Question 8.1: Tuning the control of an electrically excited synchronous ma...

(derived by Dr.Pasi Peltoniemi, LUT, 2015) This example reviews the control principles of the electrically excited synchronous machine (EESM) and considers EESM control system tuning.

Control system

Equations(8.11) through(8.28) are the equations in the d-q reference frame for an electrically excited synchronous machine rotating at rotor angular speed, that is, synchronous angular speed. This model can be used to derive model-based tuning rules for the controls used. From Figure E8.1, five different controllers are used to control the electrically excited synchronous machine (EESM) including the speed controller, the stator flux-linkage controller, the field current controller, and two current controllers.
For these controllers, the stator phase currents along with the field current are used for feedback signals. The angular speed of the machine could be measured to provide feedback, or sensorless schemes could be applied. It is assumed here that rotor electrical angular speed ω_r and the current smentioned i_UVW and i_f are measured as depicted in FigureE8.1. The angular velocity ω_c is used in the control.
EESM drive controller tuning is presented for each controller separately.
Stator current control
EESM stator current control is based on a rotor-oriented reference frame that is aligned with the field winding produced flux-linkage component L_md i_f as was shown previously in Figure8.26. The frame is also referred to as the field-oriented control frame. Stator current control is carried out in this specific frame since the inductance values are not affected by the change in rotor angle. However, the inductance values of machine are still subject to saturation, especially when operating in the field-weakening region (Pyrhönen,1998).
In this example, control laws are derived for stator current control in the rotor-oriented frame, and tuning rules are given for the controller parameters based on the system. Inductance values remain with in limits tolerated by the feedback control.
Substituting the flux-linkage expressions of Equation(8.24) and Equation(8.25) into Equation(8.14) and Equation(8.15), respectively, gives the rotor voltage equations in the

rotor-oriented reference frame.

0= R_{D}i_{D} + \frac {d}{dt}(L_{md}i_{d} + L_{D}i_{D} + L_{fD}i_{f})    (E8.1)

0= R_{Q}i_{Q} + \frac {d}{dt}(L_{mq}i_{q} + L_{Q}i_{Q})     (E8.2)

Similarly, substituting Equation(8.21) and Equation(8.22) into Equation(8.11) and
Equation(8.12) yields

u_{d} = R_{s} i_{d} + \frac {d}{dt}(L_{d}i_{d} + L_{md}(i_{D} + i_{f})) – w_{c}Ψ_{q}   (E8.3)

u_{q} = R_{s} i_{q} + \frac {d}{dt}(L_{q}i_{q} + L_{mq}i_{Q})  – w_{c}Ψ_{d}     (E8.4)

The rotor damper currents i_D and i_Q are solved from Equation(8.24) and Equation(8.25).

i_{D} = \frac {Ψ_{D} – L_{md}i_{d} – L_{fD}i_{f} } {L_{D}}     (E8.5)

i_{Q} = \frac {Ψ_{Q} – L_{mq}i_{q}  } {L_{Q}}     (E8.6)

The rotor currents can be eliminated from (E8.3) and (E8.4) using (E8.5) and (E8.6). After simplification, the result is as follows.

u_{d} = R_{s} i_{d} +L_{cc,d} \frac {d}{dt} i_{d} + \frac {L_{md}}{L_{D}}\frac {d}{dt} Ψ_{D} + L_{md} ( 1 – \frac {L_{fD} } { L_{D} } ) \frac {d}{dt} i_{f} – w_{c} Ψ_{q}     (E8.7)

u_{q} = R_{s} i_{q} +L_{cc,q} \frac {d}{dt} i_{q} + \frac {L_{mq}}{L_{Q}}\frac {d}{dt} Ψ_{Q} + w_{c} Ψ_{d}     (E8.8)

The subscript cc refers to current control.

The main idea of field-oriented current control is to enable the independent control of the d-axis stator current i_d component and the q-axis stator current i_q component. This makes it possible to independently control the flux linkage ψ and the electromagnetic torque T of the machine. However, the equations for the stator voltage components (E8.8) and (E8.9) are coupled. That is to say, the d-axis component u_d also depends on variables other than i_d. Similarly, the q-axis component u_q depends on other variables. Therefore, stator voltage components u_d and u_q can not be considered decoupled. Stator currents i_d and i_q can only be independently controlled if the stator voltage equations are decoupled.
It is possible to derive a decoupling scheme that is valid for any operation point. First,(E8.5) and (E8.6) are substituted into (E8.1) and (E8.2), and then the following derivative terms are solved from the resulting equations.

\frac {d}{dt} Ψ_{D} = – \frac {R_{D}}{L_{D}} Ψ_{D} + \frac { L_{md} R_{D} }{L_{D}} i_{d} +\frac {L_{fD} R_{D} }{L_{D}} i_{f}     (E8.9)

\frac {d}{dt} Ψ_{Q} = – \frac {R_{Q}}{L_{Q}} Ψ_{Q} + \frac { L_{mq} R_{Q} }{L_{Q}} i_{q}   (E8.10)

Substituting Equation(E8.9) into Equation(E8.7) and Equation(E8.10) into Equation(E8.8) results in the following expressions.

u_{d} = R_{cc,d} i_{d} + L_{cc,d} \frac {d}{dt}i_{d} – \frac { L_{md} R_{D} }{L^{2}_{D} } Ψ_{D} + \frac { L_{md} L_{fD} R_{D} }{L^{2}_{D} } i_{f} + L_{md} ( 1- \frac {L_{fD} }{ L_{D} } ) \frac {d}{dt} i_{f} – w_{c} Ψ_{q}   (E8.11)

u_{q} = R_{cc,q} i_{q} + L_{cc,q} \frac {d}{dt}i_{q} – \frac { L_{mq} R_{Q} }{L^{2}_{Q} } Ψ_{Q} +  w_{c} Ψ_{d}     (E8.12)

Inserting Equation(8.32) and Equation(8.33) into Equation(E8.11) and Equation(E8.12), respectively, results in these voltage equations.

u_{d} = R_{s} i_{d} + L_{cc,d} \frac {d}{dt}i_{d} – \frac { L_{md} R_{D} }{L_{D} } i_{D}  + L_{md} ( 1- \frac {L_{fD} }{ L_{D} } ) \frac {d}{dt} i_{f} – w_{c} Ψ_{q}     (E8.13)

u_{q} = R_{s} i_{q} + L_{cc,q} \frac {d}{dt}i_{q} – \frac { L_{mq} R_{Q} }{L_{Q} } i_{Q} +  w_{c} Ψ_{d}     (E8.14)

Expressed as follows, the voltage equations can be used as control laws for the EESM.

u_{d,ref} = R_{s} i_{d} + L_{cc,d} \frac {d}{dt} i_{d} + e_{d,}   (E8.15)

u_{q,ref} = R_{s} i_{q} + L_{cc,d} \frac {d}{dt} i_{q} + e_{q,}     (E8.16)

u_{d,ref } and u_{q,ref} represent the voltage references for the power electronics. The decoupling voltage terms e_{d} and e_{q} can be determined from

e_{d} = – \frac { L_{md} R_{D} }{L_{D} } i_{D}  + L_{md} ( 1- \frac {L_{fD} }{ L_{D} } ) \frac {d}{dt} i_{f} – w_{c} Ψ_{q},                                                     (E8.17)

e_{q} = – \frac { L_{mq} R_{Q} }{L_{Q} } i_{Q}  +w_{c} Ψ_{d} (E8.18)

If the field current derivative term can be considered negligible, other decoupling terms may be either measured or estimated and then used for decoupling.
Cross-coupling between stator voltage equations can be eliminated by feeding the decoupling terms (E8.17) and (E8.18) as positive feedback (measured quantities are used to calculate e_d and e_q) or as feed-forward (reference values are used to calculate e_d and e_q) to the output of the current controllers. In both cases, the system behaves like two decoupled linear first order systems from the perspective of the current control. Therefore, current controllers can be designed assuming the system has a transfer function.

G_{(s)} = \frac { I_{dq}(S) }{U_{dq} (S) } = \frac { \frac {1}{R} } {τ_{cc,dq}s+1}                     (E8.19)

where τ_{cc,dq} = \frac { I_{cc,dq} }{ R_{s} }

Simple PI controllers can be used as current controllers, because controlled variables appear as DC quantities in the dq reference frame. According to the Internal Model Control (IMC) principle, PI controller gains can be chosen as follows.

C_{PI} (S) = K_{p} + \frac {K_{i} } {s}                   (E8.20)

K_{p} = a_{cc} L_{cc,dq'} K_{i} = a_{cc} R_{s}     (E8.21)

a_cc is the desired closed loop band width of the current control. Choosing controller parameters using (E8.21), the closed loop system becomes

  G_{c1}(s) = \frac { C_{P1} (S) G(s) } { 1+ C_{P1}(s) G(s) } = \frac { a_{cc} }{s+ a_{cc} }         (E8.22)

Therefore, using the (E8.17) and (E8.18) decoupling scheme and choosing controller parameters based on the (E8.13),(E8.14), and (E8.21) machine model results in good dynamic performance. Instead of using system bandwidth, it may be more convenient to specify desired control performance using rise time t_r,cc. For a first order system, rise time and bandwidth have the following relationship.

a_{cc} = \frac { ln(9) }{ t_{r,cc} }           (E8.23)

Current rise time is limited by the voltage reserve of the inverter, so it can not be set fast arbitrarily. Severe saturation of the controller should be avoided. Obviously, current rise time also determines the torque rise time. Therefore, t_r,cc is an important design parameter.
Field current control
Another low level control in the EESM control concept is field current control. Field current control not only plays a crucial role in producing machine flux, but also in machine stability. Especially in the field-weakening range, the higher level field current reference calculation significantly impacts machine performance. This will be discussed in more detail later in Section 8.14. Here, the focus is on field current controller tuning.
As was done in the previous discussion of stator current control, the upcoming paragraphs will derive the model-based control law and consider the tuning of the appropriate controller for field current control. It is assumed that the variation in inductance values remains small enough to be manageable by the feedback control. Field current supply methods quite commonly suffer from delays with in the  control loop that limits EESM drive performance. There are compensation methods available, but as a minimum requirement the field current controller must be tuned so that field current rise time is higher than the maximum delay in the excitation system. Other wise, instability is likely to occur. In this example delay is ignored.
In FigureE8.1, the field current controller produces the field voltage reference u*_f on the control loop level, which is then produced by the excitation unit. A simplified field current control loop is shown in Figure E8.2 where the field current reference calculation is also depicted.
The dynamics of the field current are modelled as in Equation(8.13) and for field flux linkage as in Equation(8.23). Substituting Equation(8.23) into Equation(8.13) gives
u_{f} = R_{f} i_{f} + L_{f} \frac { d} { dt} i_{f} + L_{md} \frac { d } {dt} i_{d} + L_{fD} \frac { d }{dt} i_{D}     (E8.24)

The following expression describes \frac { d}{dt} i_{D}

\frac { d}{dt} i_{D} = \frac { 1} { L_{D} } ( – R_{D}i_{D} – L_{md} \frac { d}{dt} i_{d} – L_{fD} \frac { d}{dt} i_{f} )       (E8.25)

Substituting (E8.25) into (E8.24) gives

u_{f} = R_{f} i_{f} +( L_{f} – \frac { L^{2}_{ fD} } { L_{D} })  \frac { d} { dt} i_{f} + ( L_{md} – \frac { L^{2}_{fD} L_{md} } {L_{D} } ) \frac { d } {dt} i_{d} – \frac { L_{fD} R_{D} }{L_{D}} i_{D}       (E8.26)

If stator current is assumed constant during the field current transients, then

u_{f} = R_{f} i_{f} +( L_{f} – \frac { L^{2}_{ fD} } { L_{D} })  \frac { d} { dt} i_{f}  – \frac { L_{fD} R_{D} }{L_{D}} i_{D}         (E8.27)

Neglecting the damper current term, (E8.27) can be divided into a first-order system and feed-forward term. Neglecting the damper current term only affects the transient performance of the drive, because i_D=0 in steady-state. A field-current controller can now be designed assuming that the system has a transfer function.

G_{(s)} = \frac { I_{f}(S) }{U_{f} (S) } = \frac { \frac {1}{R} } {τ_{f}s+1}   (E8.28)

where τ_{f} = \frac { L_{f} – \frac { L^{2}_{fD} }{R_{f} }}{ R_{f} }

Again, simple PI controllers can be used as current controllers, because the controlled variables are DC quantities. The PI controller gains can be determined as follows.

C_{PI} (S) = K_{pf} + \frac {K_{if} } {s}           (E8.29)

K_{pf} = a_{f} ( L_{f} – \frac {L^{2}_{fD} } { L_{D} } ) ,  K_{i} = a_{f} R_{f}             (E8.30)

a_f is the desired closed loop bandwidth. Choosing controller parameters using (E8.30), the closed loop system becomes

  G_{CL}(s) = \frac { C_{P1} (S) G(s) } { 1+ C_{P1}(s) G(s) } = \frac { a_{f} }{s+ a_{f} }               (E8.31)

Here, as well as with the stator current controllers, the desired control performance is specified using the t_r,f rise time. For a first-order system, the rise time and bandwidth have the following relationship

a_{f} = \frac { ln(9) }{ t_{r,f} }         (E8.32)

A current rise time cannot be chosen fast arbitrarily, because of the limited voltage reserve of the inverter and the possible delay as previously mentioned. Generally, the excitation system saturation should be avoided.
Test simulations have been carried out for various design bandwidths. The results are shown in Figure E8.3. As expected, the responses are relatively accurate. However, significant parameter variations occur in EESM drives operating in demanding conditions. These variations must be taken into account when designing a control system.
Stator flux linkage control
Stator flux linkage is one of the primary control variables of the EESM drive. Flux linkage control is performed in a stator flux-linkage-oriented frame. In the stator flux-linkage oriented frame, as was illustrated in Figure8.28, stator flux linkage is controlled via the stator current component i_{ψ}, which is aligned with the stator flux linkage. Similarly, electromagnetic torque is controlled with the stator current component i_T which is aligned with the stator voltage. To obtain the current references for current control operating in the field-oriented dq frame, a coordinate transformation is applied as was expressed in Equation (8.183) and Equation(8.184).
The stator flux-linkage control system can be presented as shown in FigureE8.4(a). The figure illustrates that stator flux is controlled from both the stator and field winding. A significant consequence is that the power factor at the stator poles can be controlled. Flux control from the stator side is considered in the upcoming paragraphs. This flux control

operates in cascade structure with the stator current controller as shown in the figure. Assuming equivalent stator current controller dynamics, and if the coordinate transformation from the stator flux-linkage frame in to the rotor flux-linkage frame can be considered a static multiplication presenting negligible delay,the stator flux-linkage control scheme can be simplified into the diagram shown in FigureE8.4 (b).
The voltage drop across the stator resistance becomes significant only at low speeds. Therefore, it is neglected here. The open loop transfer function of the stator flux-linkage control loop can be written as follows

G_{ψ;ol}(s) = \frac {K_{pψ} ( T_{iψ} s + 1)}{T_{iψ}} \frac { a_{cc}}{s+a_{cc} s} \frac {1}{s}            (E8.33)

where K_pψ and T_iψ  are the proportional gain and integrator time constant of the stator flux-linkage controller, respectively, and a_cc is the bandwidth of the current control loop from (E8.23).
In the symmetric optimum method, the integrator time constant of the cascade controller is equal to T_i,ψ=4 T_i,cc. When the parameters shown in Table 8.2 are applied, T_i,cc=0.0129 is obtained for the current controllers. Since the bandwidth of the current control loop (a_cc) is known, setting K_pψ=1 and plotting the root locus curve (TableE8.1) reveals the effect of gain variation on the location of the system poles, and the desired system properties can be chosen.
In FigureE8.5, the root loci have been plotted for current control rise times of 5 ms and 1 ms. As the figure reveals, using faster current controllers allows bigger separation between the pole near the origin and the pole-pair that eventually becomes complex valued. Moreover, the root locus trajectories show that the current control system is robust, since the gain could be increased to infinity, and the system poles would remain in the left half plane (LHP). In practice, converter imposed limits on the performance of flux-linkage control are inevitable. Available voltage and current handling capability of the power electronics are limited. Further more, noise in the measurements could become a control problem if it increases at specific frequencies in the selected bandwidth.
A closed loop transfer function for the stator flux-linkage control loop can be derived from (E8.31).

\frac { Ψ_{s}(S)}{Ψ_{s,ref}(s) } = \frac {a_{cc} ( K_{pψ}s + K_{iψ} )}{ s^{3} + a_{cc}s^{2} + a_{cc} K_{pψ} s + a_{cc} K_{iψ} }               (E8.34)

where K_iψ = K_pψ /Ti_ψ is the integrator gain of the stator flux-linkage controller. As shown in FigureE8.5, selecting a pole location at 25Hz, where the damping is equal to 1, results in gain values of K_{pψ;1 ms} = 167 and K_pψ;5ms =116.
In FigureE8.6, (a) a step response is plotted (TableE8.2) for both designs using a transfer-function-based step response. In both cases, there is response over shoot. This over shoot becomes negligible if are ference value filter (RVF) is applied as shown in FigureE8.4. In FigureE8.6,(b) the same responses are obtained from the Simulink model that contains the detailed model of the control system and machine. The responses are similar, but the settling times of the Simulink model responses are longer. The difference is due to simplifications made in the transfer function approach.

Speed and torque control
In the control scheme, electromagnetic torque is not separately controlled, and as a result, there is no measured or estimated feedback. This is because the machine operates in the stator flux-linkage oriented frame, where electromagnetic torque can be controlled via the current component i_T . If the torque and the stator flux linkage are known, this torque-producing current component can be solved as follows.

i^{*}_{sT} = \frac { T^{*}_{e} } { \frac {3}{2} PΨ^{*}_{s} }     (E8.35)

where the superscript ∗ denotes the reference value. Therefore, asseen in FigureE8.1, the speed controller produces the torque reference T*_e, which is then divided by the stator flux-linkage reference to produce the i_T reference.
The speed control loop is similar to the stator flux-linkage control loop. Therefore, the control loop can be simplified as was done with the stator flux-linkage control loop. The simplified speed control loop is shown in FigureE8.5. Since the current reference is obtained by dividing the output of the speed controller with a stator flux-linkage reference value, it is a static value. The control loop can be simplified even further by not accounting for the division. Now, the open loop transfer function of the speed control loop can be written

G_{T;ol}(s) = \frac {K_{pψ} ( T_{iψ} s + 1)}{T_{iψ}} \frac { a_{cc}}{s+a_{cc} s} \frac {1}{Js}                  (E8.36)

where K_pw and T_iw are the proportion algain and integrator time constant of the speed controller, respectively, and J is the inertia of the machine. Since α_{cc} is known from the current controller tuning process, by setting T_iw=4T_i,cc and using the machine inertia J=0.001 kgm², the root locus of the speed control loop can be plotted (TableE8.3).
In FigureE8.7, the root loci have been plotted for current control rise times of 5 and 1 ms. Again, as was the case with stator flux-linkage controller tuning, faster current controllers allow bigger separations between the pole near the origin and the pole-pair farther away in the left half plane. Similarly, the speed control loop is also robustly stable with respect to speed controller gain variation, since the poles of the control system remain on the left half plane despite the gain value. However, as mentioned earlier for the stator flux-linkage control, increasing the bandwidth could lead to measurement noise. In fact, as seen in Figure E8.7, the damping of the complex valued pole-pair drops continuously as speed controller gain increases.
For the speed control, the closed loop transfer function can be written in the following form

\frac { w_{m}(S)}{w_{m,ref}(s) } = \frac { \frac {a_{cc}}{J} ( K_{pw}s + K_{iw} )}{ s^{3} + a_{cc}s^{2} + \frac { a_{cc} }{J} K_{pw} s +\frac { a_{cc} }{J} K_{iw} }           (E8.37)

The script given in Table E8.4 produces a step response of the speed control loop. The plotted responses are shown in Figure E8.9 (a).
Despite the gain value, some over shoot exists in the response. This overshoot can be made negligible by applying an RVF as shown in Figure E8.7. The effect of the RVF on control performance can be illustrated using the following transfer function.

  \frac { w_{m}(S)}{w_{m,ref}(s) } =\frac {1}{T_{rvf} s +1} \frac { \frac {a_{cc}}{J} ( K_{pw}s+ K_{iw} )}{ s^{3} + a_{cc}s^{2} + \frac { a_{cc} }{J} K_{pw} s +\frac { a_{cc} }{J} K_{iw} }           (E8.38)

where T_rvf is the time constant of the filter. Now, different speed controller gain values are applied when T_rvf =1 ms and the responses are shown in Figure E8.9 (b). The responses reveal the expected result. That is, response approaches the response of the RVF as the gain value is increased, and simultaneously, the overshoot becomes negligible. However, in practice, the limitations mentioned previously in the stator flux-linkage controller discussion are present, and performance is therefore constrained by these limitations.
Finally, the speed control responses are tested with the detailed Simulink model when RVF is used (T_rvf = 1 ms). A high gain value is applied (K_pw=5) for the speed controller. The result of the simulation is shown in Figure E8.7 (c). The response obtained with Simulink is in line with expectations.
EESM drive simulation using unity power factor control
One simulation has been carried out with controller tunings determined as above and applied to the drive. In the simulation, the drive operates under the unity power factor principle, which is executed by calculating the field current reference as follows per (Bühler,1997-II).

i^{*}_{f} = \frac { Ψ^{2}_{s} + L_{d} L_{q} |i_{s}|^{2} }{L_{md} \sqrt { Ψ^{2}_{s} + L^{2}_{q} |i_{s}|^{2} } }         (E8.39)

The machine parameters used correspond to those previous given in Table 8.2. The simulation results are shown in Figure E8.10.
In the simulation, the EESM drive rotates at nominal speed in no-load operation (T_L=0). At t=0.1 s, a nominal torque step takes place and a change in rotational speed is evident from Figure E8.10 (a). The drive reaches new steady-state values with in approximately
0.1 s. The new steady-state values can be calculated as shown in the following Table E8.5. The calculated steady-state values when L_d=1.17, L_q=0.57, L_md=1.05, Ψ*_s = 1, and

Table E8.5 Steady-state solutions of an EESM drive using unity power factor control in pu values

T*_e = 1 are δ_s=29.7°, I_f=1.38, I_d=0.495, I_q=0.869, Ψ_d=0.869, and Ψ_q=0.495.
Figure E8.10 reveals these values to be where the drive settles after the torque transient. The electric current values in the stator flux-oriented frame can be calculated from the inverse transformation of Equation(8.183) and Equation(8.184) as follows.

i_{dref}=-i_{\Psi ref}\cos \delta _{s}-i_{Tref}\sin \delta _{s}       (8.183)

i_{qref}=i_{\Psi ref}\sin \delta _{s}+i_{Tref}\cos \delta _{s}            (8.184)

Iψ = I_d cosδs + I_q sinδ_s =0 and I_T= -I_q sinδ_s + I_q cosδ_s = 1
The machine is only magnetized from the rotor when operated with unity power factor. The result also shows that minimum stator current operation is achieved.