The optimal current in the non-ideal
three-phase system with unsteady parameters
Krzysztof Dębowski
Silesian University of Technology, Institute of Electrical Engineering and Computer Science
Akademicka 10, 44-100 Gliwice, Poland, e-mail: Krzysztof.Debowski@polsl.pl
The symbol “ ” in the power constraint means that power condition (2) (and more particular (4)) should be fulfilled in fuzzy sense, namely, it has not to be fulfilled strictly because of unsteady parameters of the system. For the following values of membership function the corresponding values of are determined with figurative signs: the symbol means the lower limit of for ; the symbol means the upper limit
of for ; the symbol means the lower limit of for ; the symbol means the upper limit of for
; etc. By application of the above mentioned symbols the solution of the problem (optimal currents with minimum RMS values) is obtained by minimization of Lagrange’s functional formulated as follows:
The minimum of the Lagrange’s functional can be determined by means of Kuhn-Tucker conditions:
and then the optimal currents can be described as:
Lagrange’s multipliers should be determined in the numerical way. The obtained optimal currents: (9) have the minimal RMS values in fuzzy sense.
III. The example
It is considered system as in Fig.1 with source voltages:
presented in Fig.2., where RMS values of the voltage can change with intervals of ±5% and can be described as:
with membership functions as fuzzy numbers presented in Fig. 3. The impedances of the source are described in the frequency domain in the form as
follows: (fuzzy numbers presented in Fig.5 and Fig. 6).
The load impedances are described as follows: (fuzzy numbers presented in Fig. 7 and Fig. 8).
Before optimization, according to changes of the voltage source (voltages and inner impedance) and load impedances, the active power consumed by three-phase load has the values as presented in Fig. 4, where characteristic points have following values:
Pα=0, L= 17 529 W; Pα=0.5, L= 19 239 W; Pα=1, L= Pα=1, U= 21 125 W; Pα=0.5, U= 23 185 W; Pα=0, U= 25 465 W.
Before minimisation for characteristic values of active power: Pα=0, L, Pα=0.5, L, Pα=1, L= Pα=1, U, Pα=0.5, U, Pα=0, U, the corresponding norms of the current can be determined as:
where generally the norm of the current is described as:
After optimization the optimal (active) currents can be obtained as: and the norm of the current of the system:
I. INTRODUCTION.
In some papers presented before the description by means of fuzzy numbers has been applicated in relation to electrical parameters [1, 2, 3]. In monograph [4] has been presented the method of fuzzy optimization in one-phase system with unsteady parameters of the source and also in three-phase system with unsteady parameters of the load. This paper is a generalization - it is considered the system with unsteady parameters of the source as well of the load in the three-phase four-wire system.
II. FORMULATION AND SOLUTION OF THE PROBLEM
Fig. 1. The system with unsteady parameters
The considered system with unsteady parameters has been presented in Fig. 1. This system can represent the nonlinear changes of supply electrical networks as well the changes of load parameters during normal technological process e.g. for arc and resistance-arc furnaces. In the paper unsteady parameters (voltage and inner impedances of the source, impedances of the load) are described by means of fuzzy numbers. Generally fuzzy systems are based on the knowledge of membership functions. In the paper have been considered the most natural membership functions with Gaussian curves. The waveforms of source voltages are presented in Fig. 2. The considered problem can be formulated as minimization of RMS values with reference to currents of three-phase system:
(1) taking into consideration the additional power constraint: (2)
where: - the matrix of source voltages;
- voltages as fuzzy numbers with
initial phases (Fig.3);
- the matrix of source currents;
- currents of the system; - inner impedances of the source as fuzzy numbers; - the matrix of inner impedances of the source.
- the active power consumed by the load before optimization, considered as the fuzzy number.
The additional condition (2) guarantees that the active power consumed from non-ideal voltage source will be the same before and after optimization.
Generally, in this optimization determined components of optimal currents (real and imaginary parts) are real (not fuzzy) numbers and these values are considered as a compromise with reference to shapes of membership functions. In order to obtain the optimal currents in the system the appropriate compensating system will be required. The regulation system of compensating system should be parameterized to keep the optimal currents.
Taking into consideration the above mentioned symbols the minimization problem (1) can be formulated as follows:
(3) and power constraint (2) can be expressed in the form:
16000 18000 20000 22000 24000 0 4 8 12 16 20 24
Power demand in Polish national electrical system during 24h
P
t
, min minJ I 2 i ={
~ ~}
~, Re EI*T − IZZI*T ≈ P[
L1 L2 L3]
~ , ~ , ~ ~ E E E = E 3 2 1 j 3 L 3 L j 2 L 2 L j 1 L 1 L ~ ~ ; ~ ~ ; ~ ~ ψ ψ ψ e E E e E E e E E = = = 3 2 1,ψ ,ψ ψ[
IL1,IL2,IL3]
= I 3 L 3 L 3 L 2 L 2 L 2 L 1 L 1 L 1 L Ia jIb ; I Ia jIb ; I Ia jIb I = + = + = + = = L3 L2 1 L T ~ 0 0 0 ~ 0 0 0 ~ ~ ~ Z Z Z Z Z Z Z Z Z Z L3 L3 L3 L2 L2 L2 1 L 1 L 1 L ~ j ~ ~ ; ~ j ~ ~ ; ~ j ~ ~ Z Z Z Z Z Z Z Z Z R X Z R X Z R X Z = + = + = + P~(
2)
3 L 2 2 L 2 1 L 2 min min minJ I I I I i + + = =(
)
~(
)
~(
)
~ (4) ~ sin ~ cos ~ sin ~ cos ~ sin ~ cos ~ 2 3 L 2 L3 L3 2 L2 2 L2 L2 2 1 L 2 1 L 1 L 3 3 L L3 3 3 L 3 L 2 2 L 2 L 2 2 L 2 L 1 1 L 1 L 1 1 L 1 L P I I R I I R I I R I E I E I E I E I E I E b a Z b a Z b a Z b a b a b a ≈ + − + − + − + + + + + + ψ ψ ψ ψ ψ ψ ≈ E µ E Eα=0,L E µE =α =0 Eα=0,U E µE =α =0 Eα=0.5,L E µE =α =0.5 Eα=0.5,U E 5 . 0 = =α µE(
)
(
)
(
)
(
)
(
)
(
)
] sin cos sin cos sin cos [ ) 5 ( ] sin cos sin cos sin cos [ ) , , , , , , ( U , 2 3 L 2 L3 L3 L, , 2 L2 2 L2 L2 L, , 2 1 L 2 1 L 1 L L, , 3 3 L L3 U, , 3 3 L 3 L U, , 2 2 L 2 L U, , 2 2 L 2 L U, , 1 1 L 1 L U, , 1 1 L 1 L U, , 1 0 , L , 2 3 L 2 L3 L3 U, , 2 L2 2 L2 L2 U, , 2 1 L 2 1 L 1 L U, , 3 3 L L3 L, , 3 3 L 3 L L, , 2 2 L 2 L L, , 2 2 L 2 L L, , 1 1 L 1 L L, , 1 1 L 1 L L, , 1 0 , 2 3 L 2 3 L 2 2 L 2 L2 2 1 L 2 1 L 3 L 3 L 2 L 2 L 1 L 1 L α α α α α α α α α α α α α α α α α α α α α α α α ψ ψ ψ ψ ψ ψ λ ψ ψ ψ ψ ψ ψ λ λ P I I R I I R I I R I E I E I E I E I E I E P I I R I I R I I R I E I E I E I E I E I E I I I I I I I I I I I I L b a Z b a Z b a Z b a b a b a U b a Z b a Z b a Z b a b a b a L b a b a b a b a b a b a − + − + − + − − + + + + + + + − + − + − + − + + + + + + + + + + + + =∑
∑
= = [ ] [ ] 0; ) , , , , , , ( ; 0 ) , , , , , , ( ; 0 ) , , , , , , ( (7) L3; L2, L1, ; 0 ) , , , , , , ( L3; L2, L1, ; 0 ) , , , , , , ( , 3 L 3 L 2 L 2 L 1 L 1 L , 1 , 0 , 3 L 3 L 2 L 2 L 1 L 1 L , 1 , 0 3 L 3 L 2 L 2 L 1 L 1 L 3 L 3 L 2 L 2 L 1 L 1 L 3 L 3 L 2 L 2 L 1 L 1 L = ∂ ∂ = ∂ ∂ ≤ ∂ ∂ = = ∂ ∂ = = ∂ ∂∀
∀
∈ ∈ U b a b a b a U L b a b a b a L b a b a b a b b a b a b a a b a b a b a I I I I I I L I I I I I I L I I I I I I L I I I I I I I L I I I I I I I L α α α α α α β β λ λ λ λ λ λ λ λ β λ β λ (8) L3. L2, L1, ; e 1 2 1 j 1 0 , , , 1 0 , , , 1 0 , , , 1 0 , , , = ⋅ + ⋅ + ⋅ + ⋅ =∑
∑
∑
∑
= = = = β λ λ λ λ β ψ α α α β α α α β α α α β α α α β β Z U U Z L L U U L L opt R R E E I [ ] L [ ]0,1 ,U , 1 , 0 , α α α α∈∀
λ ∈∀
λ L3 L2, L1, ; j = + = β β β β opt a opt b optI I I( )
2 ~ sin( )
V;~( )
2 ~ sin(
120)
V;~( )
2 ~ sin(
120)
V~ L3 L3 L2 L2 L1 L1 t = E t e t = E t− ° e t = E t+ ° e ω ω ω ; 120 ; 230 ~ ~ ; 120 ; 230 ~ ~ ; 0 ; 230 ~ ~ 3 L ~ j 3 L 3 L 2 L ~ j 2 L 2 L 1 L ~ j 1 L 1 L = L1 = ψ = = L2 = ψ =− ° = L3 = ψ = ° ψ ψ ψ V E E e V E E e V e E E Ω + = + = = = j0~.9 ~ 3 . 0 ~ ~ 3 L 2 L 1 L Z Z Z Z Z Z Z R jX Z Ω + = + = ~O j~O ~4 j2~.5 O R X Z 0,00 0,25 0,50 0,75 1,00 215 225 235 245 µIEI IEI V Eα=0, L Eα=0.5, L Eα=1, L= =Eα=1,U Eα=0.5, U Eα=0, U 0 0,25 0,5 0,75 1 17000 18500 20000 21500 23000 24500 26000 µP P W Pα=0, L Pα=0.5, L Pα=1, L=Pα=1, U Pα=0.5, U Pα=0, U 0 0,25 0,5 0,75 1 0,28 0,29 0,3 0,31 0,32 µRz Rz Ω RZ, α=0, LRZ, α=0.5, L R=RZ, α=1, L= RZ, α=0.5, U RZ, α=0, U Z, α=1, U 0 0,25 0,5 0,75 1 0,835 0,875 0,915 0,955 µXz Xz Ω XZ, α=0, L XZ, α=0.5, L X=XZ, α=1, L= XZ, α=0.5, U XZ, α=0, U Z, α=1, U 0 0,25 0,5 0,75 1 3,5 3,65 3,8 3,95 4,1 4,25 4,4 µRo Ro Ω RO, α=0, L RO, α=0.5, L R=RO, α=1, L= RO, α=0.5, U RO, α=0, U O, α=1, U 0 0,25 0,5 0,75 1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 µXo Xo Ω XO, α=0, L XO, α=0.5, L XO, α=1, L= XO, α=0.5, U XO, α=0, U 0 0,25 0,5 0,75 1 17000 18500 20000 21500 23000 24500 26000 µP P W -250 -150 -50 50 150 250 0 5 10 15 20 µ=0 µ=0.5 µ=1 µ=0.5 µ=0 e(t) V t ms ~ eL1(t) eL2(t) eL3(t) A. 09 . 84 A; 10 . 78 A; 66 . 72 A; 67 . 67 A; 12 . 63 0.5,L 1,L 1,U 0.5, 0, L , 0 = = = = = = = = = = = = I I I I U I U Iα α α α α α 2 3 L 2 2 L 2 1 L I I I I = + + ; 68 . 27 ; 98 . 15 ; 68 . 27 ; 98 . 15 ; 0 ; 96 . 31 3 L3 2 2 L 1 L 1 L A I A I A I A I I A I L b opt a opt L b opt a opt b opt a opt = − = − = − = = = . 36 . 55 A I opt = IV. Conclusions
The obtained optimal (active) currents have the norm of current smaller than the smallest one before optimization ( ). Moreover, determined optimal currents assure reactive power compensation (optimal currents have the same phase shifts as source voltages). It means that optimal currents assure the proper flow of active power and no reactive power consumption from voltage source. In that case when the optimal currents can be kept in the system, the active power will be always obtained within the power constraint as presented in Fig. 9 (the inner Gauss function within the considered active power). The proposed methods of optimization and description by application of fuzzy numbers make possible to describe unsteady parameters of voltage sources and loads in electrical systems as well to determine the optimal current of the source in three-phase system taking into consideration the problem of minimisation of power losses. This current has the minimal RMS value and it is determined as the result of optimization method taking into considerations the additional constraint: active power generated by the three-phase voltage source is the same before and after optimization. The solution (optimal current) is determined as the crisp one (real numbers) in frequency domain. Precisely it can be determined in the numerical way.
Fig. 2. The source voltages
waveforms voltage source as the fuzzy number with Fig. 3. The exemplary RMS value of the
marked α-cuts Fig. 4. The exemplary value of
active power before optimization as the fuzzy number
Fig. 5. The resistance of the
source as the fuzzy number Fig. 6. The reactance of the source as the fuzzy number
Fig. 7. The resistance of the load
as the fuzzy number Fig. 8. The reactance of the load as the fuzzy number
Fig. 9. The obtained values of active power - the internal shape
of Gauss function A 12 . 63 , 0 = = L Iα