Modeling and stability assessment of permanent magnet machine-based DC electrical power system in more electric aircraft

Abstract

Advances in the areas of electrical and power electronic technologies play a key role in more electric aircraft (MEA), especially in power converters. Therefore, most electrical power loads on MEA are tightly controlled power converters that behave as constant power loads (CPLs). These CPLs look like a small-signal negative impedance that can significantly degrade the overall system stability. The destabilizing effect may result in poor performance until the DC bus voltage of the system does not adhere to the MIL-STD-704F standard. Thus, a stability study is very important to avoid unstable operations. A proposed MEA model which can be derived using the DQ approach is comprehensive in the introduced stability analysis methods in this article. These methods, i.e., a small-signal stability analysis using the eigenvalue theorem, a modal analysis technique called participation factor analysis, and a large-signal stability analysis via phase-plane analysis, are performed to investigate the stability margin. Moreover, the impact of key parameter variations on MEA stability is also taken into account to deliver the ways of parameter selection at the early design stages for an engineer. The MATLAB topology model and processor-in-loop (PIL) simulations validated the analytical results. The results indicate that a good agreement between theoretical, simulation, and PIL results can be achieved.

Appendices

Appendix A

The details of the classical method for vector controllers on dq-axis are as follows:

1.1 Current loop control

The schematic of the current (I_d and I_q) loop control of the proposed MEA in Fig. 1 can be shown in Fig. 

Fig. 13

Schematic of current loop control

13. Because the current loop control on d-axis and q-axis is identical, this article will only present the methodology for designing the current loop control on d-axis, wherein the controller on q-axis can use the same equation. Closed-loop transfer function of the current loop is given by (9).

$$ \frac{{I_{d} }}{{I_{d}^{ * } }} = - \frac{{K_{{{\text{pd}}}} s + K_{{{\text{id}}}} }}{{L_{d} s^{2} + \left( {R_{s} - K_{{{\text{pd}}}} } \right)s - K_{{{\text{id}}}} }} $$

(9)

It is recognized that the closed-loop transfer function for the standard second-order system can be expressed in (10).

$$ T = \frac{{\omega_{n}^{2} }}{{s^{2} + 2\zeta \omega_{n} s + \omega_{n}^{2} }} $$

(10)

Hence, the parameters K_pd and K_id can be designed by comparing the denominators of (9) and (10) so as to obtain (11)

$$ \left\{ \begin{gathered} K_{{{\text{pd}}}} = R_{s} - 2\zeta_{i} \omega_{{{\text{ni}}}} L_{d} \hfill \\ K_{{{\text{id}}}} = - L_{d} \omega_{{{\text{ni}}}}^{2} \hfill \\ \end{gathered} \right. $$

(11)

As mentioned above, both d-axis and q-axis controls are identical. Consequently, the equations for designing the controllers on q-axis are shown in (12).

$$ \left\{ \begin{aligned} K_{{{\text{pq}}}} = & R_{s} - 2\zeta_{i} \omega_{{{\text{ni}}}} L_{q} \\ K_{{{\text{iq}}}} = & - L_{q} \omega_{{{\text{ni}}}}^{2} \\ \end{aligned} \right. $$

(12)

where ω_ni = 2π × f_ni.

1.2 Voltage loop control

The schematic of the voltage (V_dc) loop control of the studied MEA is depicted in Fig. 

Fig. 14

Schematic of voltage loop control

14. Closed-loop transfer function of the voltage loop is determined in (13).

$$ \frac{{V_{{{\text{dc}}}} }}{{V_{{{\text{dc}}}}^{ * } }} = \frac{{3m\left( {K_{{{\text{pv}}}} s + K_{{{\text{iv}}}} } \right)}}{{4C_{{{\text{dc}}}} s^{2} + 3mK_{{{\text{pv}}}} s + 3mK_{{{\text{iv}}}} }} $$

(13)

Based on the current loop procedure, the equations for designing the parameters K_pv and K_iv are depicted in (14).

$$ \left\{ \begin{gathered} K_{{{\text{pv}}}} = \frac{{8\zeta_{v} \omega_{{{\text{nv}}}} C_{{{\text{dc}}}} }}{3m} \hfill \\ K_{{{\text{iv}}}} = \frac{{4C_{{{\text{dc}}}} \omega_{{{\text{nv}}}}^{2} }}{3m} \hfill \\ \end{gathered} \right. $$

(14)

where ω_nv = 2π × f_nv.

1.3 Droop control and voltage compensation

The schematic of the voltage-mode droop control and voltage compensation is demonstrated in Fig. 

Fig. 15

Schematic of droop control and voltage compensation

15. Because the studied power system in this article is a single-generator/single-bus DC distribution MEA power system, the individual droop gain (K_d) of droop controller is equal to the global droop gain (K_t) [5]. The optimal gains K_t, designed via the minimum transmission loss condition, can be expressed in (15) and (16).

$$ \left\{ \begin{gathered} {\text{if }}\frac{1}{3} \le r \le 1 \hfill \\ R_{L} \frac{1 - r}{{1 + r}} < K_{t} \le \frac{{R_{L} }}{2}\left( {\sqrt {1 + \frac{1}{r}} - 1} \right) \hfill \\ \end{gathered} \right. $$

(15)

$$ \left\{ \begin{gathered} {\text{if }}r > 1 \hfill \\ 0 < K_{t} \le \frac{{R_{L} }}{2}\left( {\sqrt {1 + \frac{1}{r}} - 1} \right) \hfill \\ \end{gathered} \right. $$

(16)

where r is ratio between the power of CPL (P_CPL) and resistive load ($P_{{R_{L} }}$).

In this article, the PI parameters for both current and voltage loops are designed by selecting ζ_i = 0.8, ζ_v = 0.8, f_ni = 2000 Hz, and f_nv = 200 Hz. As for parameters for droop control and voltage compensation are set by defining P_CPL = 10 kW and P_RL = 7 kW.

Appendix B

The details of Jacobean matrixes A, B, C, and D can be expressed as follow:

$$ a(2,5) = - \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (KK_{t} - K_{d} )}}{{L_{q} R_{L} }} + \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} P_{{{\text{CPL}}}} (K_{t} - K_{d} )}}{{L_{q} V_{b,0}^{2} }} $$

$$ a(3,1) = \frac{3}{{2C_{{{\text{dc}}}} }} \cdot \frac{1}{{V_{{{\text{dc}},0}} }}\left( { - 2K_{{{\text{pd}}}} I_{d,0} + K_{{{\text{id}}}} X_{{{\text{id}},0}} + K_{{{\text{pd}}}} I_{d}^{*} } \right) $$

$$ \begin{aligned} a(3,2) = & \frac{3}{{2C_{{{\text{dc}}}} }} \cdot \frac{1}{{V_{{{\text{dc}},0}} }}\left( { - 2K_{{{\text{pq}}}} I_{q,0} + \omega_{e} \phi_{m} - K_{{{\text{pv}}}} K_{{{\text{pq}}}} V_{{{\text{dc}},0}}} \right. \\ & + K_{{{\text{pv}}}} K_{{{\text{pq}}}} V_{b}^{*} + \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )V_{b,0} }}{{R_{L} }} \\ & + \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )P_{{{\text{CPL}}}} }}{{V_{b,0} }} \left. { + K_{{{\text{iq}}}} X_{{{\text{iq}},0}} } \right) \\ \end{aligned} $$

$$ \begin{gathered} a(3,3) = \frac{3}{{2C_{{{\text{dc}}}} }} \cdot \frac{1}{{V_{{{\text{dc}},0}}^{2} }}\left( { - K_{{{\text{pd}}}} I_{d,0}^{2} + K_{{{\text{id}}}} I_{d,0} X_{{{\text{id}},0}} + K_{{{\text{pd}}}} I_{d,0} I_{d}^{*} } \right. \hfill \\ \begin{array}{*{20}c} {- K_{{{\text{pq}}}} I_{q,0}^{2} + \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )I_{q,0} V_{b,0} }}{{R_{L} }} + \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )P_{{{\text{CPL}}}} I_{q,0} }}{{V_{b,0} }}} \\ \end{array} \hfill \\ \left. { + \omega_{e} \phi_{m} I_{q,0} + K_{iv} K_{{{\text{pq}}}} I_{q,0} X_{v,0} + K_{{{\text{iq}}}} I_{q,0} X_{{{\text{iq}},0}} + K_{{{\text{pv}}}} K_{{{\text{pq}}}} I_{q,0} V_{b}^{*} } \right) \hfill \\ \end{gathered} $$

$$ a(3,5) = \frac{{3K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )I_{q,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} R_{L} }} - \frac{{3K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )P_{{{\text{CPL}}}} I_{q,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} V_{b,0}^{2} }} $$

$$ a(8,5) = \frac{{K_{pv} (K_{t} - K_{d} )}}{{R_{L} }} - \frac{{K_{pv} (K_{t} - K_{d} )P_{CPL} }}{{V_{b,0}^{2} }} $$

$$ {\mathbf{C(x}}_{{\mathbf{0}}} {\mathbf{,u}}_{{\mathbf{0}}} {\mathbf{)}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right]_{3 \times 8} $$

$$ {\mathbf{D(x}}_{{\mathbf{0}}} {\mathbf{,u}}_{{\mathbf{0}}} {\mathbf{) = }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]_{3 \times 3} $$

$$ {\mathbf{B(x}}_{{\mathbf{o}}} {\mathbf{,u}}_{{\mathbf{o}}} {\mathbf{)}} = \left[ {\begin{array}{*{20}c} { - \frac{{K_{{{\text{pd}}}} }}{{L_{d} }}} & 0 & 0 \\ 0 & { - \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} }}{{L_{q} }}} & { - \frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} (K_{t} - K_{d} )}}{{L_{q} V_{b,0} }}} \\ {\frac{{3K_{{{\text{pd}}}} I_{d,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} }}} & {\frac{{3K_{{{\text{pv}}}} K_{{{\text{pq}}}} I_{q,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} }}} & {\frac{{3K_{{{\text{pv}}}} K_{{{\text{pq}}}} I_{q,0} (K_{t} - K_{d} )}}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} V_{b,0} }}} \\ 0 & 0 & 0 \\ 0 & 0 & { - \frac{1}{{C_{b} V_{b,0} }}} \\ 0 & 1 & {\frac{{K_{t} - K_{d} }}{{V_{b,0} }}} \\ 1 & 0 & 0 \\ 0 & {K_{{{\text{pv}}}} } & {\frac{{K_{{{\text{pv}}}} (K_{t} - K_{d} )}}{{V_{b,0} }}} \\ \end{array} } \right]_{8 \times 3} $$

$$ {\mathbf{A(x}}_{{\mathbf{o}}} {\mathbf{,u}}_{{\mathbf{o}}} {\mathbf{) = }}\left[ {\begin{array}{*{20}c} {\frac{{K_{{{\text{pd}}}} - R_{s} }}{{L_{d} }}} & 0 & 0 & 0 & 0 & 0 & { - \frac{{K_{{{\text{id}}}} }}{{L_{d} }}} & 0 \\ 0 & {\frac{{K_{{{\text{pq}}}} - R_{s} }}{{L_{q} }}} & {\frac{{K_{{{\text{pv}}}} K_{{{\text{pq}}}} }}{{L_{q} }}} & 0 & {a(2,5)} & { - \frac{{K_{{{\text{iv}}}} K_{{{\text{pq}}}} }}{{L_{q} }}} & 0 & { - \frac{{K_{{{\text{iq}}}} }}{{L_{q} }}} \\ {a(3,1)} & {a(3,2)} & {a(3,3)} & { - \frac{1}{{C_{{{\text{dc}}}} }}} & {a(3,5)} & {\frac{{3K_{{{\text{iv}}}} K_{{{\text{pq}}}} I_{q,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} }}} & {\frac{{3K_{{{\text{id}}}} I_{d,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} }}} & {\frac{{3K_{{{\text{iq}}}} I_{q,0} }}{{2C_{{{\text{dc}}}} V_{{{\text{dc}},0}} }}} \\ 0 & 0 & {\frac{1}{{L_{c} }}} & { - \frac{{R_{c} }}{{L_{c} }}} & { - \frac{1}{{L_{c} }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{{C_{b} }}} & { - \frac{1}{{R_{L} C_{b} }} + \frac{{P_{{{\text{CPL}}}} }}{{C_{b} V_{{{\text{dc}},0}}^{2} }}} & 0 & 0 & 0 \\ 0 & 0 & { - 1} & 0 & {\frac{{K_{t} - K_{d} }}{{R_{L} }} - \frac{{\left( {K_{t} - K_{d} } \right)P_{{{\text{CPL}}}} }}{{V_{b,0}^{2} }}} & 0 & 0 & 0 \\ { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - 1} & { - K_{{{\text{pv}}}} } & 0 & {a(8,5)} & {K_{{{\text{iv}}}} } & 0 & 0 \\ \end{array} } \right]_{8 \times 8} $$

Appendix C

The system parameters: R_s = 1.058 mΩ, L_s = L_d = L_q = 99 µH, $\phi_{m}$ = 0.03644 V.s/rad, p = 6, ω_e = 2π × 400 rad/s, C_dc = 1mF, cable length = 10 m, R_c = 6 mΩ, L_c = 2 µH, C_b = 0.5 mF, R_L = 10 Ω, K_pv = 3.5744, K_iv = 2807.3541, K_pd = K_pq =  − 1.9895, K_id = K_iq =  − 1563.3453, K_d = K_t = 0.06, $I_{d}^{*}$ = 0 A, $V_{b}^{*}$ = 270 V, P_RL,rated = 7 kW, and P_CPL,rated = 38 kW.