EasytoImplement Integral Numerical Simulation of Multiphase Drives under Fault Supply Condition X. Kestelyn, F. Locment, Y. Crévits and E. Semail L2EP ENSAM, 8 Bd Louis XIV, 59046 Lille France
[email protected] Abstract  This paper presents an easy way to model multiphase electrical drives in fault supply conditions. The presented technique makes it possible to simulate the drive in various configurations with keeping the same integral model established in normal mode. Simulations of a sevenleg sevenphase drive are carried out and compared to experimental measurements.
I. INTRODUCTION More and more electrical systems need a high level of reliability. Among the potential solutions, multiphase machines take a particular place to improve the reliability of AC drives [1]. Efficient current controls lead to the use of the multiple dq spaces concept [2][3]. As for threephase case, this control mode needs to transform the real electrical variables expressed into a stationary reference frame into fictitious variables expressed into multiple one or twodimensional dq rotating reference frames. For a nphase machine, the first harmonics of electrical variables can be regarded as DC components and the use of PI controllers in the current loops leads to null steady state errors [11]. When one or several phases are short or opencircuited, ripples appearing in the dq currents require a particular attention. Many authors have proposed a new model of the machine taking into account these asymmetrical connections [4][5]. These models have made it possible to deduce efficient current controls. However, a new model is built for each new case. For machines with a high number of phases, this technique becomes rapidly cumbersome and time consuming. Moreover, some authors use the derivative operation to model voltages across opencircuited phases which can lead to problems of convergence, stability and long simulation time when using numerical simulations [7]. This paper proposes a simple and easytoimplement way of integral modeling which makes it possible to achieve numerical simulations of multiphase drives under fault conditions with keeping the original model of the drive. II. MULTIPHASE DRIVE MODELLING IN NORMAL MODE Under the assumptions of balanced phases, linear magnetic behavior, no eddy currents and slots phenomenon, the electrical behavior of a nphase machine can be modeled using (1):
JJJG
JJJG JJJG d λabc
vabc = ℜ iabc +
JJJG
JJJG
JJJG
dt
(1)
with vabc , iabc , λabc stator voltage, current and flux vectors with real variables as coordinates and ℜ a nbyn diagonal stator resistance matrix. Using the multiple dq spaces concept [3][11], a vector JJJG JJJG xabc is transformed into a vector xdq using a transformation Tabc − dq from abc to dq references frames.
JJJG
JJJG
( )
xdq = Tabc − dq xdq
(2)
Equation (1) can then be rewritten as: JJJG JJG JJG d λdq (3) vdq = ℜ idq + dt JJG JJG JJJG with vdq , idq , λdq stator voltage, current and flux vectors with dq variables as coordinates. In (3), the flux vector can be broken down into two components due to the stator currents and the rotor effect: JJJG JJJG JJJG λdq = λdqs + λdqr (4) JJJG Transformation Tabc − dq is then chosen in the way that λdqs JJG depends to idq by a diagonal stator inductance matrix A dq . This choice leads to magnetically twodimensional independent systems and makes it possible to achieve efficient and simple current controls. JJJG JJG d λdqr , Using the electromotive force vector (emf) edq = dt (3) becomes for the jth twodimensional dq space n j ≤ + 1 : 2 JJG JJJG JJG didqj JJJG (5) vdqj = ℜ idqj + Ldqj + edqj dt with Ldqj the dq self inductance of the jth dq system. As (5) models a symmetrical twophase machine, the nphase machine can then be considered as equivalent to a set n of + 1 twophase fictitious machines (for n odd). 2 Machine torque T is then obtained by summing each torque Tj provided by each twophase fictitious machine:
n +1 2
n +1 2
T = ∑ T j =∑
JJJG JJG
edqj idqj
(6) Ω 1 1 If the emfs of the nphase machine are only composed of harmonics which ranks are inferior to n, transformation Tabc − dq from abc to dq vectors is chosen in a way that emfs of JJG the fictitious machines edq are constant during steady state operation. This transformation can be considered as an extended Park transformation. These previous remarks make it possible to achieve efficient current controls using classical Proportional Integral (PI) controllers [3]. Fig. 1 shows the overall control structure based on the multiple dq spaces concept. Electrical machines are in the major cases supplied by a voltage source inverter (VSI). If, for reliability considerations, the machine phases are sometimes independent, the economy of static switches often leads to wyecoupling topologies [10]. Fig. 2 depicts a nphase machine wyeconnected to a nleg VSI. Voltages across the phases are dependent on the electrical coupling mode. If a wyecoupling is considered, stator voltage k is related to VSI voltages by (7): n n−1 1 n vkN − ∑ viN + ∑ em (7) n n i =1,i ≠ k m =1 with em the emf induced in phase m by the rotor. Equation (7), usually used to model wyecoupled drives, can be rewritten as (8): JJJG JJJJG JJG JJG (8) vabc = vabcN − vz + ez
vk =
JJG JJG with vz and ez the zerosequence stator voltage and emf vectors composed of all nth order harmonics. JJJG ∗
idq1
JJJG ∗
+
JJJG

PI
vdq1
+
idq1
JJG∗
idqj
JJG ∗
vdq
JJJG ∗
+
JJG

PI
vdqj
JJJJG ∗
Tdq −abc
vabcN
Vbus VSI
JJJG
vabc
+
idqj
JJG
iabc
+
JJG
idq
Tabc −dq
Fig. 1. Multiple dq spaces concept control structure.
nphase machine
sk1
Vbus/2 N Vbus/2
s(k+1)1 ik+1
ik
vkN sk2
vk
s(k+1)2
vk+1
N’ Fig. 2. Topology of a nphase wyecoupled drive.
III. DRIVE MODELLING UNDER FAULT CONDITIONS
A. Analytical modeling If no internal fault in the machine is considered (loss of permanent magnets, broken rotor bars, turntoturn faults), the machine model established in normal mode can be used under fault supply conditions. In a case of shortcircuited phases, it is sufficient to put the voltage terminal to a null value v k = 0 . In a case of opencircuited condition, the terminal requirement is ik = 0 . However, if the machine is wyecoupled, (7) and (8) are no longer correct and stator voltages have to be recalculated. Stator voltages are dependent on the number of opencircuited phases and their locations. Moreover, the determination of transformed dq voltages needs to have an analytical expression of the open circuit voltages. This latter voltage should be calculated by differentiating the flux across the considered opencircuited phase [6]. For numerical simulation, the use of derivative operations can lead to known problems of convergence, stability and long simulation time [7]. The previous problems can be overcome by modeling the connection between the VSI and the machine terminals by a resistor whose value depends on the functioning condition. The relation between VSI voltages and stator voltages are then given by (9). vk = vSIk − RCk ⋅ ik (9) where vSIk is obtained using (8) and connection resistance RCk is null under normal condition and tends to infinite when phase k is opencircuited. Fig. 3 graphically depicts the chosen way of opencircuited conditions modeling.
JJJG
eabc
JJJJG
vabcN
+ + 
Machine model
Wyecoupling model 0 0 0 0 0
Tabc− dq
RC1 0 0 0 0
0 RC 2 0 0 0
0 0 0 0 0 ... ... 0 0 ... 1 0 0 1 0 0 1
0 0 0
... RCn
0 0 0 0 RC 3 0 0 ... 0
Tdq − abc
0 0 0 ...
1 ℜ + A abc s
JJG
iabc
It is important to note that, contrary to ℜC , matrix ℜCdq is not diagonal. It implies that in the case of opencircuited phases, the twodimensional systems are no longer independent. Moreover, according to (11), the knowledge of the different terms of matrix ℜCdq makes it possible to determine the effect of opencircuited phases over dq variables. IV. CASE OF A SEVENLEG SEVENPHASE AXIAL PERMANENT MAGNET MACHINE
Connection model
Fig. 3. Graphical description of opencircuited condition modelling.
B. Numerical simulation As this method of modeling can be regarded as a resistive companion modeling of the connection [7], the values of connection resistances RC have then to be chosen accordingly to a compromise between the expected model accuracy and the overall computation time. For linear systems, the computation step time of the numerical solver is first chosen in regard to the smallest time constant of the system. This precaution allows avoiding problems of convergence and accuracy. In normal or shortcircuited condition, the smallest time constant corresponds to the fastest current closedloop time constant (i.e. R , Ldqj first order system and associated PI control loop). In opencircuited conditions, the time constants to be considered become from the internal closedloop composed of connection resistances RC and machine electrical variables ( R , L ). A large value for connection resistances RC can lead to small step time and consequently large computation time. The previous remark leads to consider that connection resistance RCk can not tend to infinity. As a direct consequence, current ik in the opencircuited phase does not tend to zero. However, as the machine is supplied by a VSI, current ik can not have a magnitude higher than: V ( ik )max = bus (10) RC
A. Presentation of the sevenphase drive A sixpole sevenphase NN TORUS [9] machine with two external rotors has been designed for faultcondition operations. Fig. 4 shows a picture of the axial flux sevenphase machine. The stator, with Grammering windings, is soft magnetic composite made with 42 slots. Fig. 5 shows one sixth of the machine. The drive is designed for a rated torque of 65Nm and supplied by a sevenleg voltage source inverter controlled with pulse width modulation. The DC bus voltage is set to 300V and pulse width modulation switching frequency is set to 20kHz. As depicted in Fig. 1, multiple PI dq current controls are used to achieve high performances. In the case of a sevenphase machine, the drive is equivalent two a set of three fictitious twophase machines (M1, M2 and M3) and one onephase machine Mz. Due to the wyecoupling, machine Mz is not supplied.
Fig. 4. Sevenphase double rotor permanent.
MAGNETS
C. Effects of connection resistances over dq variables Using the multiple dq spaces concept, (9) becomes: JJG JJJJG JJG vdq = vSIdq − ℜCdq idq (11)
ROTORS COILS
with matrix ℜCdq calculated according to (12): ℜCdq = Tdq −abcℜCTabc−dq
(12) ℜC a nbyn diagonal matrix with connection resistor values as variables. ℜC = diag ( RC1 , RC 2 , ..., RCn ) (13)
STATOR
Fig. 5. One sixth of the machine.
C. Numerical simulation of the drive under fault conditions Fig. 7 shows the experimental and simulated phase currents when three consecutive phases are opencircuited. dq current references are kept the same as in normal mode. Connexion resistance RCk of opencircuited phases A, B and C is set to 5kΩ. This choice leads to use a step time of 1e6 second which is ten times larger than in the normal mode. This step time value makes a good compromise between the total time simulation and the desired accuracy. The comparison between experimental and simulated waveforms shows that this way of modeling leads to an accurate model slightly longer to simulate compared to the normal mode.
8 6
iF exp iG exp
0
iD sim
2
iE sim
4
iG sim
iF sim
8 0.2
0.25
0.3
0.35
Time (s)
0.4
Fig. 7. Experimental and Simulated phase currents in the case of three opencircuited phases.
D. Connection resistance versus harmonic contents of dq variables According to (12), connection matrix ℜCdq has been calculated for a sevenphase machine. Each term of this matrix has harmonic components as shown in Table I. According to (11), the value of each ℜCdq term has an impact over the dq variables. For example, using Table I it can be deduce that it exist harmonics 0 and 2 in dq variables of machine M1. Fig. 8, Fig. 9 and Fig. 10 show the experimental and simulated dq currents of the three fictitious machines and confirm the expected harmonics given by matrix ℜCdq . The use of expected harmonic content of dq variables has already been used for the control of a threephase induction machine under faultconditions [8]. TABLE I HARMONIC CONTENT OF CONNECTION RESISTANCE MATRIX ℜCdq
2
Current (A)
iE exp
2
6
3
ℜCdq
1
iA exp iB exp
0
iA sim iB sim
1
2
3 0.2
iD exp
4
Current (A)
B. Numerical simulation of the drive under normal conditions The drive is first simulated in normal mode to serve as a reference for comparison between simulated and experimental results. The simulation has been carried out with MatlabSimulink in respect with the structures shown in Fig. 1 and Fig. 3. Machine and controllers parameters have led to a step time of 1e5 second. Fig. 6 shows the experimental and simulated currents of phases A and B for the following conditions: id1, id2, iq2, id3 and iq3 references set to zero, iq1 set to 4A, all connexion resistances RCk set to 0Ω. In this case, the seven currents are sinusoidal and shifted by 2π . Since it exist a good similarity between an angle 7 simulated and experimental results, the model of the drive is considered to be correct.
0.25
0.3
Time (s)
0.35
0.4
Fig. 6. Experimental and Simulated phase currents in the case of normal condition.
M1 M2 M3
M1 0.2 2 4.6 4.6 2.4 2.4
M2 2 0.2 4.6 4.6 2.4 2.4
4.6 4.6 0.1 10 2.8 2.8
M3 4.6 4.6 10 0.1 2.8 2.8
2.4 2.4 2.8 2.8 0.6 6
2.4 2.4 2.8 2.8 6 0.6
V. CONCLUSION
8
Current (A)
6
4
id1 exp iq1 exp
2
id1 sim iq1 sim
0
2
4 0.2
0.25
0.3
Time (s)
0.35
0.4
This paper provides an easytoimplement way to model multiphase drives in various configurations using a unique integral model. This opportunity makes it possible to rapidly analyze the effect of the total number of phases, of the topology of connection or of the type of fault on machine variables. Used for numerical simulation, the connection resistances make the deduction of multiple dq variables harmonic content possible, which is essential for advanced current controls. Experimental measurements on a seven phase drive show the effectiveness of the proposed solution. VI. REFERENCES
Fig. 8. Experimental and Simulated M1 dq currents in the case of three opencircuited phases. [1] 5
[2]
4 3
Current (A)
2 id2 exp
1
[3]
iq2 exp
0
id2 sim
1
iq2 sim
[4]
2 3 4 5 0.2
[5] 0.25
0.3
Time (s)
0.35
0.4
Fig. 9. Experimental and Simulated M2 dq currents in the case of three opencircuited phases. 4
[6]
[7]
3
[8]
Current (A)
2 1
id3 exp iq3 exp
0
id3 sim
[9]
iq3 sim
1
[10]
2 3 4 0.2
[11] 0.25
0.3
Time (s)
0.35
0.4
Fig. 10. Experimental and Simulated M3 dq currents in the case of three opencircuited phases.
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