher fitness valued chromosomes are most likely selected and included in the next population of generation and then they go for recombination. Two parents are combined through crossover in order to yield i^th offspring using Equations 16 and 17. $Child (i) = r . parent (i) + (1 - r) . parent (i + 1)$ (16)

$Child (i + 1) = (1 - r) . parent (i) + r . parent (i + 1)$ (17)

where r is a random number, r ∈ [0,1].

Mutation is used to get faster convergence. It preserves genetic variety from one age group to the next and efforts to realize some stochastic unevenness of GA [29, 30].

The application of PSO for finding the best value of k using the defined fitness function (ofun) has been represented in the flowchart in Fig. 8. Similarly, GA was applied to find the best value of k using the defined fitness function (ofun).

Fig. 7

Movement of particles during the optimization process.

Fig. 8

Flowchart of PSO applied for finding the best value of k.

The values of parameters selected for PSO and GA are given in Tables 2 and 3 respectively.

Table 2

Values of PSO parameters

Parameter	Range or value
ω	0.9 to 0.4
c₁ and c₂	2 to 2.05
Population size	03
maxite	1000
Initial value of ϕ	0.1 times of initial value of k

Table 3

Simulation parameters of PSO

Description	Parameters
Population size	4
Maximum iteration	1000
Crossover probability	0.6
Mutation probability	0.01

4 Results

Figure 9 shows the 3D representation of the results obtained for the PSO convergence characteristic using MATLAB. It can be observed that the value of fitness function converges to its minimum value in before the 50th iteration. Figure 10 shows the GA results.

Fig. 9

PSO convergence characteristic.

Fig. 10

GA convergence characteristic and result.

Figure 10 shows the variation of the value of objective function with respect to the value of k during the PSO process. Similarly, Figs. 11 –14, show the variation of the values of functions (f₁, f₂, f₃ and f₄ respectively) with respect to the value of k during the PSO process.

From Fig. 15, it can be observed that the value of the fitness function is minimum near k = 1.23. The best-optimized value of k obtained using PSO is found to be 1.2364.

Fig. 11

Variation of function f1 w.r.t. k.

Fig. 12

Variation of function f1 w.r.t. k.

Fig. 13

Variation of function f3 w.r.t. k.

Fig. 14

Variation of function f4 w.r.t. k.

Fig. 15

Variation of objective function w.r.t k during the optimization process.

This result can be verified by comparing it with the result obtained by plotting the ofun w.r.t k and observing its minimum value, the value of k corresponding to the minimum value of ofun is found to be 1.236 & that with GA is 1.2363 (Table 4). This value is very close to that obtained by PSO, but since PSO gives a more précis value so it can be said that PSO is a much better technique to obtain the value of k.

Table 4

Comparison of results

Method	Obtained optimum value of k	Corresponding value of objective function
GA	1.2363	0.0972204
PSO	1.2364	0.0972200
From Plot	1.2360	0.0973

The main dissimilarity between the PSO tactics as compared with EC like GA is that PSO does not use genetic operators like crossover and mutation. Particles get updated using internal velocity; and have a memory vital to the algorithm. Also, PSO is a single direction information sharing method, as here only the ‘best’ particle gives out the information to others, and thus the process looks for the best solution. Also as compared to GA, PSO is easy to implement with few parameters to be adjusted.

There per phase equivalent circuit of the secondary side of the transformer can be expresses as shown in Fig. 12.

The developed model can be verified by estimating output voltage under-compensation as well as without compensation. Let’s assume a load current of 5A(30⁰Lag) is flowing through the secondary, an EMF of 150 V is induced in the secondary winding. Let’s assume Reference winding resistance equals to 1 ohms and reactance as per norms(X/R = 7). Results are tabulated in Table 4.

The R & L parameters are changed and the data is generated similar to that presented in Table 5. The standard deviation of compensated and non-compensated is calculated and plotted in Fig 17. It can be seen that compensated data is changed less as compared to non-compensated data in response to parameter variation.

Fig. 16

Per phase equivalent circuit of the secondary side of the transformer.

Fig. 17

Standard deviation of the seven-phase output voltage with parameter variation.

Table 5

Output voltage under compensation and no compensation

S. No.	Non-Compensated		Compensated
	Mag(Volts)	Angle(Rad)	Mag(Volts)	Angle(Rad)
1	131.1540	–0.2138	144.9880	0.0454
2	125.5663	–0.3167	143.0463	–0.0647
3	128.2623	–0.2629	144.0354	–0.0547
4	126.3370	–0.3003	143.3418	–0.0617
5	126.3370	–0.3003	143.3418	–0.0617
6	128.2623	–0.2629	144.0354	–0.0547
7	1 25.5663	–0.3167	143.0463	–0.0647

5 Conclusions

Artificial intelligence-based algorithms like Particle swarm optimization techniques like (PSO) and GA can be used effectively to model the multiphase transformer circuits that pose unbalancing as a challenge. The obtained result can be verified by comparing it with the result obtained by plotting the ofun w.r.t k and observing its minimum value, the value of k corresponding to the minimum value of ofun is found to be 1.236. This value is very close to that obtained by PSO, but since PSO gives a more precise value so it can be said that PSO is a much better technique to obtain the value of k. It can be observed that the value of the fitness function converges to its minimum value before the 50th iteration.

Footnotes

Acknowledgment

World Bank Technical Education Quality Improvement Program (TEQIP)-III Project of Rajkiya Engineering College, Ambedkar Nagar (U. P., India), financially supported this work.-224122.

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