V/f controlled Virtual PM Machine for Grid-connected Inverter
Preferred Topic 1:

Topic 4 – MEASUREMENT, SUPERVISION AND CONTROL FOR
POWER CONVERTERS
Preferred Subtopic 1: Subtopic 4b – Standard and Advanced Current / Voltage /Synchronization Control Techniques
Preferred Topic 2:
Topic 9 – POWER SUPPLIES AND INDUSTRY-SPECIFICAPPLICATIONS
Preferred Subtopic 2: Subtopic 9f – Distributed Power Supplies
Origin:University

Preference:

Dialogue Session

Keywords«DC-AC», «DC-AC converter», «Grid-connected
inverter», «Voltage control»
Abstract
The paper discusses a V/f controlled virtual PM
machine. The state equation of virtual PM
machine is approximated to second-order system.
The output power response is evaluated by thecomparison to the second-order system. The
response is checked by the simulations.
Introduction
Recently, the technics for a high-performance
motor drive and high-efficiency motors have been
actively studied and developed due to the energy
conservation. Especially, permanent magnet
synchronous motors (PMSM) is used because it isefficient. In the control for PMSM, the current
control by a field oriented control (FOC) is known
as an efficient control. The FOC requires the rotor
position sensor for a detection of the magnet poleposition. However, using the position sensor
causes an increasing a cost of motor drive systems.
Also, it is difficult to apply the position
sensors to high-speed motors. Hence, asensorless FOC is commonly used for the PMSM
driver. The sensorless FOC estimates themagnetic pole position. However, the estimation
requires a high-performance digital signal
processor. Thus, a V/f control is proposed for thecontrol of PMSM with a simple speed control.
In this paragraph, the application of the motor
drive technic to the grid-connection system is
described. The grid-connection system has
similarities with the PMSM drive system. For
example, inductance, resistance and a induced

voltage that included PMSM is considered to thecircuit elements of the grid-connection system.
This paper discusses about V/f controlled virtual
PMSM for a grid-connected system. The
proposed V/f control is fixed for a grid-connected
inverter. The V/f control outputs the inverter
voltage according to rotation speed. However, thespeed control is impossible by the V/f control
because the grid angular frequency is not
generally determined by the inverter voltage.
Thus, a virtual PMSM is used to the speed control
for synchronous to the grid. The virtual load isused in the V/f control because the output torqueof PMSM is determined by a load torque and the
speed is generally constant. As a result, the V/fcontrolled PMSM is the output power controller.
The output power control response is evaluated by
the comparison with the approximated low-ordersystem.
V/f control for grid-connected inverter
A. Comparison between PMSM and gridconnection
Figure 1 shows a circuit diagram of the gridconnection. Comparing the elements of this gridconnection system and the PMSM, the wiring
resistance R corresponds to the winding resistance,
the grid-tied inductance L corresponds to thearmature inductance, and the grid voltage
corresponds to the induced voltage. In addition,
the voltage equation is same as the surface PMSM(SPMSM) because d-axis inductance and q-axis
inductance are equal. In the other hand, the gridconnection system does not include a load torqueand a moment of inertia. Also, the grid frequency
corresponding to a electric frequency of PMSM is
uncontrollable. Thus, the rotation speed is also
uncontrollable. However, the output powercontrol of the grid-connection system is possible
by reproducing the load torque and the momentof the inertia with a controller.
B. V/f control

Figure 2 shows the relationship between the dqaxis coordinate and the -axis coordinate. The
voltage equation of PMSM in the dp-axis
coordinate is expressed as in (1),
vd   R + pL − L  id   0 
v  =  +
R + pL  iq   m 
 q   L

Vgrid

(2),

2 gridwhere Vgrid is the maximum grid line-to-line
voltage, grid is the grid angular frequency. The
voltage equation and the output power equation ofgrid-connection in the -axis coordinate areexpressed as in (3) and (4), respectively,v   R + pL −grid L  i  Vgrid  sin  v  =   L R + pL  i  +
2 cos      grid  

Vgrid iq
2

=

R

Vgrid

Vgrid ( i sin  + i cos  )2Fig. 1. The circuit configuration of the grid-connected
inverter.

qe = Vgrid

Vinv





S

d

N

Fig. 2.The relationship between the dq-axis coordinateand -axis coordinate.
grid

*

Outputpower
control

Pcommand *
Pout

(3),
(4),

where  is the deviation angle between -axis
coordinate and dq-axis coordinate, Pout is theoutput power.
The state equation of PMSM is derived by
performing a linear approximation near thesteady-state in the same way as Ref. [1]-[4], andexpressed as in (5),v*0f/V conv.
High
efficiency
control(1),where R is the winding resistance, L is thearmature inductance, p is the differential
operator, m is the flux linkage by PM,  is theelectric angular frequency. The grid-voltage
agrees the q-axis voltage vq in the gridconnection with a no-load condition. Thus, thevirtual flux linkage of the grid-connection isexpressed as in (2),Pout =L
Vdc

Generally, PMSM control uses the dq-axis
Cartesian coordinate system with the d-axis
defined as the direction of the PM flux and theq-axis defined as the direction of the induced
voltage. In this paper, the q-axis defined as the
direction of the grid voltage to apply the V/fcontrol for the grid-connection. The proposed
control principle is based on the V/f control withthe stabilization control [1]-[4], which has been
proposed for PMSM. The V/f control uses the
-axis Cartesian coordinate system with the axis defined as the direction of the inverter
output voltage and the -axis defined as the
direction 90 degrees behind the -axis.

m =

\

v

+
HPF

K1

+
*-

+


to
uvw

v*
+

inv*

-

Damping
control

1
s

q*

i

uvw
to


i

v
v

vu*
vv*
vw*

iu
iv
iw

vu
vv
vw

uvw
to


Fig. 3. The control diagram of the proposed V/f control
for the grid-connection system.

(5),px = Ax + Butwhere x = i

i

R

−

L


−0
A=
 2
P

 f m sin 
0
 J

0


0

−

R
L

−

t

1
L


B = 0

0
 0

0
1
L
0
0


i 0 
−i 0 


0 
1 

−
Pf 2 mJ   , u =  v
m
L

m
L

sin  0cos  0cos  0

0

0

−1

v

m

inv  ,




m

sin  0
L
Pf 2 mi 0 sin  0 + i 0 cos  0 ) 
(
J

0

−

L

cos  0,,

Pf is the number of pole pairs and J is the momentof inertia. Note that the subscript "0" indicates thevalue of the operating point in each variable.
Figure 3 shows the block diagram of the proposed
V/f control for the grid-connection. The dampingcontrol is used to the suppression of the torqueoscillation due to the resonance between the
moment of inertia and the armature inductance[1]-[4].

Pcommand * +The following conditions are assumed in orderto briefly discuss the stability and response.

Pout

grid
1
Grid Js-(i) No-load condition:  = 0, i = 0, i = 0

 + + *(a) The virtual inertia(ii) High-speed rotation condition: 0L >> R, 0L >> K1i(iii) The mechanical time constant issufficiently larger than the electrical time
constant: p(∆i) = 0, p(∆i) = 0

Pcommand * +
-

 =

1
2 Pf

n =

J
cos  0 K1
L

Pf m
JL

(6),
(7),

where K1 is the damping gain. As shown in Eq.(5),the damping factor  is the function of the
feedback gain K1. In other words, the proposed
V/f control is stabilized by setting K1
appropriately.
C. Output power control
Figure 4 shows the block diagrams of the output
power controllers. Fig.4(a) shows the output
power control by the virtual inertia. The state
equation is same as Eq.(5) in the case of applying
the virtual inertia. Thus, the output power control
response is derived and agrees the approximatedsecond-order system of PMSM. The naturalangular frequency n and the damping factor  areshown in Eq.(6) and Eq.(7).
Fig.4(b) shows the block diagram of the
proportional output power control. The
relationship between the torque of PMSM and theelectric angular frequency is derived by ignoringa viscous resistance, and expressed as in (8),
p =

Pf
J

(T − TL )(8),

 + + *Pout(iv) The effect of HPF on stability is small.
The natural angular frequency n and thedamping factor  are calculated from thecharacteristic equation of the second-ordersystem, which is derived by approximating thederived four-order state equation expressed asin Eq.(5) as a second-order system from theassumption of from (i) to (iv), and expressed asin (6) and (7),Kpow ergrid(b)The proportional control

Pcommand * +-LPF

Kpow ergrid + + *Pout(c)The proportional control with LPF
Fig.4. The control diagram of the output power
controller.

where T is the output torque and TL is the load
torque.
In the other hand, the relationship the output
power of the proportional control and the electric
angular frequency with the proposed V/f control
is expressed as in (9), = K power ( Pout − Pcommand ) =K powergrid(T − TL ) (9),where Kpower is the proportional gain of the output
power control on the proposed V/f control andPcommand is the output power command. As shown
Eq.(9), there is no the moment of inertia J in the
proposed V/f control. Thus, The state equation of
the grid-connected inverter does not include themoment of inertia J, and the state equation isderived by replacing J in Eq.(5). The way toreplace J is derived by the relationship between
Eq.(8) and Eq.(9), and expressed as in (10),J=

Pf
pK powergrid(10).

The natural angular frequency of the gridconnected inverter n1 is calculated from thecharacteristic equation of the first-order system,
which is derived by approximating the derived
three-order state equation expressed as in Eq.(5)and Eq.(10) as a first-order system from theassumption of from (i) to (iv), and expressed asin (11),
n1 =Pf 2 m 2 K powergrid
L

(11).

In the grid-connected system with the proposed
V/f control, the number of pole pair Pf isincluded the design of the proportional gain
Kpower. Thus, the natural angular frequency n1
is expressed as in (12),n1 =K powerVgrid 2
2grid L(12).

The control response of the output power
control without the damping control isexpressed as in (13),
Pout
n1
=
Pcommand s + n1(13).

As shown in Eq.(13), there is no the oscillation
occurs due to the resonance between the momentof inertia and the armature inductance. Fig.4(c)shows the block diagram of the proportional
control with LPF used in Ref. [5]. The control
response is derived by adding of LPF to Eq.(12)and expressed as in (14),PoutLPF n1=
Pcommand s 2 + sLPF + LPF n1(14),

where LPF is the cutoff angular frequency of LPF.
The natural angular frequency n2 and thedamping factor  are expressed as in (15) and (16), =1 LPF
2 n1

(15),

(16).
In the comparison of each methods as shown as
Fig.4, the control response should be secondorder system by the disturbance suppression
characteristics. However, the proportional output
power control improves the disturbance
suppression characteristics by the damping
control. The control response by the adding thedamping control to the proportional control isexpressed as in (17),n 2 = 2n1

Pout=
Pcommandsn1 + n1HPFK V s 2 + s  n1 + HPF − HPF grid  + n1HPF
2 grid L (17),where HPF is the cutoff angular frequency of the
HPF and KHPF is the damping gain with theproportional power control. In the case of therelationship between the angular frequency HPF>> n1, the control response of Eq.(17) is can beapproximated to Eq.(14) by the parameter design,
is expressed as in (18) and (19),

i

+
+

i

L

*

Qdq *+

Kp

-

LPF

v*

Q*

v

Fig.5 the control diagram of the high efficiency control.
Table 1 the simulation conditions.
Rated output power
Grid-tied inductanceLine resistance
Virtual inertia

P out
L
R
J

Grid anguler frequency
Grid maximum
line-to-line voltage
Damping factor
Natural anguler frequency

 grid0.08 kgm2
314 rad/s

V grid

283 V


n

0.707
14.2 rad/s

 =


K V 
1
 HPF − HPF grid 
2 n1HPF 
2grid L 

n1d = n1HPF

1 kW
10 mH(%Z:24%)
0.2 

(18),
(19),

where n1d is the natural angular frequency of the
proportional control with the damping control.
D. High efficiency control
Figure 5 shows the block diagram of the highefficiency control. The torque / current ratio isminimized by the id = 0 control with SPMSM. The
id = 0 control is achieved indirectly by using thereactive power. In the dq-axis coordinate, thereactive power Qdq is expressed as in (18),(18).
In the steady state, the reactive power thatassigned the voltage equation Eq.(1) is expressedas (19),(19).
Qdq =  L (id 2 + iq 2 ) + mid 
Qdq = vqid − vd iq

As shown Eq.(18), the reactive power that the id =0 control is achieved is expressed as (20),(20).
Qdq =  Liq 2 =  LI 2
The reactive power in the -axis coordinate Q
is expressed as in (21),Q = v i
(21).
Thus, the condition required for the id = 0 control
is expressed as in (22), L ( id 2 + iq 2 ) = v i
(22).
The function of the high efficiency control is the
suppression the reactive power to zero. Thisfunction is same as the Q-V droop control.
Simulation results
Table 1 shows the simulation conditions. The
control parameters are designed so that eachFigure 7 shows the frequency characteristic of
the proportional power control with thedamping control. The HPF cutoff angularfrequency is set the 20 times of the angularfrequency n1 designed by Eq.(12). From thecomparison of the frequency characteristics,
the response approaches the second-ordersystem from the first-order system by the damping
control. Fig.8 shows the response with theproportional control with the damping control.
The response approaches from a first-ordersystem to a second-order system. From thecomparison with Fig.6(b), the fluctuation of thegrid frequency is suppressed. The cause of the

Output power, Pout [W]Second-order system

0.1

0.2
0.3
Time [sec.]

0.4

0.5

(a) The virtual inertia.
The proportional control

Output power, Pout [W]Figuer 6 shows the output power response byeach output power controls shown Fig. 4. Fig.
6(a) shows the response with the virtual inertia.
The response with the virtual inertia isapproximated to a second-order system by
Eq.(6) and Eq.(7). From the simulation results,
it is checked that the response agrees a secondorder system. Fig.6(b) shows the response withthe proportional control. The response with theproportional control is approximated to a firstorder system as shown Eq.13. From thesimulation results, it is checked that the
response agrees a first-order system. However,the fluctuation of the grid frequency occurs on
the output power. The cause of the fluctuation
is a DC unbalanced current due to change in thevoltage amplitude of the grid-tied inductor. The
same fluctuation occurs when the power
response is a second-order system. However,
the effect of the fluctuation is large because thedisturbance suppression characteristics of the
proportional control is inferior than the one ofthe virtual inertia. Fig.5(c) shows the responsewith the proportional control with LPF. The
response with the proportional control isapproximated to a second-order system asshown Eq.14. From the simulation results, it ischecked that the response agrees a secondorder system. In addition, the similar response
to V/f controlled virtual PMSM is obtained by
the control parameter design of Eq.(15) andEq.(16).

The virtual inertia

1000
900
800
700
600
500
400
300
200
100
0
0.0

1000
900
800
700
600
500
400
300
200
100
0
0.0

First-order system

0.1

0.2
0.3
Time [sec.]

0.4

0.5

(b) The proportional control.
The proportional control with LPF

Output power, Pout [W]natural frequency n, n1, n2 and n1_d, areequal. The output power command is a stepcommand that changes to the rated output
power Pout at t = 0. The ideal control response
is designed to be a Butterworth filter of firstorder and second-order system. The output
power response by each output power control
was checked to agree the ideal response.

1000
900
800
700
600
500
400
300
200
100
0
0.0

Second-order system

0.1

0.2
0.3
Time [sec.]

0.4

0.5

(c) The proportional control with LPF
Fig. 6 The output power control response.

suppression is the improving of the disturbance
suppression characteristic by the damping control.
Figure 9 shows the dq-axis current response withthe virtual inertia by the high efficiency control.
The d-axis current is controlled to zero by the highefficiency control. Thus, the reactive power issuppressed to zero.

Conclusion

0First-order system

-20
-40

Magnitude [dB]

This paper discusses about the V/f controlled
virtual PMSM for the grid-connection system.
The output power response is designed tosecond-order system by the virtual inertia andthe damping control. In addition, The secondorder response is approximately obtained in
the case of the proportional output power
control is used with the damping control. In
the future, the disturbance suppression
characteristics will be evaluated for further
improvement of the characteristics. Also, theoutput power response and the improving of
the disturbance suppression characteristics
will be confirmed by the experiment.

-60

Second-order system

-80
-100
-120
10-1

100

101
Frequency [Hz]

102

103

Fig. 7 The frequency characteristics of the proportional
control with the damping control.
The proportional control with the damping control

Output power, Pout [W]References

1000
900
800
700
600
500
400
300
200
100
0
0.0

Second-order system
First-order system

0.1

0.2
0.3
Time [sec.]

0.4

0.5

Fig. 8 The output power control response by the
proportional control with the damping control.
With the high efficiency control

iq5

dq-axis current [A][1] J. Itoh, N. Nomura, and H. Ohsawa, “A
comparison between V/f control and positionsensorless vector control for the permanent magnet
synchronous motor”, PCC2002, pp.1310-1315,
2002.
[2] J. Itoh, T. Toi, M. Kato: "Maximum Torque per
Ampere Control Using Hill Climbing Method
Without Motor Parameters Based on V/f Control",18th European Conference on Power Electronics
and Applications (EPE'16), No. DS3d-Topic 40283, 2016[3] J. Itoh, T. Toi, K. Nishizawa: "Stabilization
Method Using Equivalent Resistance Gain Basedon V/f Control for IPMSM with Long Electrical
Time Constant", The 2018 International PowerElectronics Conference, No. 23E1-5, pp. 22292236, 2018[4] J. Itoh, T. Toi, K. Nishizawa: "Stabilization
Method for IPMSM with Long Electrical Time
Constant Using Equivalent Resistance Gain Basedon V/f Control", IEEJ Transactions on Industry
Applications, Vol. 8, No. 4, pp. 592-599 (2019)[5] N. Pogaku, M. Prodanovic, and T. C. Green,“Modeling, analysis andtesting of autonomous
operation of an inverter-based microgrid,”
IEEETrans. Power Electron., vol. 22, no. 2, pp.
613–625, Mar. 2007.

The proportional control
with the damping control

Without the high efficiency control

4

3
2
1

With the high efficiency control

0
0.0

id

0.2

0.4
0.6
Time [sec.]

0.8

1.0

Without the highefficiency control

Fig. 9 The dq-axis current response by the virtualinertia with the high efficiency control.