# Patent

## Title
Method and an apparatus for controlling an electrical machine with two or more multiphase stator windings

## Abstract

An apparatus comprises a processing system (201) configured to produce voltage control signals for controlling an electrical machine (202) comprising two or more multiphase stator windings. The processing system is configured to produce the voltage control signals on the basis of rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the multiphase stator windings and mutual inductances between the multiphase stator windings. The processing system is configured to convert phase currents of the multiphase stator windings into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of the stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents in spite of the mutual inductances between the multiphase stator windings.

## Background
FIELD OF THE INVENTION
The invention relates generally to rotating electrical machines. More particularly, the invention relates to method for controlling an electrical machine comprising two or more multiphase stator windings. Furthermore, the invention relates to an apparatus and a computer program for the purpose of controlling an electrical machine comprising two or more multiphase stator windings.
BACKGROUND
In many applications, high level of availability of an electrical drive system is required. For example, in electrical ship propulsion the availability is critical from the viewpoint of safety. The availability can be increased by adding redundancy to the electrical drive system. Due to economic reasons, redundancy by multiplying the number of complete electrical drive systems is only rarely possible. However, the redundancy can be achieved by providing an electrical machine with two or more galvanically isolated multiphase stator windings each of which being supplied with its own multiphase power stage. The electrical machine may comprise, for example, two star-connected three-phase stator windings which are shifted relative to each other by e.g. 30 electrical degrees.
Accurate control of an electrical machine is typically based on a model which models the behavior of currents and voltages of the electrical machine and sometimes also the produced torque. In conjunction with a synchronous electrical machine which may have a salient pole rotor, the currents, voltages, and flux linkages are preferably expressed in suitable rotation-converted forms in a coordinate system bound to the rotor in order to avoid position dependency of inductance parameters of the model. The rotation-converted stator currents can be controlled on the basis of differences between the rotation-converted stator currents and their target values. The target values of the rotation-converted stator currents can be formed on the basis of e.g. the desired torque. The rotation-converted stator currents and voltages are typically expressed in the d-q coordinate system whose coordinate axes are along the direct and quadrature axes of the rotor. The inherent advantage of the d-q coordinate system is that the d-component of the stator currents does not generate flux-linkage on the q-direction, and correspondingly the q-component of the stator currents does not generate flux-linkage on the d-direction. This de-coupling between the d- and q-directions significantly facilitates the control of the rotation-converted stator currents because the d- and q-components of the stator currents can be regulated with e.g. separate regulators that can be, for example, proportional-integrating “PI” regulators.
In a case of an electrical machine which comprises two or more multiphase stator windings, the situation is more complicated. The two or more multiphase stator windings have mutual magnetic couplings. Hence, for example the d-directional stator flux-linkage of one of the multiphase stator windings is dependent not only on the d-component of the stator currents of this multiphase stator winding but also on the d-components of the stator currents of the other multiphase stator windings. In a simple control principle, the above-mentioned mutual magnetic couplings are neglected and the two or more multiphase stator windings are controlled separately from each other. However, the neglecting of the above-mentioned mutual magnetic couplings weakens the accuracy of the control. On the other hand, the control gets significantly more complicated if the mutual magnetic couplings are taken into account, because this requires cooperation between regulators related to different multiphase stator windings.
SUMMARY
The following presents a simplified summary in order to provide a basic understanding of some aspects of various invention embodiments. The summary is not an extensive overview of the invention. It is neither intended to identify key or critical elements of the invention nor to delineate the scope of the invention. The following summary merely presents some concepts of the invention in a simplified form as a prelude to a more detailed description of exemplifying embodiments of the invention.
In accordance with the first aspect of the invention, there is provided a new method for controlling an electrical machine comprising two or more multiphase stator windings. The method according to the invention comprises:

    
    
        converting stator phase currents of the two or more multiphase stator windings into rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine, and
        controlling power stages connected to the multiphase stator windings on the basis of the rotation-converted stator currents and on the basis of a model of the electrical machine modeling at least inductances of the two or more multiphase stator windings and mutual inductances between the two or more multiphase stator windings,

wherein the stator phase currents are converted into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are not altered e.g. due to magnetic saturation.

    
    


The above-mentioned form of the model in which each of the stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents can be obtained, for example, by diagonalizing the inductance matrix of the traditional d-q model of the electrical machine.
The use of the form where each stator flux-linkage is dependent on only one rotation-converted stator current significantly facilitates the control of the rotation-converted stator currents because the rotation-converted stator currents can be regulated with e.g. separate proportional-integrating “PI” regulators.
In accordance with the second aspect of the invention, there is provided a new apparatus for the purpose of controlling an electrical machine comprising two or more multiphase stator windings. The apparatus according to the invention comprises a processing system configured to:

    
    
        produce voltage control signals, which are suitable for controlling the electrical machine, on the basis of rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two or more multiphase stator windings and mutual inductances between the two or more multiphase stator windings, and
        convert stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are unaltered.
    
    


The apparatus may further comprise, for example but not necessarily, power stages for providing supply voltages to the two or more multiphase stator windings on the basis of the voltage control signals.
In accordance with the third aspect of the invention, there is provided a new computer program for the purpose of controlling an electrical machine comprising two or more multiphase stator windings. The computer program according to the invention comprises computer executable instructions for controlling a programmable processor to:

    
    
        produce voltage control signals, which are suitable for controlling the electrical machine, on the basis of rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two or more multiphase stator windings and mutual inductances between the two or more multiphase stator windings,
        convert stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are unaltered.
    
    


In accordance with the fourth aspect of the invention, there is provided a new computer program product. The computer program product comprises a nonvolatile computer readable medium, e.g. a compact disc “CD”, encoded with a computer program according to the invention.
A number of exemplifying embodiments of the invention are described in accompanied dependent claims.
Various exemplifying embodiments of the invention both as to constructions and to methods of operation, together with additional objects and advantages thereof, will be best understood from the following description of specific exemplifying embodiments when read in connection with the accompanying drawings.
The verbs “to comprise” and “to include” are used in this document as open limitations that neither exclude nor require the existence of unrecited features. The features recited in depending claims are mutually freely combinable unless otherwise explicitly stated.



BRIEF DESCRIPTION OF FIGURES
The exemplifying embodiments of the invention and their advantages are explained in greater detail below in the sense of examples and with target to the accompanying drawings, in which:
FIG. 1 shows a schematic illustration of an exemplifying electrical machine according to the prior art and comprising two three-phase stator windings,
FIG. 2 shows a functional block diagram of an apparatus according to an exemplifying embodiment of the invention for the purpose of controlling an electrical machine comprising two or more multiphase stator windings, and
FIG. 3 is a schematic flow diagram of a method according to an exemplifying embodiment of the invention for controlling an electrical machine comprising two or more multiphase stator windings.



DESCRIPTION OF EXEMPLIFYING EMBODIMENTS
FIG. 1 shows a schematic illustration of an exemplifying electrical machine comprising two or more multiphase stator windings. In this exemplifying case, the electrical machine is a salient pole synchronous machine which comprises an excitation winding and/or permanent magnet material in its rotor 102. Furthermore, the electrical machine could comprise damper windings in its rotor. In this exemplifying case, a stator 101 of the electrical machine comprises two three-phase stator windings. The first one of the three-phase stator windings comprises stator phase windings 103, 104, and 105. The second one of the three-phase stator windings comprises stator phase windings 106, 107, and 108. The phase currents of the first three-phase stator winding are denoted with ia1, ib1, and ic1, and the phase currents of the second three-phase stator winding are denoted with ia2, ib2, and ic2. The phase voltages of the first three-phase stator winding are denoted with ua1, ub1, and uc1, and the phase voltages of the second three-phase stator winding are denoted with ua2, ub2, and uc2. The first and second three-phase stator windings are shifted relative to each other by an electrical angle of 2γ as illustrated by FIG. 1.
FIG. 1 presents a d-q coordinate system which is bound to the rotor 102. The d-axis of the d-q coordinate system represents the direct axis of the rotor, and the q-axis of the d-q coordinate system represents the quadrature axis of the rotor as illustrated by FIG. 1. FIG. 1 presents also a α-β coordinate system which is bound to the stator 101. The rotational position of the rotor with respect to the stator is expressed with an angle θ. The rotational speed ω of the rotor is the time derivative of θ, i.e. ω=dθ/dt. The traditional d-q model of the electrical machine is briefly discussed below as a prelude to descriptions of exemplifying embodiments of the invention.
Without limiting generality, we assume for the sake of simplicity that the rotor of the electrical machine is a permanent magnet rotor which does not comprise any excitation and damper windings. In this case, the electrical machine can be modeled with the following voltage equations expressed in the d-q coordinate system:




  
    
      
        {
        
          
            
              
                
                  
                    u
                    
                      d
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          d
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        d
                        t
                      
                    
                    -
                    
                      ωΨ
                      
                        q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      q
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          q
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        d
                        t
                      
                    
                    +
                    
                      ωΨ
                      
                        d
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      d
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          d
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      
                        d
                        t
                      
                    
                    -
                    
                      ωΨ
                      
                        q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      q
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          q
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      
                        d
                        t
                      
                    
                    +
                    
                      ωΨ
                      
                        d
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
          
          ,
        
      
    
    
      
        (
        1
        )
      
    
  




where ud1 and uq1 are rotation-converted stator voltages of the first three-phase stator windings expressed in the d-q coordinate system, ud2 and uq2 are the rotation-converted stator voltages of the second three-phase stator windings expressed in the d-q coordinate system, id1 and iq1 are rotation-converted stator currents of the first three-phase stator windings expressed in the d-q coordinate system, id2 and iq2 are the rotation-converted stator currents of the second three-phase stator windings expressed in the d-q coordinate system, and Rs is the stator resistance.

Ψd1 and Ψq1 are the d- and q-directional stator flux linkages of the first three-phase stator windings, respectively, and Ψd2 and Ψq2 are the d- and q-directional stator flux linkages of the second three-phase stator windings, respectively. These flux linkages can be modeled with the following equations:

Ψd1=Ldid1+Mdid2+ΨPMd,

Ψq1=Lqiq1+Mqiq2,

Ψd2=Ldid2+Mdid1+ΨPMd, and

Ψq2=Lqiq2+Mqiq1,  (2)

where Ld is the direct-axis inductance, Md is the direct-axis mutual inductance between the first and second three-phase stator windings, Lq is the quadrature-axis inductance, Mq is the quadrature-axis mutual inductance between the first and second three-phase stator windings, and ΨPMd is the d-directional stator flux linkage created by the permanent magnet rotor.

The model of the electrical machine defined by equations (1) and (2) can be presented in the following matrix form:




  
    
      
        
          
            u
            dq
          
          =
          
            
              
                R
                s
              
              ⁢
              
                i
                dq
              
            
            +
            
              
                L
                dq
              
              ⁢
              
                
                  d
                  
                    i
                    dq
                  
                
                
                  d
                  t
                
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              
                JL
                dq
              
              ⁢
              
                i
                dq
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              J
              ⁢
              
                
              
              ⁢
              
                Ψ
                PMdq
              
            
          
        
        ,
      
    
    
      
        (
        3
        )
      
    
  




where:





  
    
      u
      dq
    
    =
    
      [
      
        
          
            
              u
              
                d
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              u
              
                q
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              u
              
                d
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
        
          
            
              u
              
                q
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
      
      ]
    
  
  ,
  
    
      i
      dq
    
    =
    
      [
      
        
          
            
              i
              
                d
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              i
              
                q
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              i
              
                d
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
        
          
            
              i
              
                q
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
      
      ]
    
  
  ,
  
    
      L
      
        d
        ⁢
        
          
        
        ⁢
        q
      
    
    =
    
      [
      
        
          
            
              L
              d
            
          
          
            0
          
          
            
              M
              d
            
          
          
            0
          
        
        
          
            0
          
          
            
              L
              q
            
          
          
            0
          
          
            
              M
              q
            
          
        
        
          
            
              M
              d
            
          
          
            0
          
          
            
              L
              d
            
          
          
            0
          
        
        
          
            0
          
          
            
              M
              q
            
          
          
            0
          
          
            
              L
              q
            
          
        
      
      ]
    
  
  ,
  
    

  
  ⁢
  
    J
    =
    
      [
      
        
          
            0
          
          
            
              -
              1
            
          
          
            0
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            
              -
              1
            
          
        
        
          
            0
          
          
            0
          
          
            1
          
          
            0
          
        
      
      ]
    
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        Ψ
        PMdq
      
    
    =
    
      
        [
        
          
            
              
                Ψ
                PMd
              
            
          
          
            
              0
            
          
          
            
              
                Ψ
                PMd
              
            
          
          
            
              0
            
          
        
        ]
      
      .
    
  




In the above-presented equation (3), vectors are denoted with lower case bold letter and matrices with upper case bold letters. This notation will be used also in other parts of this document.
The vector of the rotation-converted stator voltages udq can be obtained from the vector of the stator phase voltages uabc=[ua1, ub1, uc1, ua2, ub2, uc2]T with the following conversion rule:

udq=T1(θ)uabc,  (4)

where θ is the angle expressing the rotational position of the rotor with respect to the stator as shown in FIG. 1, and T1(θ) is a first conversion matrix:





  
    
      
        
          
            
              T
              1
            
            ⁡
            
              (
              θ
              )
            
          
          =
          
            [
            
              
                
                  
                    
                      T
                      p
                    
                    ⁡
                    
                      (
                      
                        δ
                        =
                        
                          θ
                          -
                          γ
                        
                      
                      )
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      T
                      p
                    
                    ⁡
                    
                      (
                      
                        δ
                        =
                        
                          θ
                          +
                          γ
                        
                      
                      )
                    
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        5
        )
      
    
  




where the sub-matrices Tp are given by:





  
    
      
        
          
            
              T
              p
            
            ⁡
            
              (
              δ
              )
            
          
          =
          
            
              C
              1
            
            ⁡
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          δ
                          -
                          
                            
                              2
                              ⁢
                              π
                            
                            3
                          
                        
                        )
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          δ
                          +
                          
                            
                              2
                              ⁢
                              π
                            
                            3
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        -
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            δ
                            -
                            
                              
                                2
                                ⁢
                                π
                              
                              3
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            δ
                            +
                            
                              
                                2
                                ⁢
                                π
                              
                              3
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  




where C1 is a constant scaling coefficient.

The vector of the rotation-converted stator currents idq can be obtained from the vector of the stator phase currents iabc=[ia1, ib1, ic1, ia2, ib2, ic2]T with the following conversion rule:

idq=T1(θ)iabc.

The above-presented description is related to an electrical machine comprising two three-phase stator windings. It is, however, straightforward to generalize the above-presented equations (1)-(6) to a general case where there are N M-phase stator windings. Also in the general case, the voltage equations in the matrix form would be similar to the equation (3) shown above. The inductance matrix would Ldq be a 2N×2N matrix, the vector of the rotation-converted stator currents idq would have 2N elements, and the vector of the rotation-converted stator voltages udq would have 2N elements.
As can be seen from equations (2), the d- and q-directional stator flux linkages of the first three-phase stator winding Ψd1 and Ψq1 are dependent on also the currents id2 and iq2 of the second three-phase stator winding. Correspondingly, the Band q-directional stator flux linkages of the second three-phase stator winding Ψd2 and Ψq2 are dependent on also the currents id1 and iq1 of the first three-phase stator winding. This cross-dependency complicates the control of the currents of the electrical machine. In the matrix form equation (3), the cross-dependency is manifested by the non-zero off-diagonal elements of the inductance matrix Ldq.
In conjunction with embodiments of the present invention, the problem associated with the above-mentioned cross-dependency is avoided by using such form of the model of the electrical machine where each of the stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents in spite of the mutual inductances between the two or more multiphase stator windings when the inductance parameters of the model are not altered e.g. due to magnetic saturation.
The above-mentioned form of the model of the electrical machine can be derived, for example, from the matrix formulation presented by the equation (3).
The inductance matrix Ldq is symmetric and thus it can be converted into a diagonal form with the aid of a second conversion matrix T2:

LDQ=T2LdqT2T)  (7)

where LDQ is the diagonalized inductance matrix, and the second conversion matrix T2 is [g1 g2 . . . gn] where g1 g2 . . . gn are the eigenvectors of the inductance matrix Ldq. For example, in the case where there are two multiphase stator windings, the second conversion matrix T2 can be presented in the form:





  
    
      
        
          
            T
            2
          
          =
          
            
              C
              2
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




where C2 is a constant scaling coefficient. For example, C2=1/√2 makes the second conversion matrix orthonormal so that T2T2T=T2TT2 is a unit matrix diag(1, 1, 1, 1).

The matrix formulation represented by equation (3) can be multiplied from the left side with the second conversion matrix T2:




  
    
      
        
          
            
              T
              2
            
            ⁢
            
              u
              dq
            
          
          =
          
            
              
                R
                s
              
              ⁢
              
                T
                2
              
              ⁢
              
                i
                dq
              
            
            +
            
              
                T
                2
              
              ⁢
              
                L
                dq
              
              ⁢
              
                T
                2
                T
              
              ⁢
              
                
                  
                    d
                    
                      T
                      2
                    
                  
                  ⁢
                  
                    i
                    dq
                  
                
                
                  d
                  t
                
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              
                T
                2
              
              ⁢
              
                JL
                dq
              
              ⁢
              
                T
                2
                T
              
              ⁢
              
                T
                2
              
              ⁢
              
                i
                dq
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              
                T
                2
              
              ⁢
              J
              ⁢
              
                
              
              ⁢
              
                Ψ
                PMdq
              
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where the rule that T2TT2 is a unit matrix is utilized too.

Equation (9) can be written as:




  
    
      
        
          
            u
            DQ
          
          =
          
            
              
                R
                s
              
              ⁢
              
                i
                DQ
              
            
            +
            
              
                L
                DQ
              
              ⁢
              
                
                  d
                  
                    i
                    DQ
                  
                
                
                  d
                  t
                
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              
                J
                m
              
              ⁢
              
                L
                DQ
              
              ⁢
              
                i
                DQ
              
            
            +
            
              ω
              ⁢
              
                
              
              ⁢
              
                J
                m
              
              ⁢
              
                Ψ
                PMDQ
              
            
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




where:

    
    
        uDQ is a vector of the rotation-converted stator voltages T2udq,
        iDQ is a vector of the rotation-converted stator currents T2idq,
        Jm is a matrix for which JmT2=T2J, and
        ΨPMDQ=T2ΨPMdq.
    
    


As the inductance matrix LDQ is a diagonal matrix, each of the stator flux-linkages of this form of the model is dependent on only one of the rotation-converted stator currents in spite of the mutual inductances between the two or more multiphase stator windings. In conjunction with a current control, the voltage vector caused by the rotation of the rotor:

ωJmLDQIDQ+ωJmΨPMDQ 

can be taken into account as a feed-forward term uff that is superposed to an output of a current regulating device. Removing the above-mentioned feed-forward term from the equation (10) and then solving, in the Laplace domain, for the vector iDQ of the rotation-converted stator currents yields the following system to be controlled with the current regulating device:

iDQ(s)=(RsI+sLDQ)−1uDQ(s),  (11)

where s is the Laplace variable and I is a unit matrix having the same dimensions as the inductance matrix LDQ. The vector iDQ(S) of the rotation-converted stator currents represents the quantity to be controlled, and the vector uDQ(s) of the rotation-converted stator voltages represents the quantity with the aid of which the control is carried out. The rows of the equation (11) represent mutually independent first order systems because the inductance matrix LDQ is a diagonal matrix. For example, the nth row of the equation (11) is:





  
    
      
        i
        
          DQ
          ,
          n
        
      
      ⁡
      
        (
        s
        )
      
    
    =
    
      
        1
        
          
            R
            s
          
          +
          
            sL
            
              DQ
              ,
              n
            
          
        
      
      ⁢
      
        
          u
          
            DQ
            ,
            n
          
        
        ⁡
        
          (
          s
          )
        
      
    
  
  ,




where iDQ,n(s) is the nth rotation-converted stator current, i.e. the nth element of the vector iDQ(s), uDQ,n(s) is the nth rotation-converted stator voltage, i.e. the nth element of the vector uDQ(s), and LDQ,n is the diagonal element of the nth row, or column, of the inductance matrix LDQ.

In an exemplifying case where there are two multiphase stator windings, the inductance matrix LDQ is




  
    
      
        
          
            L
            DQ
          
          =
          
            
              [
              
                
                  
                    
                      L
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      L
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
              ]
            
            =
            
              2
              ⁢
              
                
                  C
                  2
                  2
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            L
                            d
                          
                          +
                          
                            M
                            d
                          
                        
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                            L
                            q
                          
                          +
                          
                            M
                            q
                          
                        
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        
                          
                            L
                            q
                          
                          -
                          
                            M
                            q
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        
                          
                            L
                            d
                          
                          -
                          
                            M
                            d
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        ,
      
    
    
      
        (
        
          12
          ⁢
          a
        
        )
      
    
  




where C2 is the constant scaling coefficient presented in equation (8), and the uDQ, iDQ, ΨPMDQ and Jm are:





  
    
      
        
          
            u
            DQ
          
          =
          
            
              [
              
                
                  
                    
                      u
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      u
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      u
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
                
                  
                    
                      u
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
              ]
            
            =
            
              
                C
                2
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          u
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        +
                        
                          u
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          u
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        +
                        
                          u
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          u
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        -
                        
                          u
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          u
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        -
                        
                          u
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            i
            DQ
          
          =
          
            
              [
              
                
                  
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
                
                  
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
              ]
            
            =
            
              
                C
                2
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          i
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        +
                        
                          i
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          i
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        +
                        
                          i
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          i
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        -
                        
                          i
                          
                            q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          i
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        -
                        
                          i
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        
          12
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          
            
              Ψ
              PMDQ
            
            =
            
              
                [
                
                  
                    
                      
                        Ψ
                        
                          PMD
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
              =
              
                [
                
                  
                    
                      
                        2
                        ⁢
                        
                          C
                          2
                        
                        ⁢
                        
                          Ψ
                          PMd
                        
                      
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
            
          
          ,
          and
        
        ⁢
        
          

        
        ⁢
        
          
            J
            m
          
          =
          
            J
            =
            
              
                [
                
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        
          12
          ⁢
          c
        
        )
      
    
  




The equation (10), when written in the component form, yields:




  
    
      
        {
        
          
            
              
                
                  
                    u
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        d
                        t
                      
                    
                    -
                    
                      ωΨ
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      Q
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          Q
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            Q
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        d
                        t
                      
                    
                    +
                    
                      ωΨ
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      
                        d
                        t
                      
                    
                    -
                    
                      ωΨ
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      Q
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      
                        R
                        s
                      
                      ⁢
                      
                        i
                        
                          Q
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    +
                    
                      
                        d
                        
                          Ψ
                          
                            Q
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      
                        d
                        t
                      
                    
                    +
                    
                      ωΨ
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
          
          ,
        
      
    
    
      
        (
        13
        )
      
    
  




where the flux linkages of the model are:





  
    
      
        {
        
          
            
              
                
                  
                    Ψ
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      
                        L
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁢
                      
                        i
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      Ψ
                      
                        PMD
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    Ψ
                    
                      Q
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  =
                  
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    ⁢
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    Ψ
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      L
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁢
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    Ψ
                    
                      Q
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  =
                  
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁢
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
          
          .
        
      
    
    
      
        (
        14
        )
      
    
  




As shown by the equations (14), each of the stator flux-linkages ΨD1, ΨQ1, ΨD2 and ΨQ2 of this form of the model is dependent on only one of the rotation-converted stator currents iD1, iQ1, iD2, and iQ2 when the inductance parameters LD1, LQ1, LD2, and LQ2 of the model are constant, i.e. unaltered. Magnetic saturation may cause that e.g. the rotation-converted stator current iD2 affects the value of e.g. LD1 and thus iD2 may affect indirectly to ΨD1. However, when the inductance parameters are unaltered, each stator flux linkage is dependent on only one rotation-converted stator current.
The voltages caused by the rotation of the rotor are:




  
    
      
        {
        
          
            
              
                
                  
                    u
                    
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      ,
                      ff
                    
                  
                  =
                  
                    
                      -
                      ω
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    ⁢
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      ,
                      ff
                    
                  
                  =
                  
                    
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      
                        L
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁢
                      
                        i
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    +
                    
                      ωΨ
                      
                        PMD
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      ,
                      ff
                    
                  
                  =
                  
                    
                      -
                      ω
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      L
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁢
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    u
                    
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      ,
                      ff
                    
                  
                  =
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    
                      L
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁢
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
            
          
          .
        
      
    
    
      
        (
        15
        )
      
    
  




Therefore, the mutually independent first order systems to be regulated with a current regulating device are:




  
    
      
        {
        
          
            
              
                
                  
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  =
                  
                    
                      1
                      
                        
                          R
                          s
                        
                        +
                        
                          sL
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                    ⁢
                    
                      
                        u
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  =
                  
                    
                      1
                      
                        
                          R
                          s
                        
                        +
                        
                          sL
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                    ⁢
                    
                      
                        u
                        
                          Q
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      i
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  =
                  
                    
                      1
                      
                        
                          R
                          s
                        
                        +
                        
                          sL
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                    ⁢
                    
                      
                        u
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      i
                      
                        Q
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  =
                  
                    
                      1
                      
                        
                          R
                          s
                        
                        +
                        
                          sL
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                    ⁢
                    
                      
                        u
                        
                          Q
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                  
                
              
            
          
          .
        
      
    
    
      
        (
        16
        )
      
    
  




The rotation-converted stator currents iD1, iQ1, iD2, and iQ2 are bound to the stator phase currents ia1, ib1, ic1, ia2, ib2, and ic2 via a third conversion matrix T3(θ):

iDQ=T3(θ)iabc=T2T1(θ)iabc,  (17)

iabc=T3T(θ)iDQ,  (18)

where T1(θ) is the first conversion matrix defined in equations (5) and (6). Correspondingly, the rotation-converted stator voltages uD1, uQ1, uD2, and uQ2 are bound to the stator phase voltages ua1, ub1, uc1, ua2, ub2, and uc2 via the third conversion matrix T3(θ):

uDQ=T3(θ)uabc,  (19)

An apparatus according to an exemplifying embodiment of the invention comprises:

    
    
        means for producing voltage control signals, which are suitable for controlling an electrical machine comprising two or more multiphase stator windings, on the basis of rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two or more multiphase stator windings and mutual inductances between the two or more multiphase stator windings, and
        means for converting stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when the inductance parameters of the model are unaltered.
    
    


FIG. 2 shows a functional block diagram of an apparatus 200 according to an exemplifying embodiment of the invention for the purpose of controlling an electrical machine 202 comprising two or more multiphase stator windings. In the exemplifying case shown in FIG. 2, the electrical machine comprises two multiphase stator windings that can be e.g. three-phase stator windings. The apparatus comprises a processing system 201 configured to produce voltage control signals u1 and u2 suitable for controlling the electrical machine 202. The apparatus may further comprise modulators 203 and 204 for producing switch-control signals s1 and s2 on the basis of the voltage control signals u1 and u2, and power stages 212 and 213 for producing supply voltages of the stator windings on the basis of the switch-control signals s1 and s2. Alternatively, the apparatus may comprise signal interfaces for connecting to an external system comprising modulators and power stages. The modulators 203 and 204 can be, for example, pulse width modulators “PWM”. In this case, the voltage control signals u1 and u2 are arranged to express target values of the phase voltages of the multiphase stator windings. For another example, the modulators 203 and 204 can be space-vector pulse width modulators “SVPWM”. In this case, each of the voltage control signals u1 and u2 can be expressed as a vector in the α-β coordinate system illustrated in FIG. 1.
The processing system 201 is configured to produce the voltage control signals u1 and u2 on the basis of the rotation-converted stator currents iD1, iQ1, iD2, and iQ2 which are expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two multiphase stator windings and mutual inductances between the two multiphase stator windings. The stator phase currents ia1, ib1, ic1, ia2, ib2, ic2 are converted into the rotation-converted stator currents iD1, iQ1, iD2, and iQ2 with a conversion rule corresponding to a form of the model of the electrical machine where each of the stator flux-linkages ΨD1, ΨQ1, ΨD2 and ΨQ2 of the model is dependent on only one of the rotation-converted stator currents iD1, iQ1, iD2, and iQ2 when the inductance parameters of the model are unaltered. A conversion rule of the kind mentioned above is presented with equation (17). A functional block 205 shown in FIG. 2 represents the conversion of the stator phase currents into the rotation-converted stator currents.
In the exemplifying embodiment illustrated in FIG. 2, the processing system 201 is configured to direct regulator functions 206, 207, 208, and 209 to differences eiD1, eiQ1, eiD2 and eiQ2 between the rotation-converted stator currents and iQ2 and their target values iD1,ref, iQ1,ref, iD2,ref, and iQ2,ref as illustrated in FIG. 2. The regulator functions 206-209 can be implemented with, for example, mutually independent proportional-integrating “PI” regulators or with mutually independent proportional-integrating-derivative “PID” regulators. The outputs of the regulator functions 206-209 are superposed with voltages uD1,ff, uQ1,ff, uD2,ff, and uQ2,ff caused by the rotation of the rotor and expressed in the coordinate system bound to the rotor. The voltages uD1,ff, uQ1,ff, uD2,ff, and uQ2,ff are given by the equations (15). A functional block 210 shown in FIG. 2 represents the forming of the voltages uD1,ff, uQ1,ff, uD2,ff, and uQ2,ff on the basis of the rotation-converted stator currents iD1, iQ1, iD2, and iQ2, the rotational speed ω of the rotor, the inductance parameters of the model, and the flux linkage ΨPMD1 generated by the permanent magnet material of the rotor. The outputs of the superposition represent voltage target values uD1,ref, uQ1,ref, uD2,ref, and uQ2,ref which are desired values of the rotation converted stator voltages uD1, uQ1, uD2, and uQ2. The processing system 201 is configured to convert the voltage target values uD1,ref, uQ1,ref, uD2,ref, and uQ2,ref into voltage values expressed in a suitable coordinate system bound to the stator of the electrical machine so as to produce the voltage control signals u1 and u2. The voltage target voltages uD1,ref, uQ1,ref, uD2,ref, and uQ2,ref can be converted, for example, into voltage values corresponding to the stator phase voltages with the aid of the equation (20), i.e.

[u1u2]T=C3T3T(θ)[uD1,ref,uQ1,ref,uD2,ref,uQ2,ref]T,  (21)

where C3 is a constant scaling coefficient, and the voltage control signals u1 and u2 are assumed to be row-vectors of values proportional to the target phase voltages of the two multiphase stator windings. It is also possible to convert the voltage target values uD1,ref, uQ1,ref, uD2,ref, and uQ2,ref into voltage values expressed in the α-β coordinate system illustrated in FIG. 1. A functional block 211 shown in FIG. 2 represents the conversion of the voltage target values uD1,ref, uQ1,ref, uD2,ref, and uQ2,ref into the voltage values expressed in the coordinate system bound to the stator.

In an apparatus according to an exemplifying embodiment of the invention, the processing system 201 is configured to determine the target values of the rotation-converted stator currents iD1,ref, iQ1,ref, iD2,ref, and iQ2,ref at least partly on the basis of a desired torque Tref. By a straightforward analysis it can be shown that the torque Te produced by the electrical machine can be estimated with the following equation:

Te=CTp(ΨPMD1iQ1+(LD1−LQ1)iD1iQ1+(LD2−LQ2)iD2iQ2),  (22)

where p is the number of pole pairs of the electrical machine and CT is a constant scaling coefficient.

In the exemplifying case shown in FIG. 2, the electrical machine comprises two multiphase stator windings. In this case, the rotation-converted stator currents iD1, iQ1, iD2, and iQ2 are according to the equations (12b):




  
    
      [
      
        
          
            
              i
              
                D
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              i
              
                Q
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        
          
            
              i
              
                D
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
        
          
            
              i
              
                Q
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
      
      ]
    
    =
    
      
        C
        2
      
      ⁡
      
        [
        
          
            
              
                
                  i
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                +
                
                  i
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
            
          
          
            
              
                
                  i
                  
                    q
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                +
                
                  i
                  
                    q
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
            
          
          
            
              
                
                  i
                  
                    q
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                -
                
                  i
                  
                    q
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
            
          
          
            
              
                
                  i
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
                -
                
                  i
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
            
          
        
        ]
      
    
  
  ,




where C2 is the constant scaling coefficient defined in the equation (8) and id1 is the direct-axis component of a first current space-vector created by the stator phase currents ia1, ib1, and ic1 of the first one of the two multiphase stator windings, iq1 is the quadrature-axis component of the first current space-vector, id2 is the direct-axis component of a second current space-vector created by the stator phase currents ia2, ib2, and ic2 of the second one of the two multiphase stator windings, and iq2 is a quadrature-axis component of the second current space-vector.

The target values of the rotation-converted stator currents iD2 and iQ2 are advantageously, but not necessarily, selected to be zeros, i.e. iD2,ref=iQ2,ref=0. This selection is actually an attempt to achieve symmetrical loading between the two multiphase stator windings because id1=id2 and iq1=iq2 when iD2,ref=iQ2,ref=0. In this case, the reference value iQ1,ref can be given as:




  
    
      
        
          i
          
            
              Q
              ⁢
              
                
              
              ⁢
              1
            
            ,
            ref
          
        
        =
        
          
            
              T
              ref
            
            
              
                C
                T
              
              ⁢
              
                p
                ⁡
                
                  (
                  
                    
                      Ψ
                      
                        PMD
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    +
                    
                      
                        (
                        
                          
                            L
                            
                              D
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                          -
                          
                            L
                            
                              Q
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        )
                      
                      ⁢
                      
                        i
                        
                          
                            D
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          ,
                          ref
                        
                      
                    
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        23
        )
      
    
  




The reference value iD1,ref can be selected, for example, so that the overall stator current is minimized and the stator voltage has a desired value. A functional block 214 shown in FIG. 2 represents the determining of the target values of the rotation-converted stator currents iD1,ref, iQ1,ref, iD2,ref, and iQ2,ref at least partly on the basis of the desired torque Tref.
In an apparatus according to an exemplifying embodiment of the invention, the processing system 201 is configured to use a measured position angle θ of the rotor in the conversion of the stator phase currents of the multiphase stator windings into the rotation-converted stator currents.
In an apparatus according to an exemplifying embodiment of the invention, the processing system 201 is configured to estimate the position angle θ of the rotor on the basis of one or more electrical quantities related to the electrical machine and to use the estimated position angle in the conversion of the stator phase currents of the multiphase stator windings into the rotation-converted stator currents. Methods for estimating the position angle θ of the rotor can be found, for example, from the following publications: T. Halkosaari: “Speed Sensorless Vector Control of Permanent Magnet Wind Power Generator—The Redundant Drive Concept,” in Wind Power. InTech, 2010, 558 p., and A. Piippo, M. Hinkkanen, and J. Luomi: “Sensorless Control of PMSM Drives Using a Combination of Voltage Model and HF Signal Injection” In Conference Record of the 39th IEEE-Industry Applications Society (IAS) Annual Meeting, vol 2, pp 964-970, Seattle, Wash., USA.
The processing system 201 can be implemented with one or more processor circuits, each of which can be a programmable processor circuit provided with appropriate software, a dedicated hardware circuit such as, for example, an application specific integrated circuit “ASIC”, or a configurable hardware circuit such as, for example, a field programmable gate array “FPGA”. For example, some of the functional blocks 205-211, and 214 can be implemented with a dedicated or configurable hardware circuit and some of these functional blocks can be implemented with one or more programmable processor circuits, or all of these functional blocks can be implemented with one or more programmable processor circuits or with dedicated or configurable hardware circuits. The present invention is not limited to any methods of implementation.
FIG. 3 is a schematic flow diagram of a method according to an exemplifying embodiment of the invention for controlling an electrical machine that comprises two or more multiphase stator windings. The method comprises, in a phase 301, converting stator phase currents of the two or more multiphase stator windings into rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine. The conversion is carried out using a conversion rule that corresponds to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents in spite of the mutual inductances between the two or more multiphase stator windings when the inductance parameters of the model are unaltered. The method further comprises, in a phase 302, controlling power stages connected to the multiphase stator windings on the basis of the rotation-converted stator currents and on the basis of the model of the electrical machine.
In a method according to an exemplifying embodiment of the invention, the stator phase currents of two multiphase stator windings are converted into four rotation-converted stator currents, and inductance coefficients between the stator flux-linkages and the rotation-converted stator currents are proportional to: Ld+Md, Lq+Mq, Ld−Md, and Lq−Mq, where Ld is a direct-axis inductance, Md is a direct-axis mutual inductance between the two multiphase stator windings, Lq is a quadrature-axis inductance, and Mq is a quadrature-axis mutual inductance between the two multiphase stator windings.
In a method according to an exemplifying embodiment of the invention, the four rotation-converted stator currents are proportional to id1+id2, iq1+iq2, iq1−iq2, and id2−id1, respectively, where id1 is the direct-axis component of a first current space-vector created by the stator phase currents of the first one of the two multiphase stator windings, iq1 is the quadrature-axis component of the first current space-vector, id2 is the direct-axis component of a second current space-vector created by the stator phase currents of the second one of the two multiphase stator windings, and iq2 is the quadrature-axis component of the second current space-vector.
In a method according to an exemplifying embodiment of the invention, the target values of the rotation-converted stator currents which are proportional to iq1−iq2 and to id2−id1 are zeroes so as to achieve symmetrical loading between the two multiphase stator windings.
A method according to an exemplifying embodiment of the invention comprises measuring a position angle of the rotor and using the measured position angle in the conversion of the stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents.
A method according to an exemplifying embodiment of the invention comprises estimating a position angle of the rotor on the basis of one or more electrical quantities related to the electrical machine and using the estimated position angle in the conversion of the stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents.
In a method according to an exemplifying embodiment of the invention, the controlling of the power stages comprises:

    
    
        regulating rotation-converted stator voltages expressed in the coordinate system bound to the rotor on the basis of differences between the rotation-converted stator currents and their target values, and
        converting the rotation-converted stator voltages into stator voltages expressed in a coordinate system bound to the stator of the electrical machine.
    
    


In a method according to an exemplifying embodiment of the invention, the regulating of the rotation-converted stator voltages comprises:

    
    
        processing the differences between the rotation-converted stator currents and their target values with a regulator, and
        superposing, to outputs of the regulator, voltages caused by rotation of the rotor and expressed in the coordinate system bound to the rotor.
    
    


The regulator may comprise, for example, a separate proportional-integrating “PI” or proportional-integrating-derivative “P I D” regulator for each of the differences between the rotation-converted stator currents and their target values.
A method according to an exemplifying embodiment of the invention comprises determining the target values of the rotation-converted stator currents at least partly on the basis of a desired torque.
A computer program according to an exemplifying embodiment of the invention comprises software modules for the purpose of controlling an electrical machine comprising two or more multiphase stator windings. The software modules comprise computer executable instructions for controlling a programmable processor to:

    
    
        produce voltage control signals, suitable for controlling an electrical machine comprising two or more multiphase stator windings, on the basis of rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two or more multiphase stator windings and mutual inductances between the two or more multiphase stator windings,
        convert stator phase currents of the two or more multiphase stator windings into the rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when the inductance parameters of the model are unaltered.
    
    


The software modules can be e.g. subroutines or functions implemented with a suitable programming language and with a compiler suitable for the programming language and the programmable processor.
A computer program product according to an exemplifying embodiment of the invention comprises a computer readable medium, e.g. a compact disc “CD”, encoded with a computer program according to an embodiment of invention.
A signal according to an exemplifying embodiment of the invention is encoded to carry information defining a computer program according to an embodiment of invention.
The specific examples provided in the description given above should not be construed as limiting. Therefore, the invention is not limited merely to the embodiments described above.

## Claims

1. A method for controlling an electrical machine comprising two multiphase stator windings, the method comprising:
controlling power stages connected to the multiphase stator windings on the basis of four rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two multiphase stator windings and mutual inductances between the two multiphase stator windings,
converting stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are unaltered, and
obtaining a position angle of the rotor and using the obtained position angle of the rotor in the conversion of the stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents,

wherein inductance coefficients between the stator flux-linkages of the model and the rotation-converted stator currents are proportional to: Ld+Md, Lq+Mq, Ld−Md, and Lq−Mq, where Ld is a direct-axis inductance, Md is a direct-axis mutual inductance between the two multiphase stator windings, Lq is a quadrature-axis inductance, and Mg is a quadrature-axis mutual inductance between the two multiphase stator windings.

2. A method according to claim 1, wherein the method comprises measuring a position angle of the rotor and using the measured position angle in the conversion of the stator phase currents of the two multiphase stator windings into the rotation-converted stator currents.

3. A method according to claim 1, wherein the controlling of the power stages comprises regulating rotation-converted stator voltages expressed in the coordinate system bound to the rotor on the basis of differences between the rotation-converted stator currents and their target values, and converting the rotation-converted stator voltages into stator voltages expressed in a coordinate system bound to the stator of the electrical machine.

4. A method according to claim 3, wherein the regulating of the rotation-converted stator voltages comprises processing the differences between the rotation-converted stator currents and their target values with a regulator and superposing, to outputs of the regulator, voltages caused by rotation of the rotor and expressed in the coordinate system bound to the rotor.

5. A method according to claim 4, wherein the regulator comprises a separate proportional-integrating or proportional-integrating-derivative regulator for each of the differences between the rotation-converted stator currents and their target values.

6. A method according to claim 3, wherein the method comprises determining the target values of the rotation-converted stator currents at least partly on the basis of a desired torque.

7. A method according to claim 1, wherein the four rotation-converted stator currents are proportional to id1+id2, iq1+iq2, iq1−iq2, and id2−id1, respectively, where id1 is a direct-axis component of a first current space-vector created by stator phase currents of a first one of the two multiphase stator windings, iq1 is a quadrature-axis component of the first current space-vector, id2 is a direct-axis component of a second current space-vector created by stator phase currents of a second one of the two multiphase stator windings, and iq2 is a quadrature-axis component of the second current space-vector.

8. A method according to claim 7, wherein the target values of the rotation-converted stator currents which are proportional to iq1−iq2 and id2−id1 are zeroes so as to achieve symmetrical loading between the two multiphase stator windings.

9. An apparatus comprising a processing system configured to:
produce voltage control signals suitable for controlling an electrical machine comprising two multiphase stator windings, the processing system being configured to produce the voltage control signals on the basis of four rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two multiphase stator windings and mutual inductances between the two multiphase stator windings, and
convert stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are unaltered, and
said processing system being configured to obtain a position angle of the rotor and to use the obtained position angle of the rotor in the conversion of the stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents,

wherein inductance coefficients between the stator flux-linkages of the model and the rotation-converted stator currents are proportional to: Ld+Md, Lq+Mq, Ld−Md, and Lq−Mq, where Ld is a direct-axis inductance, Ma is a direct-axis mutual inductance between the two multiphase stator windings, Lq is a quadrature-axis inductance, and Mq is a quadrature-axis mutual inductance between the two multiphase stator windings.

10. An apparatus according to claim 9, wherein the processing system is configured to use a measured position angle of the rotor in the conversion of the stator phase currents of the two multiphase stator windings into the rotation-converted stator currents.

11. An apparatus according to claim 9, wherein the processing system is configured to:
direct regulator functions to differences between the rotation-converted stator currents and their target values,
superpose, to outputs of the regulator functions, voltages caused by rotation of the rotor and expressed in the coordinate system bound to the rotor, the outputs of the superposition representing voltage target values expressed in the coordinate system bound to the rotor, and
convert the voltage target values to a coordinate system bound to the stator of the electrical machine so as to produce the voltage control signals suitable for controlling the electrical machine.

12. An apparatus according to claim 11, wherein the processing system is configured to direct a separate proportional-integrating or proportional-integrating-derivative regulator function to each of the differences between the rotation-converted stator currents and their target values.

13. An apparatus according to claim 11, wherein the processing system is configured to determine the target values of the rotation-converted stator currents at least partly on the basis of a desired torque.

14. An apparatus according to claim 9, wherein the four rotation-converted stator currents are proportional to id1+id2, iq1+iq2, iq1−iq2, and id2−id1, respectively, where id1 is a direct-axis component of a first current space-vector created by stator phase currents of a first one of the two multiphase stator windings, iq1 is a quadrature-axis component of the first current space-vector, id2 is a direct-axis component of a second current space-vector created by stator phase currents of a second one of the two multiphase stator windings, and iq2 is a quadrature-axis component of the second current space-vector.

15. An apparatus according to claim 14, wherein the processing system is configured to regulate the rotation-converted stator currents which are proportional to iq1−iq2 and id2−id1 to zeros so as to achieve symmetrical loading between the two multiphase stator windings.

16. An apparatus according to claim 9, wherein the apparatus further comprises modulators for producing switch-control signals on the basis of the voltage control signals, and power stages for providing supply voltages to the multiphase stator windings on the basis of the switch-control signals.

17. A non-transitory computer readable medium encoded with a computer program comprising computer executable instructions for controlling a programmable processor to:
produce voltage target values, suitable for controlling an electrical machine comprising two multiphase stator windings, on the basis of four rotation-converted stator currents expressed in a coordinate system bound to a rotor of the electrical machine and on the basis of a model of the electrical machine modeling at least inductances of the two multiphase stator windings and mutual inductances between the two multiphase stator windings, and
convert stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents with a conversion rule corresponding to a form of the model of the electrical machine where each of stator flux-linkages of the model is dependent on only one of the rotation-converted stator currents when inductance parameters of the model are unaltered, and
obtain a position angle of the rotor and use the obtained position angle of the rotor in the conversion of the stator phase currents of the two multiphase stator windings into the four rotation-converted stator currents,

wherein inductance coefficients between the stator flux-linkages of the model and the rotation-converted stator currents are proportional to: Ld+Md, Lq+Mq, Ld−Md, and Lq−Mq, where Ld is a direct-axis inductance, Md is a direct-axis mutual inductance between the two multiphase stator windings, Lq is a quadrature-axis inductance, and Mq is a quadrature-axis mutual inductance between the two multiphase stator windings.