Abstract— The functional models derived in previous work for stability analysis of electric power systems for “more electric” vehicular technologies is validated by time domain simulations of a test system using both the derived functional models and behavioral models in which details of the converter switching process are fully represented. The analytically derived functional model is employed to perform small-signal stability analyses and to assess the influence of operating conditions and leading design parameters on system stability.

Index Terms—Vehicular power systems, Power system stability, Linearization.

I. INTRODUCTION “More electric” technologies in naval [1]-[3], automotive

[4]-[5], and aerospace [6]-[7] industries are gaining increasing interest thanks to their potential to reduce the environmental impact and costs of transportation systems. Power electronics controlled machine drives for power generation, propulsion, actuation and environmental control allows the achievement of increased reliability and efficiency compared with their mechanical and hydraulic counterparts currently in use [2]-[5], [8]. Future “more electric” vehicular electrical power systems will require the interconnection of a wide range of electrical components, resulting in a significantly more complex electrical distribution system with multiple distributed loads, most of which are supplied and controlled by power electronic converters. However, tightly regulated motor drives, DC/DC and DC/AC converters pose a significant constraint on the safe

operation of the electrical power system due to their inherent constant power characteristic which in turn results in destabilizing negative impedance behaviour [9]-[10].

A detailed understanding of the power system dynamic behaviour and of its stability margins is a prerequisite for the safe operation of the power system and its optimization. The derivation of a framework for state-space averaged modelling of a typical power system architecture, Fig. 1, proposed for the “more electric” aircraft, has been reported in [11]-[13]. It consists of a number of power electronics controlled motor drives loads fed by a common DC bus powered by a variable frequency synchronous generator whose output is rectified by an 18-pulse autotransformer rectifier unit (ATRU).

In this paper, a comprehensive validation of the developed functional model is undertaken by means of time domain simulations. Comparison of the simulation results obtained with both the developed average valued functional model and detailed behavioural model in which switching transients are fully represented are presented and discussed.

Small-signal stability of the power system is performed by evaluating the eigenvalues of the system Jacobian matrix, and sensitivity calculations are undertaken in order to assess how operating conditions and system design and control parameters influence its modes and stability behaviour. Instability phenomena predicted by the eigenvalue analysis is verified by the results obtained from time domain simulations using both the functional and behavioural models.

Fig. 1 Schematic of hybrid power system for “more electric” aircraft

II. TIME DOMAIN SIMULATIONS AND MODEL VALIDATION The block diagram in Fig. 2 summarizes the governing

equations and their inter-dependency of the state variables of

the representative system under investigation. In order to validate the developed functional models, detailed behavioral models for all power system components have been established in SaberTM simulation environment, as shown in

Stability Analysis of “More Electric” Aircraft Hybrid Electrical Power Systems

Antonio Griffo1 and Jiabin Wang2 1 Department of Electrical and Electronic Engineering, the University of Bristol, United Kingdom

2 Department of Electronic and Electrical Engineering, the University of Sheffield, United Kingdom Email: a.griffo@bristol.ac.uk, j.b.wang@sheffield.ac.uk

Fig. 3. Thorough time-domain simulations have been carried out to validate the analytically derived functional models by comparing the resulting waveforms with those obtained using the detailed behavioral models. The results of a test case are presented in Figs. 4-9. The synchronous generator is initially open-circuited under AVR control and its output is connected to the AC bus at 10ms when the rated bus voltage is established. At 50 ms the motor drive load is applied with its speed demand gradually increased from 0 to 300 rad/s over 0.25s and a load torque of 165Nm. The load torque demand is further increased to 333 Nm at 0.4s. A step increase of 10% load power demand at 0.5s eventually drives the system to instability. Motor drive current control loop bandwidth fc is set to 2kHz, while the speed control loop bandwidth fs=1/3fc. The AC voltage controller is designed to yield ζ=0.8, ωn=100rad/s. Other parameters are listed in Appendix. Figs. 4-5 compare the generator output voltages Vd,Vq and currents Id

gen, Iqgen,

respectively while Fig. 6 compares the field winding voltage and currents Vfd, Ifd. Similar comparisons of DC bus voltage and current are shown in Fig. 7, while motor drive load voltages and currents are compared in Figs. 8-9, respectively. As can be seen, an excellent agreement between the transient waveforms of the state variables predicted by the functional and behavioral models is observed, albeit the switching

harmonics are not present in the waveforms of the functional model. It is worth noting that simulation of the functional model took less than 5s to complete, compared with roughly 10 min required for the simulation of the behavioral model on the same computer.

VA

Vfd

Igenerator Irectifier

imotor

Synchronous generator eqs.

AC voltage equations eq.

ATRU eq.

DC capacitor eq.

Motor drive eqs.

AC voltage regulator eq.

Fig. 2 Simulation block diagram for power system employing average valued functional model.

Fig. 3 Saber schematic of power system simulation model

D−axis

voltage(V

)

0.0

100.0

200.0

300.0

400.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Q−axis

voltage(V

)

100.0

200.0

300.0

400.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6100

200

300

400

Q-a

xis

volta

ge (V

)

0 0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

time (s)

D-a

xis

volta

ge (V

)

(b)

Fig. 4 D-Q axis generator voltages, obtained with behavioral model (a) and functional model (b)

D−axis

current(A

)

−400.0

−300.0

−200.0

−100.0

0.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Q−axis

current(A

)

−300.0

−200.0

−100.0

0.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6-200

-150

-100

-50

0

Q-a

xis

curr

ent (

A)

0 0.1 0.2 0.3 0.4 0.5 0.6-400

-300

-200

-100

0

time (s)

D-a

xis

curr

ent (

A)

(b)

Fig. 5 D-Q axis generator currents, obtained with behavioral model (a) and functional model (b)

Field

windingcurrent

0.0

20.0

40.0

60.0

80.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Field

windingvoltage(V

)

0.0

20.0

40.0

60.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

Fiel

d w

indi

ng v

olta

ge (V

)

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

time (s)

Fiel

d w

indi

ng c

urre

nt (A

)

(b)

Fig. 6 Generator field winding voltage and current obtained with behavioral model (a) and functional model (b)

DCvoltage(V)

500.0

520.0

540.0

560.0

580.0

600.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

DCcurrent(A)

0.0

100.0

200.0

300.0

400.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

DC

cur

rent

(A)

0 0.1 0.2 0.3 0.4 0.5 0.6500

550

600

time (s)

DC

vol

tage

(V)

(b)

Fig.7 DC bus voltage and current obtained with behavioral model (a) and functional model (b)

D−axis

voltage(V

)

−200.0

−150.0

−100.0

−50.0

0.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Q−axis

voltage(V

)

0.0

100.0

200.0

300.0

400.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

Q-a

xis

volta

ge (V

)

0 0.1 0.2 0.3 0.4 0.5 0.6-200

-150

-100

-50

0

time (s)

D-a

xis

volta

ge (V

)

(b)

Fig. 8 D-Q axis motor voltage obtained with behavioral model (a) and functional model (b)

D−axis

current(A

)

−10.0

−5.0

0.0

5.0

10.0

time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Q−axis

current(A

)

100.0

200.0

300.0

400.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6100

200

300

400

Q-a

xis

curr

ent (

A)

0 0.1 0.2 0.3 0.4 0.5 0.6-10

-5

0

5

10

time (s)

D-a

xis

curr

ent (

A)

(b )

Fig. 9 D-Q axis motor current obtained with behavioral model (a) and functional model (b)

III. SMALL-SIGNAL STABILITY ANALYSIS Small-signal stability analysis has been performed by

computing the eigenvalues of the system Jacobian matrix and the influence of the operating conditions and main design parameters on stability margins has been subsequently assessed. Modal analysis based on the calculation of participation factors has been applied in order to measure the contribution of each mode/eigenvalue into state dynamics [15]-[16]. Table I reports the identified eigenvalues, listed in order of decreasing magnitude, along with the states that show the largest participation into the associated modes. The calculation was made against the steady state operation when the generator frequency was 400Hz, and the drive output power was 110kW with motor drive current control loop bandwidth fc =1kHz. The other parameters of the system are the same as those previously stated in section II.

Modes #1 to #5 are found to be well damped in all operating conditions. They are mainly related to synchronous generator damper windings dynamics and AC capacitor. Modes #8 to #11 are also relatively well damped and are mostly associated with the current and speed control loops of the motor drive. Modes #14-and #15 are essentially related to the AC voltage regulator, while modes #16 and #17 are related to the time constant of the motor d-q axis windings.

It has been verified that the system stability mostly depends on the two pairs #6,7 and #12,13 of complex eigenvalues, which are closely associated with DC bus dynamics and AC bus voltage regulator, respectively.

TABLE I SYSTEM’S MODES AND DOMINANT STATES

Mode # Eigenvalue Dominant states

1,2 -27e3±j83e3 Vd,Vq,ψd,ψq 3,4 -1e3±j73e3 Vd,Vq,ψd,ψq

5 -22000 ψd,ψq,ψkD,ψkQ 6,7 -348±j4700 Udc,idc,ψd,ψq,

8 -6246 idm,xD

9 -3532 iqm,ωm

10 -2098 ωm,x, iqm

11 -1586 ωm, x, iqm

12,13 -99±j298 ψfd,ψkD 14 -178 ψkQ,ψq 15 -102 xG, Vfd 16 -50 xQ, iq 17 -51 xD, id

By way of example, Fig. 10 shows the eigenvalue loci when the motor drive output power is varied from 90kW to 150kW with other parameters and operating condition being the same as stated previously. The remaining eigenvalues whose real part is less than -250 are not shown. As will be seen, the mode #6,7 with a natural frequency of ~800Hz becomes unstable when the load power is greater than 120kW. In order to illustrate the destabilizing effects of negative impedance characteristic of the tightly regulated motor drive load, Fig. 11 shows the eigenvalues loci when the current control loop bandwidth fc of the motor drive load is varied from 500Hz to 2000Hz while the speed control loop bandwidth is set to (1/3)fc. Again it is evident that the ~800Hz mode #6,7 become unstable when fc is above 1.8kHz. However, the lower frequency pair #12,13 associated with the AC bus regulator is virtually left unaffected by the variations in fc. It can also be seen that the eigenvalues of modes, #8 to #11, which are closely related to the current and speed control loops, moves to the left as the control bandwidth increases. These modes are nevertheless well damped although a complex pair is present. It should be noted that the instability due to high bandwidth drive controllers predicted by the eigenvalue analysis is in agreement with the results obtained from of time domain simulations shown in Figs. 4-9 when fc=2kHz.

Conversely, the dependency of the low frequency pair eigenvalues, i.e., modes #12 and 13 of on the AVR control parameters is illustrated in Fig. 12 for different combinations of the closed loop damping ratio ζ and bandwidth ωn. However, these variations have little effect on the other eigenvalues.

The influence of synchronous operating frequency on the system stability is shown in Fig. 13. As the frequency increases, the stability margins associated with modes #6,7, i.e. the DC bus dynamic behavior improves. This is due to the fact that the damping effect introduced by the commutation overlap of the ATRU increases with frequency. However, an opposite trend is observed for the eigenvalues of mode #11,12 which is related to the AVR control gains. This is because the control gains should ideally be inversely proportional to the generator operating frequency. However, for a classic AVR, fixed gains designed against a nominal operating frequency of 400Hz with a damping ratio of 0.8 and natural frequency of 50Hz are assumed in the eigenvalue analysis. Consequently, as the

operating frequency increases, the effective loop gain of the AVR increases which leads to a more oscillatory response but the system remains stable over the frequency range ωs/2π=250~800Hz.

-200 -100 0 100 200 300

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

Real λi

Imag

λi

Load power=90÷150kW

Fig. 10 Influence of load power on eigenvalues

-4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0-6000

-4000

-2000

0

2000

4000

6000

Real λi

Imag

λi

fc=500÷2000Hz

Fig. 11 Influence of current control bandwidth fc of the motor drive on eigenvalues

Figure 14 shows the stabilizing effect of increasing the DC

bus capacitance Cdc on modes #6,7. It is also evident that the variation of Cdc has little effect on modes #12,13. The opposite trend is observed in Fig. 15 when the DC filter inductance Ldc, is increased from 5 to 40 μH, which implies that modes #6,7 is less stable with increase in the inductance. On contrary, an improvement in stability margins for modes #6,7 is visible in Fig. 16 when the equivalent AC commutation inductance Lac is increased. This is again due to the fact that the damping associated with the commutation overlap increases with the increase in Lac which represents the combined effect of the AC bus cable inductance and leakage inductance of the ATRU.

An increase in the AC capacitance CAC have been found, as shown in Fig. 17, to reduce the natural frequencies of modes #1,2 and #3,4 with their damping ratios being increased.

-120 -100 -80 -60 -40 -20 0-400

-300

-200

-100

0

100

200

300

400

Real λi

Imag

λi ωn=50rad/s

ζ=0.4 ζ=0.2ζ=0.6ζ=0.8

ωn=200rad/s

Fig. 12 Loci of low frequency eigenvalues as a function of generator control parameters

-600 -400 -200 0 200-6000

-4000

-2000

0

2000

4000

6000

Real λi

Imag

λ i

ωs/2π=250÷800Hz

-600 -400 -200 0 200-6000

-4000

-2000

0

2000

4000

6000

Real λi

Imag

λ i

ωs/2π=250÷800Hz

Fig. 13 Influence of generator operating frequency ωs on eigenvalues

-400 -200 0 200 400 600

-6000

-4000

-2000

0

2000

4000

6000

Real λi

Imag

λi

Cdc=100÷600μF

Fig. 14 Influence of DC bus capacitance Cdc on Eigenvalues

However, the AC capacitance also affects the eigenvalues of modes #6,7 and #12,13 as shown in the inset in Fig. 17.

While modes #6,7 are more stable by an increase in CAC, the stability margin of modes #12,13 are adversely affected. The underlining cause of this problem lies in the fact that the design of the AVR controller assumes open-circuit, and the effect of the capacitive load is not taken into account.

-200 -150 -100 -50 0 50 100-6000

-4000

-2000

0

2000

4000

6000

Real λi

Imag

λi

Ldc=5÷40μH

Fig. 15 Influence of DC filter inductance Ldc on Eigenvalues

-4000 -3000 -2000 -1000 0-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

Real λi

Imag

λi

Lac=1÷20μH

Fig. 16 Influence of AC commutation inductance Lac on Eigenvalues

-3 -2 -1 0 1 2x 104

-8

-6

-4

-2

0

2

4

6

8

x 104

Real λi

Imag

λi

-100 -50 0 50-5

0

5CAC=10÷250μF

x103

Fig. 17 Influence of AC capacitance CAC on eigenvalues

IV. SENSITIVITY ANALYSIS In order to quantitatively understand how design and

parameter choice can influence system dynamics, the sensitivities of eigenvalues with respect to variations in the most influential design and operational parameters have been calculated. The sensitivity of eigenvalue λi with respect to parameter γ, normalized with respect to the eigenvalue magnitude, can be calculated as [17]-[19]:

T

T1 i i

i

i i i

λ γγ λ

∂⋅ ⋅

∂ ∂=∂ ⋅

Aw v

w v (16)

where vi and wi are the right and left eigenvectors, respectively, corresponding to the i-th eigenvalue λi of the modal matrix A:

T Ti i i

i i i

λ

λ

⋅ =⎧⎪⎨

⋅ =⎪⎩

A v v

w A w. (17)

Real and imaginary parts of sensitivities of eigenvalues are shown in the histograms of Figs. 18-23. Only sensitivities of modes #6 to #17 of Tab. 1 are shown as they are the most relevant to the system stability studies. Fig. 18 shows the sensitivities of modes to changes in the synchronous frequency ωs. It is clear that the real part of sensitivity of mode #6-7 is negative, implying that an increase in ωs decreases the real part of the eigenvalue pair. On the other hand the real part of sensitivity of mode #12-13 is positive, indicating that this eigenvalue pair moves towards the right on the complex plane as the frequency increases. Both calculations are consistent with the behavior observed in the eigenvalue loci of Fig. 13. Similar calculations have been carried out for the sensitivities of eigenvalues against changes in the DC bus capacitance Cdc, the DC filter inductance Ldc, the AC bus capacitance Cac, the AC inductance Lac, and the control bandwidth of the motor drive, as illustrated in Figs. 19-23. All numerically calculated sensitivities are consistent with the result of small-signal analyses reported in the previous section. As will be seen, modes #15-#17 are not affected by variation of these parameters as the sensitivity associated with them is virtually zero.

6 7 8 9 10 11 12 13 14 15 16 17

-6

-4

-2

0

2

x 10-4

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-5

0

5x 10-4

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 18 Sensitivity of real and imaginary parts of eigenvalues to variations of synchronous operating frequency ωs

6 7 8 9 10 11 12 13 14 15 16 17-100

-50

0

50

100

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-2000

-1000

0

1000

2000

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 19 Sensitivity of real and imaginary parts of eigenvalues to variations of DC capacitor Cdc

6 7 8 9 10 11 12 13 14 15 16 17-4000

-2000

0

2000

4000

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-5000

0

5000

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 20 Sensitivity of real and imaginary parts of eigenvalues to variations of DC inductor Ldc

6 7 8 9 10 11 12 13 14 15 16 17-4000

-2000

0

2000

4000

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-5000

0

5000

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 21 Sensitivity of real and imaginary parts of eigenvalues to variations of AC capacitor Cac

6 7 8 9 10 11 12 13 14 15 16 17-3

-2

-1

0

1

2x 104

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-1

-0.5

0

0.5

1x 104

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 22 Sensitivity of real and imaginary parts of eigenvalues to variations of AC commutation inductance Lac

6 7 8 9 10 11 12 13 14 15 16 17-20

-15

-10

-5

0

5x 10-4

Rea

l par

t sen

sitiv

ity

6 7 8 9 10 11 12 13 14 15 16 17-1

-0.5

0

0.5

1x 10-5

Mode #

Imag

inar

y pa

rt se

nsiti

vity

Fig. 23 Sensitivity of real and imaginary parts of eigenvalues to variations of motor drive control bandwidth

V. CONCLUSION As vehicular power systems grow in complexity and power

ratings a clear understanding of systems dynamic behavior is of paramount importance for their design and safe operation. An accurate and computationally efficient model of a typical power system for ‘more electric’ aircraft has been established. Through detailed simulations the developed model has been shown to accurately capture the dynamic behavior of power system transients. The utilities of the linearized model for small-signal stability analysis has been demonstrated and validated. Comprehensive and quantitative analyses of the influence of design parameters and operating conditions on stability margins have been undertaken. It has been shown that the system stability is essentially dominated by two complex modes associated with the DC and AC bus voltage transients. The modeling and stability assessment technique provide an effective tool for design optimization of the passive filter and for evaluation of the stability robustness against parametric uncertainties. They can be readily extended to more complex, multiple loads architectures or adapted to other vehicular or

terrestrial power systems that include power electronics controlled components.

VI. APPENDIX Parameters used in simulations are listed in Tabs. II-III

TABLE II SYNCHRONOUS GENERATOR PARAMETERS

Rfd=3.62mΩ RkD=33mΩ RkQ=44mΩ Rs=11mΩ Lfd=159μH LkD=50.4μH LkQ=15μH Ls=25μH Lmd=738μH Lmq=660μH Time constant of the exciter Tex = 10ms.

TABLE III MOTOR DRIVE PARAMETERS

R=65mΩ Lq=1.28mH Ld=1.28mH J=0.1 kgm2 B=6e-4Nms Ke=0.764 Vs Kt=1.13Nm/A PWM switching frequency=10kHz

TABLE IV PASSIVE COMPONENTS

Ldc=25μΗ Cdc=200μF RL-dc=30Ω CAC=10μF Lac=2.7μΗ Rac=5mΩ

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