Electric' Aircraft - IEEE Xplore

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, China

Stability Assessment of AC Hybrid Power Systems for ‘More Electric’ Aircraft L. Han, J. Wang, A. Griffo, and D. Howe Department of Electronic and Electrical Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, United Kingdom

Abstract—This paper analyses the small-signal stability of an AC hybrid power system for ‘more-electric’ aircraft. It consists of a synchronous variable frequency generator which supplies frequency insensitive AC loads, and high voltage DC (HVDC) loads via a 12-pulse transformerrectifier unit. Various sub-system dynamics are derived and combined to form a complete system dynamic equation. The small-signal stability of the system under various operating conditions is assessed by evaluating eigenvalues of the Jacobian matrix of the linearised system. It has been shown that the system stability deteriorates as the power drawn by the motor drive load increases, and the load filter capacitance decreases. Keywords: power system stability, “more electric” aircraft

I.

INTRODUCTION

The increasing use of electrical power in military and civilian aircraft will result in significantly more complex electrical distribution systems with multiple distributed power electronic converters to supply various static and dynamic loads. Therefore, ‘more-electric’ aircraft power systems will be highly non-linear and time-varying, and contain dynamics in time-scales of differing orders of magnitude, and will impose increased demands as regards harmonic generation, integrity and reliability. However, power electronic converters generally exhibit a constant power characteristic by virtue of their regulation capability. Hence, they behave as negative impedance loads [1]-[3]. The interaction of such loads with the power source/generator sub-system can cause instability problems, due to the combined effect of the energy

storage components (inductance/capacitance), non-linear or actively controlled loads, and actively controlled subsystems and their interactions. To safeguard power network stability and to optimise the system architecture and performance, it is essential that the dynamic behaviour, which is influenced by a variety of design, control and operational parameters, is fully understood, and comprehensively assessed. This paper analyses the small-signal stability of the AC hybrid power systems for ‘more-electric’ aircraft shown in Fig. 1. It consists of a synchronous variablefrequency generator whose output is connected to the AC bus which directly feeds frequency insensitive loads such as de-icing loads denoted as Rac. The power system includes a local high voltage dc (HVDC) bus which is derived from the AC bus via a 12-pulse Transformer Rectifier Unit (TRU). The output of the TRU supplies power electronic loads, denoted as PCPL, such as Electrohydraulic actuators (EHA)/Electromechanical actuators (EMA) or Electric Environmental Control System (ECS) via R-L-C filters. Resistive loads RL may also be present at the HVDC bus. To improve power quality on the AC side, a capacitor filter may be employed on the AC bus, or an 18-pulse autotransformer rectifier unit (ATRU) may be used. Further, for the simplicity, the effect of cable which connects various power electronic loads is neglected. Thus the combined effect of power electronic loads are represented by a single constant power load PCPL.

Fig. 1 Schematic of a representative AC hybrid power system for ‘more electric’ aircraft

C 2008 IEEE 978-1-4244-1849-7/08/$25.00○

2008 IEEE Vehicle Power and Propulsion Conference, Harbin, China

Vm*

Vd Vq

V d2 + V q2 V m

K pS +

K iS E fd s

1 sTex + 1

Va

V fd

Vb Vc

Fig. 2 Block diagram of voltage regulator and exciter

II. MODELLING OF SUB-SYSTEMS The dynamics of the AC hybrid power systems are governed by the synchronous generator and its control, the TRU, and the HVDC filters and loads as well as the AC loads and capacitor filter. The dynamics of each subsystem will be described as follows. A. Excitation system of synchronous generator Figure 2 shows the block diagram of a typical excitation system [4] for a synchronous generator. It consists of an automatic voltage regulator (AVR) that produces the field excitation voltage demand, and an exciter that uses a diode rectifier bridge to provide dc excitation current to the synchronous machine field winding. The generator terminal voltage Vm is compared with the desired reference voltage Vm*, and a proportionalintegral control is employed to regulate the output (AC bus) voltage. The exciter is represented by a first-order delay function with a time constant Tex which models the time delay between the exciter field voltage Vfd and the demand Efd. The voltage behind transient reactance model [5] is employed to design the proportional and integral gains KpS and KiS of the voltage regulator. The nonlinear state-space equations which represent the dynamic regulation behaviour of the excitation system in Fig. 2 are expressed by: K pS K pS * ⎧ dV fd 1 1 = − V fd + xG − Vd2 + Vq2 + Vm ⎪⎪ dt Tex Tex Tex Tex ⎨ ⎪ dxG = − K V 2 + V 2 + K V * iS d q iS m ⎪⎩ dt

(1)

where xG is the internal state of the voltage PI controller. Equations in (1) are linearized as: K pSVd 0 ⎧ dΔV fd 1 1 =− ΔV fd + ΔxG − ΔVd ⎪ dt T T TexVm* ex ex ⎪ K pSVq 0 K pS * ⎪ Vm − ΔVq + ⎨ * TexVm Tex ⎪ K iSVq 0 ⎪ dΔxG K iSVd 0 * ⎪ dt = − V * ΔVd − V * ΔVq + K iS ΔVm m m ⎩

behaviour of the synchronous generator. It has Vd and Vq, and the excitation voltage Vfd, as the inputs, stator and rotor flux linkages, ψd, ψq, ψfd, ψkD and ψkQ, as state variables. The stator currents, Id and Iq, as outputs, and the rotor currents, Ifd, IkD and IkQ, as intermediate variables, are derived as: ⎡ I d ⎤ ⎡ LLd − d ⎢I ⎥ ⎢ 0 ⎢ q⎥ ⎢ ⎢ I fd ⎥ = ⎢ LL fd − d ⎢ ⎥ ⎢ ⎢ I kD ⎥ ⎢ LLkD − d ⎢ I kQ ⎥ ⎢ 0 ⎣ ⎦ ⎣

0

LLd − fd

LLd − kD

LLq − q 0

0

0

LL fd − fd

LL fd − kD

0 LLkQ − q

LLkD − fd 0

LLkD − kD 0

0 ⎤ ⎡ψ d ⎤ LLq − kQ ⎥⎥ ⎢⎢ ψ q ⎥⎥ 0 ⎥ ⎢ψ fd ⎥ ⎥⎢ ⎥ 0 ⎥ ⎢ψ kD ⎥ LLkQ − kQ ⎥⎦ ⎢⎣ψ kQ ⎥⎦

(3) where the components of the matrix can be found from the inverse of the dq axis-inductance matrix of a synchronous machine [5][6]. Thus the linearized small signal state space equation is given by: ⎧ dΔ ψ d ⎪ dt = ΔVd + Rs LLd − d Δψ d + ωs Δψ q + Rs LLd − fd Δψ fd ⎪ + Rs LLd − kD Δψ kD ⎪ Δ d ψ q ⎪ = ΔVq − ωs Δψ d + Rs LLq − q Δψ q + Rs LLq − kQ Δψ kQ ⎪ dt ⎪ dΔψ fd ⎪ = ΔV fd − R fd LL fd − d Δψ d − R fd LL fd − fd Δψ fd ⎨ dt ⎪ − R fd LL fd − kD Δψ kD ⎪ dΔ ψ kD ⎪ = − RkD LLkD − d Δψ d − RkD LLkD − fd Δψ fd ⎪ dt − RkD LLkD − kD Δψ kD ⎪ ⎪ dΔψ kQ = − RkQ LLkQ − q Δψ q − RkQ LLkQ − kQ Δψ kQ ⎪ ⎩ dt

(4) (2)

where Vd0 and Vq0 are the d- and q-axis output voltages of the generator. B. Synchronous generator A full-order synchronous machine model in the d-q axis reference frame is used to represent the dynamic

where, Rs, RkD, RkQ, and Rfd are the stator resistance, dand q-axis damper winding resistances, and excitation winding resistance, respectively. ωs is the angular frequency of the generator. C. 12-pulse Transformer Rectifier Unit A state-space average valued model of the 12-pulse TRU, which includes the effect of the AC cable inductance, transformer leakage inductance, the inter-phase reactor (IPR) and the dc filter inductor, has been reported in [7]. The TRU model contains the average valued dc inductor


current Idc as a state variable, and takes the peak AC bus voltage Vm and the dc capacitor voltage Udc as inputs. The state-space equation is given by: dI dc = f1 (μ , I dc ) + f 2 (μ , Vm ) + f 3 (μ , U dc ) dt

(5)

where μ is the commutation angle given by: μ = cos−1 1 − ωs LI dc / 3Vm , L is the AC cable and transformer leakage inductance and the definition for f1, f2 and f3 are given in [7]. The outputs of the TRU are the inductor current Idc and AC current IRD and IRQ represented in the AC bus dq-axis reference. IRD and IRQ are related to Idc and Vm by:

(

)

⎧ 1 3 1 I RD '' − I RQ'' ) ( I RD ' + ⎪⎪I RD = nv 2 2 ⎨ 1 1 3 ⎪I = (I + I + ) I ⎪⎩ RQ nv RQ ' 2 RD'' 2 RQ''

⎧ 3 3Vm ⎛ 1 3⎞ I dc cos μ + ⎜ cos 2μ − cos μ + ⎟ ⎪ I RD ' = ⎪ π πωs L ⎝ 4 4⎠ ⎨ V 3 3 1 1 ⎞ ⎛ m ⎪I = − I sin μ − ⎜ sin 2μ − sin μ + μ ⎟ ⎪⎩ RQ ' π dc πωs L ⎝ 4 2 ⎠ ⎧ π 3 I dc cos( μ + ) ⎪ I RD '' = π 6 ⎪ π π μ 3 3⎤ ⎪ 3Vm ⎡ 1 ⎪+ πω L ⎢ 4 cos(2μ + 6 ) − cos( μ + 6 ) − 4 + 8 ⎥ ⎪ s ⎦ ⎣ ⎨ π 3 ⎪I = − I dc sin( μ + ) ⎪ RD '' π 6 ⎪ 3V ⎡ 1 π π 3μ 3 ⎤ m ⎪− + ⎥ ⎢ sin( 2μ + ) − sin( μ + ) + 6 6 4 8⎦ ⎪⎩ πωs L ⎣ 4

(6)

(7)

(8)

where nv is the voltage ratio of the three winding transformer. The total AC load current, including the effect of the AC resistive load Rac is given by: I Ld = I RD + Vm / Rdc ;

I Lq = I RQ

(9)

It should be noted that the d-axis of the synchronous machine model is referred to the rotor axis while the daxis of the AC bus dq reference is selected to coincide with the peak of the AC bus voltage. The angular difference δ between the q-axis of the rotor reference and the d-axis of the AC bus reference is given by

δ = tan −1 (Vd / Vq ) ; Vm = Vd2 + Vq2

(10)

Thus the AC load currents expressed in the rotor dq reference frame are given by: I ds = I Ld sin δ − I Lq cos δ I qs = I Ld cos δ + I Lq sin δ

(11)

D. AC bus filter capacitor and HVDC load filter capacitor The state-space equation for the AC capacitor voltage expressed in the rotor dq reference system is given by:

dVd I d I ds = − + ω sVq dt Cb Cb dVq I q I qs = − − ω sVd dt Cb Cb

(12)

Similarly the capacitor voltage at the HVDC loads is governed by: dU dc I dc U dc P = − + CPL dt C 2 C 2 RL C 2U dc

III.

(13)

SYSTEM DYNAMICS

Combining equations (1-13) and performing small signal perturbation around an operating point yields the linearized state space equations of the following form: K pSVd 0 ⎧ dΔV fd 1 1 =− ΔV fd + ΔxG − ΔVd ⎪ Tex Tex TexVm* ⎪ dt K pSVq 0 K pS ⎪ − ΔVq + ΔVm* ⎪ TexVm* Tex ⎪ ⎪ dΔxG = − K iSVd 0 ΔV + K iSVq 0 ΔV + K ΔV * d q iS m ⎪ dt Vm* Vm* ⎪ dΔψ d ⎪ = ΔVd + Rs LLd −d Δψ d + ωs Δψ q + Rs LLd − fd Δψ fd ⎪ dt ⎪ + Rs LLd −kD Δψ kD ⎪ dΔ ψ q ⎪ = ΔVq − ωs Δψ d + Rs LLq −q Δψ q + Rs LLq −kQ Δψ kQ ⎪ dt ⎪ dΔψ fd ⎪ dt = ΔV fd − R fd LL fd − d Δψ d − R fd LL fd − fd Δψ fd ⎪ − R fd LL fd − kD Δψ kD ⎪ ψ Δ d ⎪ kD = − RkD LLkD −d Δψ d − RkD LLkD − fd Δψ fd ⎪ dt ⎪ − RkD LLkD −kD Δψ kD ⎨ ⎪ dΔψ kQ = − RkQ LLkQ −q Δψ q − RkQ LLkQ −kQ Δψ kQ ⎪ ⎪ dt ⎪ dΔVd = − H1 ΔV + ⎛⎜ − H 2 + ω ⎞⎟ΔV + LLd −d Δψ d s⎟ q d ⎜ C ⎪ dt Cb Cb b ⎝ ⎠ ⎪ LLd − fd LL H ⎪ + Δψ fd + d −kD Δψ kD − 5 ΔI DC ⎪ Cb Cb Cb ⎪ Δ LL d V ⎛ ⎞ H H q q −q ⎪ = ⎜⎜ − 3 − ωs ⎟⎟ΔVd − 4 ΔVq + Δψ q ⎪ dt Cb Cb ⎝ Cb ⎠ ⎪ LLq − kQ H ⎪ + Δψ kQ − 6 ΔI DC ⎪ Cb Cb ⎪ A V ⎪ dΔI dc = A3Vd 0 ΔVd + 3 q 0 ΔVq + ( A1 + A2 + A4 )ΔI dc ⎪ dt Vm Vm ⎪ + A5 ΔU DC ⎪ ⎪ dΔU dc = ΔI dc − ΔI dc + PCPL ΔU dc ⎪ dt C2 C2 RL C2 U DC 0 2 ⎩

(

)

(14) where Vd0, Vq0, Udc0 are AC bus and DC capacitor voltages at the operating point. The definition for H1~H6 and A1~A5 are given in appendix. As can be seen, there are, in total, 11 states in the system, of which, 5 states are associated with the synchronous generator, 2 states with


the generator control, 2 states with the AC capacitor voltages, one state for dc output current, and one state for the DC capacitor voltage. It is also evident that there are complex coupling between these states via the transformer rectifier unit. IV.

SMALL SIGNAL STABILITY ANALYSIS

The small signal stability of the system can be analysed by evaluating the eigenvalues of the Jacobian matrix of the system equations (14). By way of example, Fig. 3 shows the distribution of the eigenvalues of the Jacobean matrix when a constant power load of 80kW and a resistive load RL = 50kW are connected to the HVDC bus and a 50kW resistive load is present at the 400Hz AC bus whose phase-to-neutral voltage is 230V rms. The voltage ratio of the three-winding transformer was set to 1:1:1; whilst the equivalent leakage inductance L and resistance r referred to the secondary and tertiary circuits are 158μH and 2mΩ; rdc=1mΩ, Lipr+Ldc=40μH, C2=1mF and Cb = 0.01mF. The time delay of the exciter response is set to 8ms and the voltage control bandwidth is 90rad/s. 2000 1500 1000

Vfd and is mainly influenced by the exciter control parameter Tex. The real eigenvalues λ6 and λ5 are essentially related to the state variables ψkD, ψkQ and dependent on the parameters of the generator d- and qaxis circuits. The conjugate pair of eigenvalues λ3 & λ4, which has a low natural frequency, is mainly related to the state variables xG and ψfd, and depends on the voltage controller and the field circuit parameters, KpS, KiS, Lfd and Rfd. The conjugate pair of eigenvalues λ1 & λ2, which has a relatively high natural frequency, has a strong relation with the state variables Idc and Udc, and is predominantly determined by the equivalent ac inductance L and resistance r, the dc inductance Lipr +Ldc and resistance rdc, the dc filter capacitance C2 and the HVDC loads PCPL and RL . Figure 4 shows the variation of the high natural frequency eigenvalues with the constant power load PCPL and dc filter capacitance C2 when the AC resistive load are not present and Cb =0.1mF, the other conditions being the same as those stated previously. As will be seen, the system stability deteriorates as the power drawn by the constant power load increases, and the load filter capacitance decreases. The system becomes unstable when PCPL>~115kW and C2 = 0.5mF. The influence of other parameters, such as the generator control bandwidth, the AC filter capacitance, the transformer leakage and cable inductance and the generator operating frequency, on the eigenvalue distribution and system stability can be similarly studied, but are not presented due to space limit. V.

Imag(rad/s)

500

Time domain simulation using detailed and average valued models have been undertaken. Figure 5 compares the magnitude of the ac bus voltage, the d-q currents of the generator, the dc voltage and current obtained from the detail model and the average value model of the 12pulse TRU under the conditions stated in Fig. 3. A step change in constant power load from 80kW to 110kW occurs at 0.02s. As will be seen, the averaged value model can capture the key features of the transient responses of the AC hybrid power system supplied by the synchronous generator.

0 -500 -1000 -1500 -2000 -600

-500

-400

-300 Real(rad/s)

-200

-100

TIME DOMAIN SIMULATION

0

Fig. 3 Eigenvalues sensitivity map 3000

340 Vm(V)

2000

330 320

1000

0

C2 = 0.5mF

I d(A)

Imag(rad/s)

310

C2 = 1.0mF C2 = 2.0mF

-1000

400 350 300 250 200

C2 = 4.0mF Iq(A)

170

-2000

160 150 140

-400

-350

-300

-250

-200 -150 Real(rad/s)

-100

-50

0

50

100

Fig. 4 Influence of constant load power on eigenvalues

Udc (V)

-3000 -450

530 510 490 470

There are in total 11 eigenvalues, four of which related to the damping winding parameters of the generator and the AC filter capacitor are on the far right of the s-plane and hence not shown. The seven eigenvalues in Fig. 3 consists of three real and two conjugate pairs. The real eigenvalue λ7 has a strong relation with the state variables

I dc(A)

450 300 250 200 150 100

0

0.02

0.04

0.06 Time(s)

0.08

Fig. 5 Comparison of transient responses

0.1

0.12


Figure 6 shows the transient response of the generation terminal voltage and currents, and the dc voltage and current with the variation of the constant power load PCPL. The dc filter capacitance C2 is 0.5mF and the ac filter capacitance Cb = 0.1mF. Initially, a 100 kW PCPL load is connected to the HVDC bus and the other conditions are same as those for Fig. 5. PCPL is increased to 110kW at 0.1s, and to 120kW at 0.3s. It can be seen that the system becomes unstable when PCPL is increased to 120kW, which is also predicted in Fig. 4. Thus, the time-domain simulation results are consistent with those obtained by eigenvalue analysis. Detailed analysis and comprehensive evaluation of the time/frequency domain simulation results will be reported in the full paper.

[2] [3]

[4]

[5] [6]

[7]

Vm(V)

340

L. Han, J. Wang, and D. Howe, “Small signal stability studies of a 270V DC more electric aircraft power system”, Proceedings of IEE PEMD2006, Dublin, 4-6 April, 2006, pp. 197-201. A. Emadi, and M. Ehsani, “Negative impedance stabilising controls for PWM DC/DC converters using feedback linearization techniques”, Proceedings of 35th Intersociety Energy Conversion Engineering Conference and Exhibition, 2000, pp. 613-620. R. Schaefer and K. Kim, “Excitation control of the synchronous generator”, IEEE Industry applications magazine, March/April, 2001, pp. 37-43. P.Kundur, “Power system stability and control”, ISBN 0-07035958-X, McGaw-Hill, Inc., 1994 J. Jatskevich, S. D. Pekarek and A. Davoudi, “Parametic averagevalue model of synchronous machine-rectifier systems”, IEEE Trans. Energy Conversion, vol. 21, no. 1, 2006, pp. 9-18. L. Han, J. Wang and D. Howe, “State-space average modelling of 6- and 12-pulse diode rectifiers”, Proc. of 12th European Conference on Power Electronics and Applications (EPE) 2007, Aalborg, Denmark, Paper ID 639. 2007.

330 320

APPENDIX

310

I d(A)

450

A. Definition for A1 ~ A5

400 350

A1 to A5 are associated with the linearization of (5) and dependent on the commutation overlapping angleμ. They are given by:

300

I q(A)

120 100 80

(a)

Udc(V)

60 540 520 500 480 460 440

I dc (A)

300 250 0

0.1

0.2

0.3 Time(s)

0.4

0.5

0.6

Fig. 6 Transient responses to step changes in PCPL

VI.

A3 =

CONCLUSION

State-space equations which govern the dynamic behavior of an AC hybrid power system for “more electric” aircraft have been derived. Small-signal stability of the system has been assessed by evaluating the eigenvalues of the Jacobian matrix of the linearized statespace equations at a given operating point. This allows the influence of the design, control and operational parameters of the power system on stability to be quantified. It has been shown that system stability margin is dependent on the operating frequency of the generator, and that system stability deteriorates as the constant power loads (motor drive loads) increase and the dc filter capacitance decreases. Time domain simulations using detailed models have validated the results obtained from the eigenvalue analysis. REFERENCES [1]

⎛R R ⎞6 ∂f1 R ⎛R R ⎞6 ∂μ = −⎜⎜ 1 − 2 ⎟⎟ μ 0 − 2 − ⎜⎜ 1 − 2 ⎟⎟ I dc 0 π π ∂I dc L L L L L ∂ I dc 2 ⎠ 2 2 ⎠ ⎝ 1 ⎝ 1 ⎡⎛ 1 3 + 2 3 1 3 ⎞ ∂f 6 ⎟ cos μ 0 A2 = 2 = Vm 0 ⎢⎜⎜ − ∂I dc π L2 2 ⎟⎠ 4 ⎣⎢⎝ L1 π ⎞⎤ ∂μ 1 3 ⎛ cos⎜ μ 0 − ⎟⎥ − L2 2 6 ⎠⎦ ∂I dc ⎝

A1 =

350

200

when μ ≤ π/6,

A. Emadi, B. Fahimi, and M. Ehsani, “On the concept of negative impedance instability in more electric aircraft power systems with constant power loads”, Proceedings of 34th Intersociety Energy Conversion Engineering Conference, SAE Journal, 1999

∂f 2 6 ⎡⎛ 1 3 + 2 3 1 3 ⎞ ⎟ sin μ 0 = ⎢⎜⎜ − ∂Vm π ⎢⎣⎝ L1 4 L2 2 ⎟⎠ 1 3 ⎛ π ⎞ 1 3⎤ sin ⎜ μ 0 − ⎟ + − ⎥ 6 ⎠ L2 4 ⎦ L2 2 ⎝

⎛ 1 1 ⎞6 ∂μ ∂f 3 = −⎜⎜ − ⎟⎟ U dc 0 ∂I dc ∂I dc ⎝ L1 L2 ⎠ π ⎛1 1 ⎞6 1 ∂f A5 = 3 = −⎜⎜ − ⎟⎟ μ0 − L L L π ∂U dc 2 ⎠ 2 ⎝ 1

A4 =

(b) When π/6 ≤ μ ≤ π/3, A1 =

A2 =

⎛R R ⎞6 ⎛ 2R R ⎞ ∂f1 = −⎜⎜ 3 − 1 ⎟⎟ μ 0 − ⎜⎜ 1 − 3 ⎟⎟ ∂I dc ⎝ L3 L1 ⎠ π ⎝ L1 L3 ⎠ ⎛R R ⎞6 ∂μ − ⎜⎜ 3 − 1 ⎟⎟ I dc 0 π L L I dc ∂ 1 ⎠ ⎝ 3 ⎡⎛ 1 3 1 3 + 2 3 ⎞ ⎛ ∂f 2 π 6 ⎟ cos⎜ μ 0 − ⎞⎟ = Vm 0 ⎢⎜⎜ − ⎟ ⎝ ∂I DC π L L 4 4 6⎠ 1 ⎠ ⎣⎢⎝ 3 ⎤ ∂μ 1 3 + cos μ 0 ⎥ L3 4 ⎦ ∂I dc


π ∂f 2 6 ⎡⎛ 1 3 1 3 + 2 3 ⎞ ⎛ ⎟ sin⎜ μ 0 − ⎞⎟ = ⎢⎜⎜ − ⎟ ∂Vm π ⎢⎣⎝ L3 4 L1 4 ⎠ ⎝ 6⎠ 1 3 + 2 3 1 3⎤ 1 3 + − sin μ 0 + ⎥ L3 4 L1 L3 8 ⎦ 8

A3 =

⎛ 1 1 ⎞6 ∂μ ∂f 3 = −⎜⎜ − ⎟⎟ U dc 0 L L ∂ I dc π ∂I dc 1⎠ ⎝ 3 ⎛ 1 1⎞6 ⎛2 1⎞ ∂f A5 = 3 = −⎜⎜ − ⎟⎟ μ 0 − ⎜⎜ − ⎟⎟ ∂U dc ⎝ L3 L1 ⎠ π ⎝ L1 L3 ⎠

3

3 1 B3 − B 4 2 2 1 3 B 6 = B 2 + B3 + B4 2 2

B5 = B1 +

where, ⎛3 ⎞ ⎞ ⎛7 R1 = ⎜ r + rdc ⎟ ; R 2 = (r + rdc ) ; R3 = ⎜ r + rdc ⎟ ⎝4 ⎠ ⎠ ⎝8

(

⎛7 ⎞ L1 = ⎜ L + Ldc + + Lipr ⎟ ; L2 = L + Ldc + + Lipr ⎝8 ⎠

)

⎛3 ⎞ L3 = ⎜ L + Ldc + + Lipr ⎟ ⎝4 ⎠

∂μ = B0 = ∂I dc

⎛ ⎞ ωL 1 − ⎜⎜1 − s I dc 0 ⎟⎟ 3Vm 0 ⎝ ⎠

2

3Vm 0

3

B2 = − −

(cos μ 0 − I dc 0 sin μ 0 B0 ) + 3Vm0 3

⎛ 1 ⎞ ⎜ − sin 2μ 0 + sin μ0 ⎟ B0 ⎠

(sin μ0 + I dc 0 cos μ0 B0 )

3Vm 0 ⎛ 1 1⎞ ⎜ cos 2μ 0 − cos μ 0 + ⎟ B0 2⎠

πωs L ⎝ 2

π π 3⎛ ⎞ ⎜ cos( μ0 + ) − I dc 0 sin( μ 0 + ) B0 ⎟ π ⎝ 6 6 ⎠ π π 1⎤ 3Vm 0 ⎡ 1 + − sin(2 μ0 + ) + sin( μ 0 + ) − ⎥ B0 πωs L ⎢⎣ 2 6 6 4⎦ 3⎛ π π ⎞ ⎜ sin( μ0 + ) + I dc 0 cos( μ 0 + ) B0 ⎟ 6 6 π ⎝ ⎠ 3Vm 0 ⎡ 1 3⎤ π π − ⎢ cos(2 μ 0 + ) − cos( μ0 + ) + ⎥ B0 6 6 4 ⎦ πωs L ⎣ 2

B4 = −

D6 = D 2 +

1 3 D3 + D4 2 2

then

(

πω s L ⎝ 2

π

3 1 D3 − D 4 2 2

H1 = I Ld 0 cos δ 0 + I Lq 0 sin δ 0

B. Definition for H1 ~ H6 Let

π

D5 = D1 +

ωs L

1

r is the AC cable and transformer winding resistance, rdc and Ldc is the resistance and inductance of the DC inductor, respectively, and Lipr is the inductance of the inter-phase reactor. μ0, Idc0, Udc0 and Vm0 are the steady state values of μ, Idc, Udc and Vm under a given operating condition. Vm0 = Vm*

B3 =

πωs L ⎝ 4

⎡1 π π μ0 3 3 ⎤ + ⎢ cos(2 μ0 + ) − cos( μ 0 + ) − ⎥ 6 6 4 8 ⎦ πωs L ⎣ 4 3μ 0 3 ⎤ π π 3 ⎡1 D4 = − + ⎥ ⎢ sin( 2 μ0 + ) − sin( μ 0 + ) + πωs L ⎣ 4 6 6 4 8⎦

D3 =

A4 =

B1 =

3 ⎛1 3⎞ ⎜ cos 2 μ0 − cos μ 0 + ⎟ 4⎠ 3 ⎛1 1 ⎞ D2 = − ⎜ sin 2 μ 0 − sin μ 0 + μ 0 ⎟ πωs L ⎝ 4 2 ⎠ D1 =

)

Vq 0

(V )

2

m0

⎛ sin δ 0 ⎞ Vd 0 ⎟ + ⎜⎜ D5 sin δ 0 − D6 cos δ 0 + Rac ⎟⎠ Vm 0 ⎝ Vd 0 H 2 = − I Ld 0 cos δ 0 + I Lq 0 sin δ 0 (Vm0 )2 ⎛ sin δ 0 ⎞⎟ Vq 0 + ⎜ D5 sin δ 0 − D6 cos δ 0 + ⎜ R ac ⎟⎠ Vm 0 ⎝ Vq 0 H 3 = (− I Ld 0 sin δ 0 + I Lq 0 cos δ 0 ) (Vm0 )2 ⎛ cos δ 0 ⎞⎟ Vd 0 + ⎜ D5 cos δ 0 + D6 sin δ 0 + ⎜ R ac ⎟⎠ Vm 0 ⎝ V H 4 = −(− I Ld 0 sin δ 0 + I Lq 0 cos δ 0 ) d 0 2 (Vm0 ) ⎛ cos δ 0 ⎞⎟ Vq 0 + ⎜ D5 cos δ 0 + D6 sin δ 0 + ⎜ R ac ⎟⎠ Vm 0 ⎝ H 5 = (B5 sin δ 0 − B6 cos δ 0 )

(

)

H 6 = (B5 cos δ 0 + B6 sin δ 0 )

where, Vd0, Vq0, ILd0, ILq0, and δ0 are the steady state values of the d, and q axis voltages, load currents and the load angle of the synchronous generator, respectively.