Proceedings of the 14th International Middle East Power Systems Conference (MEPCON’10), Cairo University, Egypt, December 19-21, 2010, Paper ID 231.
Optimized Sizing of High Speed PM Generator for Renewable Energy Applications Adel El Shahat and Ali Keyhani Member, IEEE, Fellow, IEEE Department of Electrical and Computer Engineering The Ohio State University, Columbus, Ohio 43210, USA {ahmed.210 & keyhani.1}@osu.edu
Hamed M. El Shewy Electrical Power and Machines Department Faculty of Engineering, Zagazig University Zagazig, Egypt
[email protected] machine has better mechanical behaviour [31]. Seven configurations of both radial-flux and axial-flux machines give comparisons among PM wind generators of different topologies from 1 kW to 200 kW. Optimization of design data are done and verified by finite element analysis and commercial generator test results [32]. Aspects of PM motor technology and design of brushless PM machines are introduced in [12] and [13]; to be used in this paper.
Abstract-- This paper proposes optimized sizing of High Speed Permanent Magnet Synchronous Generators (HSPMSGs) for renewable energy applications to be driven by micro-turbines. These designs have more significant improvement in weights and volumes than usual or classical. This paper presents at the beginning a step by step sizing procedure of (HSPMSGs) to be used in optimization functions formulations. Then, Unconstrained Optimization for Total Losses Minimization is presented. Simple constrained optimized Total Mass with bounded constraints is introduced. Finally, Minimum Mass Genetic Sizing, Constrained by Minimum Losses is proposed. The optimization variables are the same in every optimization process. The optimized and genetic results are well depicted by some variables 3D figures for initial and detailed sizing. The simulation of generators sizing is performed using MATLAB Optimization Toolbox, and Genetic Algorithm.
II. BASIC SELECTIONS NdFeB (Neodymium Iron-Boron) is preferred because it is cheaper and more readily available. So, NdFeB magnets are selected for use in PMG, with some conservatively assumed values [13-15]. TM19, 29 gauge electrical silicon steel is selected for the PMG stator and rotor because it is economical [13], [16], its thin laminations minimize power losses due to the circulating eddy current, and because it has a saturation flux density of about 1.8 T [13], [14].
Index Terms-- Genetic, High-speed, Optimization, generators, Size.
I.
INTRODUCTION
High speed PM generators provide a substantial reduction in size and weight, also higher in power density; because as a machine’s speed increases, its size decreases for a given output power. The size, weight, and cost are from the major factors for successful design. The increase of the rotating speed of electrical machines is a way to improve their power / mass ratio and thus the dimensions and weight are reduced and the total efficiency is increased [1–5]. For high-speed applications, the rotor aspect ratio, defined as length-todiameter, is a critical parameter. Stator core losses may be minimized by using laminated steel in stator construction and by not generating frequencies that are too high. The main applications of PMSG are for power generation as part of renewable energy resources and main generators for aircraft, etc. [2–9]. The sizing of HSPMSG design must address system topology for good power/volume, low cost, and superior efficiency [1–14]. Optimum design of high speed PM alternator is proposed to be used in distributed power generation system with experimental tests are conducted to verify the FEM predictions [8], [9]. The high speed permanent magnet (PM) machine has been widely used in distributed power generation. The high speed generator distributed generation system in comparison with PM Doubly Fed Reluctance Generator for the same application has better electromagnetic property and the PM doubly fed reluctance
III. MACHINE DESIGN PARAMETERS A. Stator Mechanical Design Slotted stators consist of openings around the stator for the armature windings, as shown in Fig. 1 (a) are selected.
Fig. 1 Slotted Stator Design (a) and Stator Slot Geometry (b).
Slots are trapezoidal, but assumed here to be rectangular, as shown in Fig. 1 (b). They contain form-wound windings so that the depression width is the same as the top slot width. Slotting is used because of its advantages [14]. The initial design of the generator assumes a three-phase machine with 36 slots [13].
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B. Rotor Mechanical Design For high-speed applications, the rotor aspect ratio, defined as length-to-diameter (L/D), is a critical parameter. PM machines offer flexibility in selecting pole sizes, which allows for smaller diameters. A normal L/D ratio for a wound rotor machine is 0.5 – 1.0, and 1 – 3 for a PM machine [17]. The rotor radius and rotational speed also determine the tip speed of the machine, which is the surface velocity of the rotor. v tip = r ω
slot, along with insulation material. A slot fill factor (λs) is used to determine how much of the slot’s cross-sectional area is occupied by winding material, as in Eq. 6. λ
=
Winding Area Total Slot Area
where ωm = angular speed (rad/sec); r = rotor radius (m) The upper limit on tip speed is between 100-250 m/s, depending on the design of the of machines. In this design, a range of tip speed is taken to be (50:250).
G. Machine Calculated Parameters Each phase of the machine is modelled, as shown in Fig. 2.
C. Number of Poles and Magnets Pole Design An even number of poles is always used, (here pole pairs number = 3). Assuming a constant mechanical rotation speed, electrical frequency is given as. N ( 2 P ) = 120
Fig. 2 A Per Phase Electrical Model.
(2 )
f
Ra: Armature resistance; Ls: Synchronous inductance; Ea: Back e.m.f voltage and Va : Terminal voltage.
N = speed (rpm); P = pole pairs; f = electrical frequency (Hz) Magnet poles skew factor is selected to reduce cogging torque and smooth out variations in air gap reluctance, flux, and voltage waveforms. k
sn
=
sin( n θ s )
θ
H. Winding Resistances Copper phase windings resistance is calculated in Eq. 7.
(3 )
Ra =
s
2
D. Magnetic Dimensions
A ac =
The magnetic dimensions that affect a PM machine are air gap and magnet height. Air gap flux density (Bg) represents in Eq. 4. The radial air gap is made as small as possible to maximize the air gap flux density, minimize the flux leakage, and to produce a lower reluctance value. g
=
hm B hm + g
r
l
σ * A ac
(7 )
where l: length of conductor, σ: winding conductivity Aac: winding cross – sectional area
where θs: Skew angle, rad E; n: Harmonic number
B
(6 )
Typically, machines contain two coils sides per slot, making the winding a double-layer design [13]. Overall, slot fill factors vary in value from 0.3 – 0.7, depending on the number and size of the conductors in the slots, as well as the amount of labour utilized. A slot fill factor of 0.5 is assumed.
(1 )
m
s
As * λ s 2* NC
(8 )
where As: slot Area, NC: turns per coil But the above stator resistance equation may be used as in low frequencies applications, so it has to be developed. Since the machine rotates at high speed, and high frequency and so the skin depth may be affected. In conductors that carry high frequency currents, skin effect can become an issue and affect the operation of the machine. Skin effect is caused by eddy currents in the windings themselves due to the changing magnetic field. These eddy currents force the current flowing in the conductor to crowd to the outer edges of the conductor. This in turn causes the current to flow through a smaller cross – sectional area and increase the resistance of the conductor. It is well known that, when conductive material is exposed to an ac magnetic field, eddy currents are induced in the material in accordance with Lenz’s law. The power loss resulting from eddy currents which can be induced in the slot conductors appears as an increased resistance in the winding. To understand this phenomenon, let us consider a rectangular conductor as shown in fig. 3. The average eddy current loss in the conductor due to a sinusoidal magnetic field in the y direction is given approximately by Hanselman [12].
(4 )
where hm: Magnet height (mm); g: Air gap (mm); Br: Magnet Remnant Flux Density (T) Magnets losses are reduced, using smaller magnets. For uniform magnetic fields, the magnet height is usually larger than the air gap, by a factor 5 – 10. E. Slots Per Pole, Per Phase Three-phase machines are typically used in this study as the standard choice for most motors and generators. Another important design parameter is the number of slots per pole, per phase (m), as in Eq. 5. N s (5 ) m = 2 * P * q
F. Stator Windings The pitch of a winding (α) refers to the angular displacement between the sides of a coil. The breadth of a stator winding results from the coils occupying a distribution of slots within a phase belt. In smaller machines, coils are composed of round insulated wires that are placed in the stator
Fig. 3 Rectangular conductor geometry
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=
P ec
1 σ L ω c h 3 ω 2 u 02 H 12
2 m
(9 )
k
Hm: the turn field intensity value; u0: perm. of free space. Since skin depth is defined as 2
δ =
Lω ch3 H 6 σδ 4
2 m
2 s
d Lh n Pe = ( s )I 9 σδ 4 ω s
2
sl
=
ρn L k cp ω s d
s
= R sl + R ec = R sl (1 + Δ e )
e
≡
R ec 1 ds 2 h 2 = ( ) ( ) 9 δ δ R sl
= k
pn
* k
k
bn
)]
(19 )
(20 )
= ω 0λ
Bg =
(23 )
k l Cφ * Br u 1 + k r * rec PC
(21)
hm ge * Cφ
(24 )
a
(25 )
= 2* P * Nc
s
( 15 )
ge: effective air gap; ws: average slot width; wt: tooth width Here, a leakage factor (Kl ~ 0.95) and a reluctance factor (Kr~1.05) are both used for surface magnets. The presence of the slots in the stator also affects the air gap flux density because of the difference in permeance caused by the slots. Carter’s coefficient (kc) is used to account for this effect [12]. k c = [1 −
1
τs
g + 1) * (5 * ws ws
]−1
(27 )
The terminal voltage (Va) is calculated from the internal voltage (Ea), and the synchronous reactance voltage drop. The armature resistance is usually ignored because it is much smaller than synchronous reactance. The voltage is found as a relation in output power (Pwr), e.m.f, and reactance from the resulting quadratic equation. Va =
2
BB =
The breadth factor explains the effect of the windings occupying a distribution or range of slots within a phase belt. The breadth factor is derived in Eq. 18. n * m * γ sin( ) 2 = n * γ m * sin( ) 2
1 − np
) +
where Nc : Turns per coil; Na : Number of armature turns (each slot has 2 half coils) (26 ) τ = ws + wt ; g e = k c * g
Short-pitching is an important means for eliminating harmonics and improving the power quality of the machine. The pitch factor is shown in Eq. 17. n *α n *π (17 ) k pn = sin( ) * sin* ( ) 2
− R v2
np + 1
(22)
N
(14 )
(16 )
bn
1 − np
− R1
PC: permeance coeff.; CΦ: flux concentration factor (Am/Ag)
I. Winding and Magnet Factors Winding are short-pitched and have breadth associated with them. To account for these effects, a winding factor (kw) is utilized, as in Eq. 16. wn
* (R1
np + 1 2
2 * Rs * Lst * Na * kw * ks * B1 P 4 Pθ B1 = * Bg * k g * sin( m ) π 2 where θm: magnet physical angle
PC =
This result shows that the resistance increases not only as a function of the ratio of the conductor height to the skin depth but also as a function of the slot depth to the skin depth. Thus, to minimize ac losses, it is desirable to minimize the slot depth as well as the conductor dimension. For a fixed slot crosssectional area, this implies that a wide but shallow slot is best.
k
np ) * (R np + 1
where urec: recoil permeability; Br: remnant flux density
In this equation, Δe = Rec/Rsl is the frequency-dependent term. Using Eq. (13) and Eq. (12), this term simplifies to Δ
2 np s
( 13 )
L: the slot length; kcp: the conductor packing factor, is the ratio of cross sectional area occupied by conductors to the entire slot area and ρ: electrical resistivity (Ω.m). Using Eq. (13), the total slot resistance can be written as R st
* [(
2 np
λ=
I: rms conductor current; ωs: Slot width (m); ds: Slot depth (m) The slot resistance of a single slot containing ns conductors connected in series is R
− Ri
s
Ea
(12 )
2 s
R
np − 1
J. Flux, and Voltage For useful voltage, only the fundamental components are used to determine the internal voltage (back e.m.f) of the generator, as shown in Eq.s 20, 21, and 22.
(11 )
Using this expression it is possible to compute the ac resistance of the slot conductors. If the slot conductors are distributed uniformly in the slot, and substituting the field intensity into Eq. (11) and summing over all ns conductors gives a total slot eddy current loss of 2
Ri 2 np
Rs: outer magnetic boundary, R2: outer boundary of magnet; Ri: inner magnetic boundary, R1: inner boundary of magnet
Eq. (9) can be written as Pec =
=
np * R np − 1 ( 10 )
ω u 0σ
gn
2 2; X s Pwr − E a CC 3
− BB +
=
2 X 9
BB 2 2 s
2
P wr
− 4 CC
(28 )
2
K. Machine Inductances In a slotted PM machine, there are three distinct components of inductance: the largest, air gap inductance slot leakage inductance, and the smallest, end-turn inductance. The total inductance for the phase is the sum of the three inductances, ignoring other small factors. (29 ) L s = L ag + L slot + L e ; X s = ω0 * L
(18 )
where m: slots per pole per phase; γ: coil electrical angle The magnetic flux factor equation [12], for the slotted stator and surface magnet configuration is shown in Eq. 19.
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The air gap inductance is given by Eq. 30. λ
L ag =
2
=
i
2
u * R s * L st * N a * k wn 4 q * * 0 2 nπ n 2 * P 2 * (g + hm )
PR c ( ω ) =
(30 ) 8 c Fe k Fe (
L am = 2 * P * L st * Perm * N s p * N c
(32 )
2
2
2
Las = 2 * P * Lst * Perm*[4 * N c (m − N s p ) + 2 * N s p * N c ](33)
A slot permeance per unit length is shown in Eq. 34. Perm
=
1 h h * s + d 3 w st wd
e
=
u
0
* N
c
* N 2
2 a
*τ
s
* ln(
τ
s * π 2 * A
)
(35 )
The coefficient of friction can be approximated by Eq. 41.
s
C
L. Basic Losses Losses in a machine consist of core losses, conductor losses, friction and windage losses, and rotor losses. Rotor losses will be discussed later. Stator core losses, per weight, can be greater than normal in machines because of higher frequencies. These losses are minimized by using laminated steel in stator construction and by not generating frequencies that are too high. Core losses consist of hysteresis and eddy current losses. The best way to approximate core losses is to use empirical loss data. An exponential curve fitting is applied to the empirical data for M-19, 29 gauge material, in order to obtain an equation for estimating core losses, as in Eq. 36, with constant values in [19]. PC = P0 * (
B εB f ) * ( )ε B0 f0
f
)
2 a
For rotors operating at high speed, the friction and windage in air can cause losses which result in inefficiency and heat production. These losses are calculated, using the power necessary to overcome the drag resistance of a rotating cylinder, as given by Eq. 40 [20]. Pwind = C f * π * ρ air * ω 3 * R 4 * L st (40 )
(34 )
The end turn inductance is introduced in Eq. 35, assuming the end turns are semi-circular, with a radius equal to one-half the mean coil pitch. L
ω0
1 .5
2
Where Rc: core resistance, cFe: correction factor for iron loss calculation, bst: stator tooth width, kFe: specific iron loss, mst: stator teeth mass, βslot: slot angle, hsy: stator yoke height When this core loss resistance is depicted in an equivalent circuit, it should be noted that the resistance is frequency dependent. The conductor losses are found, using Eq. 39. Pa = q * I a2 * R a (39 )
(31 )
L slot = L as − L am ; ( 3 phase )
1
( 37 )
ω 3 π L st N ( 38 ) p β slot 2 1 2 1 2 ( ) { m st ( ) + m sy ( ) } B0 b st h sy 2
R c (ω ) =
The slot leakage inductance is presented in Eq. 31. Assume the slot is rectangular with slot depressions, as in Fig. 1, and assume (m) slots per pole per phase, with a standard double layer winding.
V a2 Rc
f
≅ 0 . 0725 * Re y − 0 .20
(41 )
where Rey: Reynold’s Number IV. SIZING METHOD For the basic sizing calculations, an air-cooled generator is assumed with 10 psi [13], [21]. The machine power equation is utilized to derive the rotor radius and stack length of the machine, as in Eq. 42. (42 ) Pwr = 2 * π * r * L st * v tip * τ where r = rotor radius; Lst = stack length; τ = shear stress (psi) The L/D ratio is substituted for Lst. Using shear stress, rotor tip speed, and machine power rating range, the power equation is calculated to obtain rotor radius and stack length, while matching the desired rotational speed of the machine with a L/D ratio. Using a pole pair value of 3, a slot height of 10 mm, and a slot fill fraction of 0.5, the frequency is found. Once the basic sizing of the machine is complete, in-depth analysis is conducted to obtain the overall performance within the power range of 5: 500 kW generators. Using the equations presented in previous sections, all the detailed parameters can be obtained. The lengths, volumes, masses, and overall generator parameters are calculated, using basic geometric equations and relationships. A 15% service mass fraction is added to the total mass estimate to account for the additional services associated with machines cooling [14], [21]. Once the mass of each of the stator parts is known, core losses are estimated. The calculation of lengths, volumes, and weights are presented. The mass of armature conductors, core mass, magnet mass, and shaft mass are calculated to give the total mass value. Finally, stator resistance, terminal voltage, current, loss types, input power, and efficiency are calculated.
(36 )
where P0: Base power; B0: Base flux density; εB: Flux density exponent; f0: Base frequency; εf: Frequency exponent The above commonly used equation considering hysteresis and eddy-current loss is not completely satisfactory, because the measured iron loss is much higher than theoretically calculated. This is so because it assumes a homogenous magnetization of the laminations, which is not a valid representation of what happens during the magnetization process. The loss caused by the movements of the magnetic domain walls is higher than the loss calculated with the commonly used equation. The difference between measured and calculated loss is called the excess loss or the anomalous loss. Sometimes, this anomalous or excess loss is considered as a third contribution to the iron loss. Great efforts have been made to calculate this excess loss, because of the complexity of the domain patterns. For reasons mentioned before, it is useful to represent the core loss by core loss resistance, which is placed in equivalent circuit. The core loss resistance is connected across the voltage Va. Therefore, the power dissipated in this resistance is [22-30].
V. UNCONSTRAINED NONLINEAR OPTIMIZATION
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A. Trust-Region Methods Many of the methods used in Optimization Toolbox solvers are based on trust regions, a simple yet powerful concept in optimization. To understand the trust-region approach to optimization, consider the unconstrained minimization problem, minimize f(x). Suppose we are at a point x in n-space and we want to improve. The basic idea is to approximate f with a simpler function q, which reasonably reflects the behavior of function f in a neighborhood N around the point x. This neighborhood is the trust region. A trial step s is computed by minimizing over N. This is the trust - region sub - problem, min s {q(s), s ∈ N} [33]. The current point is updated to be x + s if f (x + s) < f (x); otherwise, the current point remains unchanged and N, the region of trust, is shrunk and the trial step computation is repeated. The key questions in defining a specific trust – region approach to minimizing f(x) are how to choose and compute the approximation q (defined at the current point x), how to choose and modify the trust region N, and how accurately to solve the trust – region sub – problem. In the standard trust-region method, the quadratic approximation q is defined by the first two terms of the Taylor approximation to F at x; the neighborhood N is usually spherical or ellipsoidal in shape. Mathematically the trust – region sub – problem is typically stated min{
1 T s Hs + s T g _ such 2
_ that
Ds
≤ Δ
where m = Σx(i)/n, i = 1...n, and calculate f(r). 4. If f(x(1)) ≤ f(r) < f(x(n)), accept r and terminate this iteration. Reflect 5 .If f(r) < f(x(1)), calculate the expansion point s s = m + 2(m – x(n+1)), and calculate f(s). a. If f(s) < f(r), accept s and terminate the iteration. Expand b. Otherwise, accept r and terminate the iteration. Reflect 6. If f(r) ≥ f(x(n)), perform a contraction between m and the better of x(n+1) and r: a. If f(r) < f(x(n+1)) (i.e., r is better than x(n+1)), calculate c = m + (r – m)/2 and calculate f(c). If f(c) < f(r), accept c and terminate the iteration. Contract outside Otherwise; continue with Step 7 (Shrink). b. If f(r) ≥ f(x(n+1)), calculate cc = m + (x(n+1) – m)/2 and calculate f(cc). If f(cc) < f(x(n+1)), accept cc and terminate the iteration. Contract inside Otherwise; continue with Step 7 (Shrink). 7. Calculate the n points v(i) = x(1) + (x(i) – x(1))/2 and calculate f(v(i)), i = 2,...,n+1. The simplex at the next iteration is x(1), v(2),...,v(n+1). Shrink C. Total Losses Minimization Sizing Using the previous technique the total losses could be minimized. The first step is to choose the optimizing variables x1, x2, x3, these variables here are L/D ratio, Rotor radius, and Stack length respectively. These variables will be stayed also with constrained ones, also with Genetic Algorithm. Second step is to formulate the total losses function to be minimized as fitness or objective function in the form of MATLAB m – file. Then, third step, using optimization tool box GUI with a proper choice for initial variables values also the accuracy with suitable number of iteration. Some sizing relations examples for this optimization problem are presented in the following figures. These examples at 500 kW, and 400 kW with tip speeds of 250, 200 and 150 m/s. Using x1, x2, and x3; all detailed parameters could be obtained. The following figures show the examples.
( 43 )
where g is the gradient of f at the current point x, H is the Hessian matrix (the symmetric matrix of second derivatives), D is a diagonal scaling matrix, Δ is a positive scalar, and . is the 2-norm. Good algorithms exist for solving previous equation; such algorithms typically involve the computation of a full Eigen system and a Newton process applied to the secular Equation [33] 1 1 − =0 Δ s
(44)
Such algorithms provide an accurate solution to the equation. However, they require time proportional to several factorizations of H.
(45)
+P +P PTotal _ Losses = P Core Conductor Wind
Minimized Total Losses
B. Fminsearch Algorithm Fminsearch uses the Nelder-Mead simplex algorithm as described in [34]. This algorithm uses a simplex of n + 1 points for n-dimensional vectors x. The algorithm first makes a simplex around the initial guess x0 by adding 5% of each component x0(i) to x0, and using these n vectors as elements of the simplex in addition to x0. (It uses 0.00025 as component i if x0(i) = 0.) Then the algorithm modifies the simplex repeatedly according to the following procedure. 1. Let x(i) denote the list of points in the current simplex, i = 1,...,n+1. 2. Order the points in the simplex from lowest function value f(x(1)) to highest f(x(n+1)). At each step in the iteration, the current worst point x(n+1) is discarded, and another point is accepted into the simplex (or, in the case of step 7 below, all n points with values above f(x(1)) are changed). 3. Generate the reflected point r = 2m – x(n+1),
7000 6000 5000 4000 3000 5 4.8
250
4.6 4.4
5
x 10
200
4.2 4
Output Power (Watt)
150
Tip Speed (m/s)
Fig. 4 Total Minimum Losses with Tip Speed and Output Power relations 3
X1
2.5 2 1.5 1 5
4.8 5
x 10
4.6
250 4.4
Output Power (Watt)
4.2
200 4
150
Tip Speed (m/s)
Fig. 5 X1 with Tip Speed and Output Power relations
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When trying to use unconstrained optimization with this case, from the beginning the maximum value for L/D ratio exceeds its upper limit. So, some simple constraints are used in this case as bounds from 0.0 to 3.0. The function used here is presented as follow.
0.044 0.042
X1
0.04 0.038 0.036 5
4.8
250
4.6
4.4
5
200
4.2
x 10
4
A. fmincon Active Set Algorithm In constrained optimization, the general aim is to transform the problem into an easier sub – problem that can then be solved and used as the basis of an iterative process. A characteristic of a large class of early methods is the translation of the constrained problem to a basic unconstrained problem by using a penalty function for constraints that are near or beyond the constraint boundary. In this way the constrained problem is solved using a sequence of parameterized unconstrained optimizations, which in the limit (of the sequence) converge to the constrained problem. These methods are now considered relatively inefficient and have been replaced by methods that have focused on the solution of the Karush-Kuhn-Tucker (KKT) equations. The KKT equations are necessary conditions for optimality for a constrained optimization problem. If the problem is a socalled convex programming problem, that is, f(x) and Gi(x), i = 1,...,m, are convex functions, then the KKT equations are both necessary and sufficient for a global solution point. The Kuhn-Tucker equations can be stated as
150
Output Power (Watt)
Tip Speed (m/s)
Fig. 6 X2 with Tip Speed and Output Power relations 0.25
X3
0.2 0.15 0.1 0.05 5
4.8
250
4.6
4.4
5
200
4.2
x 10
4
150
Output Power (Watt)
Tip Speed (m/s)
Fig. 7 X3 with Tip Speed and Output Power relations 4
x 10
Speed (rpm)
6 5 4 3 5
4.8
4.6
5
250 4.4
x 10
4.2
200 4
150
Output Power (Watt)
Tip Speed (m/s)
Overall Machine Length (m)
Fig. 8 Speed (rpm) with Tip Speed and Output Power relations 0.35 0.3 0.25
m
0.2 5
4.8
4.6
5
∇f ( x*) + ∑ λi .∇Gi ( x*) = 0 i =1
250 4.4
x 10
4.2
200 4
λi .Gi ( x*) = 0, i = 1,........., me λi ≥ 0, i = me + 1,.............,m, (46) The first equation describes a canceling of the gradients between the objective function and the active constraints at the solution point. For the gradients to be canceled, Lagrange multipliers (λi, i = 1,...,m) are necessary to balance the deviations in magnitude of the objective function and constraint gradients. Because only active constraints are included in this canceling operation, constraints that are not active must not be included in this operation and so are given Lagrange multipliers equal to 0. This is stated implicitly in the last two Kuhn-Tucker equations. The solution of the KKT equations forms the basis to many nonlinear programming algorithms. These algorithms attempt to compute the Lagrange multipliers directly. Constrained quasi-Newton methods guarantee super linear convergence by accumulating second-order information regarding the KKT equations using a quasi-Newton updating procedure. These methods are commonly referred to as Sequential Quadratic Programming (SQP) methods, since a QP sub problem is solved at each major iteration (also known as Iterative Quadratic Programming, Recursive Quadratic Programming, and Constrained Variable Metric methods). The 'active-set' algorithm is not a large-scale algorithm.
150
Output Power (Watt)
Tip Speed (m/s)
Overall Machine Diameter (m)
Fig. 9 Machine Length with Tip Speed and Output Power relations 0.17 0.165 0.16 0.155 0.15 5
4.8
4.6
5
250 4.4
x 10
4.2
200 4
150
Output Power (Watt)
Tip Speed (m/s)
Fig. 10 Machine Diameter with Tip Speed and Output Power relations
T o tal M ass (kg )
30 25 20 15 10 5
4.8
4.6
5
250 4.4
x 10
4.2
200 4
150
Output Power (Watt)
Tip Speed (m/s)
Fig. 11 Total Mass with Tip Speed and Output Power relations
Efficiency
1 0.995 0.99 0.985 5
4.8 5
4.6
250 4.4
x 10
Output Power (Watt)
4.2
200 4
150
Tip Speed (m/s)
Fig. 12 Efficiency with Tip Speed and Output Power relations
VI. SIMPLE CONSTRAINED OPTIMIZATION
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B. Large-Scale vs. Medium-Scale Algorithms An optimization algorithm is large scale when it uses linear algebra that does not need to store, nor operates on, full matrices. This may be done internally by storing sparse matrices, and by using sparse linear algebra for computations whenever possible. Furthermore, the internal algorithms either preserve sparsity, such as a sparse Cholesky decomposition, or do not generate matrices, such as a conjugate gradient method. Large-scale algorithms are accessed by setting the Large Scale option to on, or setting the Algorithm option appropriately (this is solver-dependent). In contrast, medium-scale methods internally create full matrices and use dense linear algebra. If a problem is sufficiently large, full matrices take up a significant amount of memory, and the dense linear algebra may require a long time to execute. Medium-scale algorithms are accessed by setting the Large Scale option to off, or setting the Algorithm option appropriately (this is solver-dependent). Don’t let the name “large-scale” mislead you; you can use a large-scale algorithm on a small problem. Furthermore, you do not need to specify any sparse matrices to use a large-scale algorithm. Choose a medium-scale algorithm to access extra functionality, such as additional constraint types, or possibly for better performance.
0.04
X2
0.03 0.02 0.01 5
3
X3
0.1
2
4
200
2
1
150
Output Power (Watt)
Tip Speed (m/s)
Fig. 16 X3 with Tip Speed and Output Power relations 5
x 10
Speed (pm)
2 1.5 1 0.5 0 5
4
250
3
200
2
5
x 10
1
Output Power (Watt)
150
Tip Speed (m/s)
Overall Machine Length (m)
Fig. 17 Speed (rpm) with Tip Speed and Output Power relations 0.5 0.4 0.3 0.2 0.1 5
4
250
3
200
2
5
x 10
1
150
Output Power (Watt)
Tip Speed (m/s)
Overall Machine Diameter (m)
Fig. 18 Overall Machine Length with Tip Speed and Output Power 0.16
0.14
0.12
0.1
0.08 5
4.5
4
3.5
3
2.5
2
5
x 10
1.5
1
150
Output Power (Watt)
160
170
180
190
200
210
220
230
240
250
Tip Speed (m/s)
Fig. 19 Overall Machine Diameter with Tip Speed and Output Power
(47 )
10000 8000 6000 4000 2000 5
4
3
5
x 10
250 2
Output Power (Watt)
200 1
250
3
5
x 10
Total Losses (Watt)
Minimum Mass (kg)
5
Tip Speed (m/s)
0.15
0.05 5
250
200 1
150
Tip Speed (m/s)
Fig. 20 Total Losses with Tip Speed and Output Power relations
150
Output Power (Watt)
150
0.2
10
x 10
1
0.25
20
4
200
2
Fig. 15 X2 with Tip Speed and Output Power relations
30
0 5
250
3
Output Power (Watt)
C. Total Mass Minimization Sizing The previous technique with the described function above could be used for minimizing the total mass. The optimizing variables x1, x2, x3, these variables are the same. Then formulate the total mass function to be minimized as fitness function in the form of MATLAB m – file. Later, using optimization tool box GUI with a proper choice for initial variables values also the accuracy with suitable number of iteration. Also, it should be notified that, Large scale (trust region) method does not currently solve this type of problem, but using medium scale (line search) instead. Some illustrating examples are proposed to show the validity of this minimization. M Total = M Core + M Magnet + M Shaft + M Conductor + M Service
4 5
x 10
Tip Speed (m/s)
Fig. 13 Minimum Mass with Tip Speed and Output Power relations Efficiency
1 3
X1
2.9 2.8 2.7
0.96 0.94 5
2.6 5
x 10
0.98
4
3
5
Output Power (Watt)
250 2
1
4
3
5
x 10
200 150
Output Power (Watt)
250 2
200 1
150
Tip Speed (m/s)
Fig. 21 Efficiency with Tip Speed and Output Power relations
Tip Speed (m/s)
Fig. 14 X1 with Tip Speed and Output Power relations
VII.GENETIC ALGORITHM HSPMSG SIZING
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lower, as a nonlinear constraint with total mass function as objective function to be minimized. This also will reduce the machine cost. The following introduce the detailed result of this prescribed example in TABLE I:
A. Description of the Nonlinear Constraint Solver The genetic algorithm uses the Augmented Lagrangian Genetic Algorithm (ALGA) to solve nonlinear constraint problems. The optimization problem solved by the ALGA algorithm is min x f (x), such that ci (x) ≤ 0, i = 1 ….. m (48) ceqi (x) = 0, i = m+1 … mt A.x≤ b Aeq . x = beq lb ≤ x ≤ ub, Where c (x) represents the nonlinear inequality constraints, ceq (x) represents the equality constraints, m is the number of nonlinear inequality constraints, and mt is the total number of nonlinear constraints. The Augmented Lagrangian Genetic Algorithm (ALGA) attempts to solve a nonlinear optimization problem with nonlinear constraints, linear constraints, and bounds. In this approach, bounds and linear constraints are handled separately from nonlinear constraints. A sub problem is formulated by combining the fitness function and nonlinear constraint function using the Lagrangian and the penalty parameters. A sequence of such optimization problems are approximately minimized using the genetic algorithm such that the linear constraints and bounds are satisfied. A sub - problem formulation is defined as m
mt
i =1
i = m+1
Θ( x, λ, s, ρ ) = f ( x) − ∑ λi si log(si − ci ( x)) + ∑ λici ( x) +
ρ
mt
∑ c (x)
2 i =m+1
i
2
Output Power 400000 Min. Mass 16.9503 Tooth Width 0.0053
(49)
where the components λi of the vector (λ) are nonnegative and are known as Lagrange multiplier estimates. The elements si of the vector (s) are non – negative shifts, and ρ is the positive penalty parameter. The algorithm begins by using an initial value for the penalty parameter (Initial Penalty). The genetic algorithm minimizes a sequence of the sub – problem, which is an approximation of the original problem. When the sub - problem is minimized to a required accuracy and satisfies feasibility conditions, the Lagrangian estimates are updated. Otherwise, the penalty parameter is increased by a penalty factor (Penalty Factor). This results in a new sub – problem formulation and minimization problem. These steps are repeated until the stopping criteria are met [35], [36].
TABLE I DETAILED SIZING PARAMETERS Tip Speed X1 = 1.5317 200 X2 = 0.0388 X3 = 0.1189 Total Loss Rotor 4.5192e+003 Diameter 0.0776 Slot Top Width Slot Bottom 0.0053 Width 0.0045
End Turn Travel (one end) 0.0289 Average Slot Width 0.0049
Width of Slot and Tooth 0.0102
Magnetic Flux 0.0641
Internal Voltage 700.6569
Slot Area 4.9057e005 Armature Conductor Area 1.2264e005 Core Outside Radius 0.0803 Core Mass 5.7961 Arm. Resistance 0.0098 Teeth Loss 966.9290 Current 203.8567
Total Inductance 1.4327e-005 Mass of Armature Conductor 2.3611
End Length (half Coil) 0.0908
Air Gap Inductance 1.0987e-005 Total Reactance 0.2215 Overall Mach. Length. 0.1767
Speed(rpm) 4.9205e+004 Stator Core Back Iron Depth 0.0091 Magnetic Gap Factor 1.1404 Air Gap Magnetic Flux Density 0.8566 Slot Leakage Inductance 3.3814e-006 Armature Conductor Length 7.2104 Core Inside Radius 0.0712
Overall Mach. Diameter 0.1605
Back Iron Mass 3.9461
Teeth Mass 1.8500
Magnet Mass 2.2489 Tooth Flux Density 1.7132
Shaft Mass 4.3333 Back Iron Flux Density 1.2237 Term. Voltage 684.0544 Efficiency 0.9888
Service Mass 2.2109 Core Back Iron Loss 1.0107e+003 Wind Loss. 1.3199e+003
Total Core Loss 1.9776e+003 Conductor Loss 1.2216e+003
VIII.
B. Min. Mass Genetic Sizing, Constrained by Min. Loss
End Length (Axial direction) 0.0289 Eff. Air Gap 0.0024
Frequency 2.4602e+003
CONCLUSIONS
Unconstrained optimization for the minimization of total loss is performed; this function of minimizing total losses is implemented and results are presented. The optimizing variables are rotor length to diameter ratio, rotor radius, and stack length, for each of the functions, in both constrained and unconstrained optimization and in the genetic algorithm. The constrained optimized total mass with some constraints is set up to keep the total mass to be minimized. It should be noted that the large-scale trust-region method does not currently solve this type of problem, but using medium scale line search has provided an acceptable performance. The results of the genetic algorithm are presented with the same optimization variables, as before, but the fitness functions, in which the constraints are varied. We have also presented minimum mass genetic sizing, constrained by minimum losses. The same optimizing variables and the same bounds are
One case from the previous examples is proposed, using the total mass function (m.file) as a fitness function in GA with nonlinear constraint as minimum losses. The same optimizing variables with a simple constraints that are [1 0 0] as lower limit, and [3 1 1] as upper limit, also using non – linear constraint function with the aid of minimum losses values obtained from the part of minimum losses sizing one. The example here, selected randomly from the previous parts at tip speed = 200 m/s, and output power = 400 kW. The non – linear constraint function, which governs the optimization process, is implemented using the minimum losses value at the same values of tip speed, and output power. This design proposes machine sizing for minimum mass, under the condition of minimum total losses equal or under the minimum total losses value. This means make a benefit from limiting the machines losses at its absolute minimum value or
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used in the genetic example, with non-linear constraint function of minimum losses. This is done under the condition of minimum total losses, equal to, or less than, the minimum total losses value. We have observed that this will have the benefit of limiting machines losses. In our study, we have found that a noticeable improvement appears in the performance parameters.
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