Proceedings of the 14th International Middle East Power Systems Conference (MEPCON’10), Cairo University, Egypt, December 19-21, 2010, Paper ID 287.
Synchronous Motor Design using Particle Swarm Optimization Technique R. A. El-Sehiemy and M. I. Abd-Elwanis
A. B. kotb and M. Elwany
Department of Electrical Engineering University of Kafrelsheikh Kafrelsheikh, Egypt {elsehiemy&mohamed.soliman4}@eng.kfs.edu.eg
Department of Electrical Engineering University of Al-Azhar Cairo , Egypt
[email protected]
Abstract- This paper investigates an optimization procedure for the design of a synchronous motor (SM) using a particle swarm optimization (PSO) procedure. The PSO is proposed to minimize the motor volume and to maximize the motor output power. The proposed procedure has two stages for motor design. In the first stage, the stator parameters are optimized while in the second stage, the field and damper winding are designed. The proposed algorithm is efficiently compared with the practical experience-based method. The proposed procedure is efficiently design the SM based on the 4pole proto-type synchronous machine which is produced at 27 military production factories. The proposed procedure leads to more economical motor compared to the practical experience based method. Also, the proposed procedure maximizes the SM developed apparent power and reduces the field/damper windings in terms of their conductors' diameters and number of conductor per slot. Keywords: Particle swarm optimization technique, synchronous motor (SM).
I.
INTRODUCTION
Synchronous motors are being increasingly used in different industry sectors in new applications or as alternatives to induction motors in current applications. This is due to their many advantages including high efficiency, compactness, fast dynamics and high torque to inertia ratio. Synchronous motors with extra features of mechanical robustness, capability of flux weakening and high speed operation are particularly suitable as variable speed drives. The realization of these merits depends greatly on motor configuration. Therefore, a great deal of attention has been focused on the optimal design of synchronous motors in recent years. In the literature, different optimization procedures were carried with different objectives, depending on the prospective application of the motor and the user’s desire. Comparative studies of minimization techniques for optimization of ac machines design was presented with greet progress in mathematical tools leads to use these optimization tools for the electric machines design in reference [1-10]. The optimization procedures are applied for IM [1]. The practical consideration was presented for the IM design in [2]. The SM optimal design using Immune algorithm [3], genetic [4-6] finite element [7]. The optimized design problem in [8] considered the torque capability and low magnetic volume while in [9] a multi-objective optimal design procedure for the interior permanent SM with improved core formula was presented. A design optimization
795
is performed on interior type permanent magnet synchronous motors to achieve high torque development capability with low permanent magnet consumption. A multi-objective optimization is performed in search for optimum magnet dimensions and location. The design optimization results in a motor structure superior to original motor specifications. For many reasons, experience based design methods cannot find optimal design solutions when dealing with nonlinear systems. Also, these methods do not guarantee a global solution for nonlinear systems; stochastic search algorithms may provide a promising alternative to these traditional approaches. An intelligent model parameter identification method using particle swarm optimization (PSO). PSO is a relatively new stochastic optimization technique developed in the mid-1990s. The particle swarm optimization procedure was presented in [10-11] for IM design in [10] and for the parameter identification of the SM. In past several years, PSO has been successfully applied in many research and application areas [12]. It is demonstrated that PSO gets better results in a faster, cheaper way compared with other methods. Another reason that PSO is attractive is that there are few parameters to adjust. One version, with slight variations, works well in a wide variety of applications. Particle swarm optimization has been used for approaches that can be used across a wide range of applications, as well as for specific applications focused on a specific requirement. In this paper, a multistage design procedure of a synchronous motor is proposed. In the first stage, the stator parameters are optimized using particle swarm optimization technique for two objective functions. These objective functions are the motor volume minimization and maximization of the developed apparent power. After the SM stator parameters are optimized, the rotor filed and damper are designed. Two studied cases are considered and compared with the experience based SM design method. The effect of the in fed voltage of damper and field windings are considered for their diameter and their number of conductors per slot.
II. RPOBLEM FORMULATION A. Synchronous Motor Output Equation
large slot pitch. The slot pitch should be less than 25 mm for low voltage machines. For salient pole machines, the number of slots /pole/ phase is usually in the range 2-4. c) Armature conductor constraint The armature conductor size is dependent on the current passes through it. The conductor current can be computed from: S (kV A ) × 103 I z = I ph = (6) 3E ph If there is a set of parallel baths, the permissible current density in the armature conductor is assumed to be with 3 -5 A/mm2. The cross-section area for armature conductors is computed from: qs = I z / J mm2. s Where, J s = current density in armature conductors, A/mm2.
The output equation for synchronous machine can be expressed in term of its main dimensions, specific magnetic and electric loadings and speed; the equation describing this relationship is known as output equation as:
2
S (kVA) = C D L n
o
Where,
C
o
= 11B
av
(1)
s
acK w × 10
−3
L=stator core length m, Bav=magnetic loading, ac=electric loading, D: stator conductor diameter in mm. and kw= winding factor. The SM volume can be written as: 2 V(m) = D L
(2)
(3 )
The conductor diameter (dc) can be computed as:
d cs =
B. SM DESIGN AS AN OPTIMIZATION PROBLEM The SM design problem can be expressed as an optimization problem as: Min f (x ) s .t . g (x ) = 0 (4) h (x ) ≤ 0 Where, f(x) is the objective function, g(x), h(x) represent the equality and inequality constraints, respectively and x is the vector of the control variables of the SM. These control variables are the stator diameter, length, flux density, the ratio of pole length to pole pitch ( γ )
There are two suggested optimal based design procedures, The first design procedure aimed at minimizing the motor volume (Equation 3) while the second procedure aimed at maximizing the apparent power at the air gap (Equations 1 and 2). Both objectives are achieved while the motor constraints are considered.
AT
¾
a
= 2.7
K T ph w A I ph p
The Full load field mmf can calculated from: AT f = 2 ATa
(8)
(9)
¾ For a certain excitation voltage Vf, About 15 to 20 % of this voltage is kept in reserve. Then, the voltage across each field coil is: (0.8 to 0.85)V f E = (10) f p ¾ The mean turn length of filed winding is obtained as :
2) SM design Constraints a) Air gab length constraint The length of air gap (lg) greatly influences the performance of a synchronous machine. A large air gap offers a large reluctance to the path of flux produced by the armature mmf and thus reduces the effect of armature reaction. This leads to a small value of synchronous reactance and high value of SCR. The ratio of lg to pole pitch should satisfy the following constraint: l g τ p ≤ 0.02 (5)
Lmf = 2 L + 2.3τ
pr
+ 24cm
(11)
Where, τ =effective span of coils. p ¾ The voltage across each field coil is computed from:
b) Armature Slot constraint
πD = ) depends upon the voltage s s of the machine. For high voltage machines which are normally built in large capacities, it is desirable to use a
τ
(7)
C. SM ROTOR DESGIN PROCEDURE The rotor contains the damper and field windings. The design of these windings is carried out as follows: 1) Filed winding The rotor winding is distributed in slots. The pole pitch is so chosen that undesirable harmonics are not produced in the flux density wave. The width of rotor slots is limited by stresses at the root of the teeth and by hoop stress in the end retaining rings. Rotor current density may be about 2.5 A/mm2 for cooled machines. However, in modern direct cooled generators the rotor current densities may be as high as 9.5 to 14 A/mm2. Rotor winding design steps: ¾ Computing the Armature mmf per pole as:
1) SM design Objectives
The slot pitch
2 qc π
s (τ
796
E
f
= I = I
L
T I 2 = 0 .8 5 1 I 1 T2
f L
ρ
f
= ρ
R
f
mf a
T mf f a f AT
The area of the rotor conductors is found out by assumes suitable value for current density. The current density ( J ) 2 in the rotor is chosen almost equal to that in the stator. Rotor conductor area is computed from:
(12)
fl
a 2 = I 2 J 2 mm2. The damper diameter of conductor is then computed from:
f
Where, I = field current A, f T = number of turns in each field coil, f a = area of field conductors mm2, and f ρ = resistively ohm /m ¾ The area of field conductors can be calculated from: a
=ρ
f
L
mf E
AT
fl
mm2
d = 2.0 a 2 π
III.
(13)
)
(14)
= ( 2 p A T f l ) (T f ) The number of field conductor is computed from: Z
f
=
2 pA T
fl
(15)
J f af
And the number of field conductor per slot is computed by: 2 pA T fl Z = (16) s J f af s r 2) Damper winding The design of damper winding is dependent on the purpose for which it is provided. In synchronous generator, it is provided to suppress the negative sequence field and to damp the oscillations when the machine starts hunting, while in a synchronous motor its function is to provide starting torque and to develop damping power when the machine starts hunting. The rotor voltage on open circuit between slip rings should not exceed 500 voltages for small machine. The damper turns T2 is computed as follows:
T2 =
E 2 k ω1 T ph E1 k ω2
(19)
PARTICLE SWARM OPTIMIZATION TECHNIQUE
Particle Swarm Optimization (PSO) was invented by Kennedy and Eberhart in 1995 while attempting to simulate the choreographed, graceful motion of swarms of birds as part of a study investigating the notion of “collective intelligence” in biological populations. Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in 1995 [12], inspired by social behavior of bird flocking or fish schooling. PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA). The system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike GA, PSO has no evolution operators such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. In PSO, a set of randomly generated solutions (initial swarm) propagates in the design space towards the optimal solution over a number of iterations based on large amount of information about the design space that is assimilated and shared by all members of the swarm. Modification of the swarm agent positions is realized by the position and transition information. Each agent transition can be simulated by two dimensional referred to the available information's about self and group experiences. A basic PSO version based on the collected information of self and group experience according to the agents positions. The basic PSO version was presented as [12, 13]:
f
af
mm
Where, J =current density in rotor conductors. 2 Round conductor are used for small motors. But for large motor it becomes necessary to use bar conductors.
¾ For the field current density Jf, the field winding conductors area can be computed from the following equation: ⎛ ⎞ a tf = (2 A T f l ) ⎜ E f 1 ⎟ ⎝ R f af ⎠
= (2 p A T f l ) (I f
(18)
best Δx k = v .Δx k + c1r1 (xbest k − x k ) + c2 r2 (gxk − x k )
x k +1 = x k + Δx k
(17)
(20)
Where,
Where, E1 is the stator voltage per phase and E2is rotor voltage per phase at standstill. The rotor current per phase ( I 2 ) is computed from considering the full load rotor mmf is
xk x
is the vector of control variables at iteration k.
best k
is the vector of personal best of control variables at iteration k.
about 85% of stator mmf as:
797
Case 1) Designs of the SM using the proposed MPSO procedure based on minimize the motor volume. Case 2) Design of the SM using MPSO method based on maximizes the output power. Table 1 shows the results obtained using the experience based design procedure (case 0) compared to the other two optimized cases for the tested motor. In this table, Cases 1 and 2 leads to more reduction in the motor volume with reduction of 21.84 % compared with the case 0. Case 2 increases the output power compared to the other two cases. Also, cases 1 and 2 leads to more reduction in the required total ampere conductor (AC) with a reduction of 0.88%. The field and damper parameters are varied according the objective functions considered in the first stage of the design based on PSO technique. For the first studied case (Case 1), Figures 1and 2 show the variation of field winding diameter and field conductor per slot versus the field voltage, respectively. It is cleared that, the field winding diameters increase linearly with the field voltage while, the field winding conductor per slot decreases with the field voltage increased. This means that: the higher field voltage leads to decrease the field conductor per slot. Figures 3- 4 show the variation of damper conductor per slot and damper diameter versus damper voltage, respectively. It is cleared that from these figures, the damper conductor per slot increase linearly with the damper voltage while, the damper winding conductor per slot decreases with the damper voltage increased. This means that: the higher damper voltage leads to decrease the damper diameter. For the second studied case (Case 2), Figures 5 and 6 show the variation of damper conductor per slot and damper diameter versus the damper voltage for Case 2. It is cleared that, the damper winding turns and conductor per slot increase linearly with the damper voltage while, the damper winding conductor per slot decreases with the damper voltage increased. This means that: the higher damper voltage leads to decrease the damper diameter The field winding diameter increase linearly with the field voltage as shown in Figure 7 while, the field winding conductor per slot decreases with the field voltage increased as shown in Figure 8. This means that: the higher field voltage leads to decrease the field conductor per slot.
gx kbest is the vector of global best of control variables at iteration k.
x k +1 is the vector of control variables at iteration k+1. The velocity is updated at iteration k for the control variables using equation (21) as: v k = v k m ax − (v k m ax − v k min ) × k Iter m ax (21) The large number of inertia coefficient (v) leads to more global solution. The learning coefficients c1 and c2 are the factors which PSO technique optimizes different objective functions on the basis of personal and group experiences and each agent tries to modify its position the updating formula (20). The updating formula in (20) for agent position and transition information is limited by the minimum and maximum transition values as: T kmin ≤ Δx k ≤ T kmax (22) Where the maximum and the minimum transition in (22) are computed from:
⎧T kmax = k m (x kmax − x kmin ), ⎨ min max min ⎩T k = −k m (x k − x k )
(23)
IV. PROPOSED SOLUTION METHODOLOGY
The PSO-based design procedure steps are: 1. Defining the motor limits and constraints and the PSO algorithm coefficients (learning – inertia). 2. Solving the stator side problem by solving the PSO-based design problem to obtain the best stator design considering two objectives namely, developed power (apparent power) and the stator volume. For the motor volume minimization, Equation 2 is considered as an objective function with the stator constraints (Equations 5-7). For the developed apparent power maximization, Equation 3 is considered as an objective function with the stator constraints (Equations 5-7). 3. For each studied case, design the field/damper windings are performed based on the optimal stator parameters in the previous step. 4. Again, redesign the damper/field windings for different in fed voltages based on the stator results obtained in step 2. 5. Performing a comparison with the experience based design for the synchronous motor for varied range.
TABLE 1 A COMPARISON BETWEEN SM DESIGN RESULTS USING DIFFERENT OPTIMIZATION TECHNIQUES FOR THE TESTED MOTOR
V. NUMERICAL EXAMPLE
studied cases
Variables
Three studied cases are considered in this paper to design the SM for a 4-pole proto-type synchronous machines which is produced at 27 military production factory by modifying the rotor of a 4-pole 380V, star connection, 50Hz, squirrel-cage induction motor. The modified SM starts as an IM and the continuous operation will be synchronous. Three studied cases are considered: Case 0) Design of the SM using conventional method.
Control variables Rated power Volume Damper winding
798
γ B (web/m2) D (m) AC L (m) S kVA Vm3 Ts E2 (V) Tr
case 0
case 1
case 2
1.6 0.54 0.0885 26500 0.111 3.3 0.00087 246 202.312 255
1.54 0.85 0.08 26268 0.1 3.37 0.00068 184 101.04 84
1.54 0.845 0.081 26268 0.103 3.374 0.00068 184 144 121
0.94
1.1
0.918
Zs damper
56
7
10
Vf (V)
220
23.38
220
diameter mm Zs Zf
1
0.79 104 1256
0.756 209 2513
150 3604
damper conductor per slot
Field winding
diameter mm
0.04
15 10 5 0
0
50 100 150 200 damper voltage per phase (volt)
0.03 15
0.01 0
0
10
20 30 field voltage (volt)
40
50
Figure 1: Field winding diameter verses field voltage for Case 1 2.5
5
0
0
1.5 1
50 100 damper voltage per phase (volt)
150
Figure 5 Damper conductors per slot versus damper voltage for Case 2
0.5 0
0
50
100 150 200 field voltage (volt)
250
Figure 2: Field conductor per slot versus field voltage for Case 1
3 2.5
4 3 2 1 0
0
50 100 damper voltage per phase (volt)
150
2 Figure 6: Damper winding diameter versus field voltage for Case 2
1.5
0.05 field winding diameter
damper winding diameter
10
2
damper winding diameter
field conductor per slot
250
Figure 4 Damper conductors per slot versus damper voltage for Case 1
0.02
damper conductor per slot
field winding diameter
0.05
20
1 0.5
0
50 100 150 200 damper voltage per phase (volt)
250
Figure 3: Damper winding diameter versus damper voltage for Case 1
0.04 0.03 0.02 0.01 0
0
10
20 30 40 field voltage (volt)
50
Figure 7: Field winding diameter versus field voltage for Case 2
799
[8]
field conductor per slot
2.5 2
[9]
1.5
[10]
1
[11]
0.5
0
10
20 30 40 field voltage (volt)
50
[12] [13]
Figure 8: Field conductors per slot versus field voltage for Case 2
VI. CONCLUSIONS This paper has been efficiently solved the problem of synchronous motor design using a particle swarm optimization technique. The results obtained with the designed procedure are compared with experience-based method. The proposed PSO technique offers some advantages over deterministic methods as: 1) Minimizing the motor volume from 870 cm3 using the conventional method to 680 cm3 using the proposed optimized design procedure. 2) Maximizing the apparent power compared to the conventional design method. 3) The total ampere conductor using the proposed optimization technique is saved by 0.88%. 4) The increased damper voltage leads to more reduction in damper winding diameter. While, The increased field voltage leads to more reduction in the field conductor per slot.
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[5] [6] [7]
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