Stability Improvement of Onboard HVdc Grid and Engine Using an Advanced Power Generation Center for the More-Electric Aircraft

In the high-power settings of engine, such as maximum take-off, more power should be extracted from the high-pressure spool of engine (HPS) than the low-pressure spool of engine (LPS) to avoid the overspeed and potential instability of the HPS. However, as revealed in this article, extracting more power from the HPS will degrade the onboard high-voltage direct current (HVdc) grid stability for the conventional single dc bus power generation center. To address this conflict, in this article, an advanced power generation center (APGC) incorporating an extra back-to-back (BTB) converter is introduced to improve the stability for both engine and HVdc grid. The BTB converter connects the HP power generation channel and the LP channel, providing an additional power flow path between the HP channel and LP channel. A transfer function-based impedance model and a state-space model of the HVdc grid are proposed to study the HVdc grid stability. The analytical findings of the HVdc grid stability and stability improvement using the APGC architecture have been verified through simulation and experimental results.


I. INTRODUCTION
T HE more-electric aircraft (MEA) concept is one of the major trends for the modern aerospace industry. Greenhouse gases emission and fuel consumption can be significantly reduced, and low maintenance cost can be achieved with MEA technology [1]. Existing onboard components which consume mechanical, pneumatic, and hydraulic energies are replaced by their electrical counterparts [2]. Consequently, electrical power (EP) generation systems with sufficient power supply capability should be deployed in MEA.
For most commercial aircrafts, their engines are designed with a twin-spool structure: a LPS and a HPS. The LPS connects the fan, low-pressure compressor (LPC), and low-pressure turbine (LPT). The HPS links the axial and radial high-pressure compressors (HPCs), and high-pressure turbine (HPT). Conventionally, generators are coupled to the HPS because the HPS features a relatively high and constant speed. This enables engineers to minimize the weight and size of the coupled HPG [3].
However, this single generator architecture is not optimal considering the significant power demand of MEA. It has been revealed that if too much power is extracted only from the HPS, compressor surge, which is a critical threat to engine operation will occur [4]. Although solutions such as excessive bleedings between compressor stages or oversizing the HPT can address this issue, the actions will cost extra fuel and result in undesirable thrust.
To cope with the high-power demand without undesirable fuel waste or thrust, an effective way is to add an extra LPG, converting the mechanical power (MP) of the LPS into EP. Considering the different speed range of the LPS and HPS (the range of the HPS is typically 1:2, e.g., 10 000-20 000 r/min. That of the LPS is 1:4, e.g., 1000-5000 r/min [5]), it indicates different fundamental ac frequencies of the LPG and HPG. Consequently, it is unrealistic to directly construct an ac bus due to the frequency conflict. Alternatively, one can independently control the LPG and HPG via two rectifiers and then connecting their dc terminals to a common dc bus [6].
Therefore, this gives a dual-generator-single-dc-bus power generation center (PGC) as shown in Fig. 1 [7]- [11]. In Fig. 1, the two generators are independently regulated by two rectifiers, denoting as the LPR and HPR, respectively. A variety of onboard loads and energy management components can absorb power from this dc bus [12]. It has been verified that this PGC can not only efficiently exploit the power of engine, but also improve the onboard dc grid redundancy and power quality [7]- [11].
However, the PGC in Fig. 1 suffers from the "power coupling effect" (PCE) between the MP of engine spools and the EP delivered by rectifiers. As can be seen from the red arrows in Fig. 1, the EP of the HPR is extracted from the MP of the HPS. The EP of the LPR comes from the MP of the LPS. The feature that EP is coupled to the MP is defined as PCE, which is HP Channel: HPR's EP = HPS's MP LP Channel: LPR's EP = LPS's MP.
The PCE makes the HVdc grid stability, and the engine stability shows the opposite trend. Improving one of the stabilities will degrade the other. For improving stability of the HVdc gird, this article proves that the LPR is expected to feed more EP than the HPR to the dc bus. Due to the PCE, this means that more MP is extracted from the LPS than the HPS. However, in the high-power settings of engine, such as maximum take-off (MTO) and top of climb, the HPS should provide more MP than the LPS to avoid the overspeed and potential instability of the HPS [13], [14]. Apparently, the PGC with PCE in Fig. 1 cannot handle this conflict.
To address the PCE and achieve stability improvement for both engine and HVdc grid, an APGC with a BTB converter is introduced in Fig. 2. The BTB converter connects the ac terminals of the LPG and HPG. It can transfer power between the HP and LP power generation channels. From the viewpoint of power, as the red arrows shown in Fig. 2, the APGC has the following feature: Hence, this BTB converter can be regarded as a power decoupling bridge. More MP can be extracted from the HPS than the LPS for preventing overspeed of the HPS. On the other hand, the LPR can share more power than the HPR to improve the HVdc grid stability. This favorable feature makes this APGC more flexible in terms of power flow and the stability of both engine and HVdc gird can be enhanced.
The APGC shown in Fig. 2 was first presented in our previous work [11], which focuses on the elimination of field  weakening operation of the HPG at a high speed. A 5% efficiency improvement is achieved under 10-kW load condition. This article will focus on the power decoupling characteristic of the APGC. The main findings and contributions of this article are highlighted as follows.
1) A new concept named PCE is defined and analyzed for the first time. 2) An onboard HVdc grid equivalent circuit is built.
By analyzing this equivalent circuit, it reveals that increasing the proportion of the LPR's power can improve stability of the HVdc grid. 3) Simulation and experiments are carried out, proving that the APGC can address the PCE and enhance both the engine and HVdc grid stability. The rest of this article is organized as follows. In Section II, an adaptive control design for the rectifiers is carried out. The derived dc current dynamics is further used for stability analysis in Section III. In Section III, a dc side equivalent circuit is established for assessing the HVdc grid stability. In Sections IV and V, control performances of the APGC and stability improvement for engine and HVdc grid are validated in simulations and experiments. Section VI concludes this article.

II. ADAPTIVE CONTROL DESIGN FOR THE RECTIFIERS
The control scheme for both the HP and LP rectifiers are given in Fig. 3. Since both the HPR and LPR supply power to a common dc bus, suitable power sharing strategy should be applied. In this study, a decentralized droop control [15] is adopted for the power sharing purpose. The dc current reference is generated depending on the dc bus voltage and the predefined V -I droop characteristic, which is shown in the following: where k D is the droop gain. i ref dc is the dc current command. v dc and v ref dc are the actual and nominal dc voltage, respectively. The system dynamic behavior is strongly related to the type of generators. Due to high power density, high efficiency and wide speed range, permanent magnet generators (PMGs) appear as a favorable option for the MEA application [16]. The electrical dynamics of PMG in dq rotary frame are shown as follows [17]: where u d , u q are the dq-axes stator voltages. L d , L q are the dq-axes stator inductance. R s is the stator resistance. ψ f is the flux linkage of the rotor permanent magnet. ω e is the electrical angular speed in rad/s. The electrical dynamics in (2) can be transformed into the Laplace domain as follows: Considering the fact that time constant of the mechanical system is much larger than that of the electrical system due to the large spool inertia of the engine, speed ω e can be regarded as unchanged during successive electrical control period. Hence, the increment of variables in (3) can be expressed as follows: Due to the presence of the BTB converter, the currents of generators are different from that of the rectifiers. Here the LP side power generation channel is considered. The schematic is shown in Fig. 4. To remove the PCE, the BTB converter transfers power from the HP channel to the LP channel in the high-power settings of engine. Hence, the following relationships can be obtained: i aLPR =i aL +i aBTBL i bLPR = i bL + i bBTBL i cLPR =i cL +i cBTBL (5) where i xL is the phase currents of the LPG; i xLPR are the phase currents of the LPR; i xBTBL represent the ac phase currents of the BTB L converter. x = a, b, c.
A current synchronization strategy is used to regulate the power of BTB converter. Details are presented in [11]. By applying this strategy, the BTB converter mainly transfers active power. Hence, the following relationships in steady-state in dq rotary frame can be obtained: (6) where i dL , i qL are the dq-axes stator currents of the LPG. i dLPR , i qLPR are the dq-axes currents of the LPR. i qBTBL is the q-axis current of the BTB L converter. Subsequently, according to Fig. 4, the ac side power and dc side power of the LPR can be expressed as follows: The relation in (7) can be further linearized as follows: where the superscript " − " means a stable operating point. ε q is a lumped term incorporating i qBTBL . According to the control scheme in Fig. 3, the control block diagram for the LPR can be obtained in Fig. 5, where G c (s) is the inner-current loop dynamics, G iq_dc (s) is the dynamics from i qL to i dc . C LP is the capacitance of the dc bus capacitor. 1/(C LP s +P/v 2 dc ) is the transfer function from the dc current i dc to the dc voltage v dc , whereP is the load power.
For designing suitable voltage-loop proportional-integral (PI) parameters, the key is to obtain G iq_dc (s). Given that i dL = 0 control is applied for the surface mounted PMG, G iq_dc (s) can be derived on the basis of (4) and (8) Denote the transfer function of the PI regulator as G pi (s), the dc current dynamics from i ref dc to i dc can be derived as follows: Given that the dynamics of the outer voltage-loop is slower than that of the inner current-loop, the term (ε q −ī dc v dc )/v dc i q in (9) can be regarded as a disturbance with slow dynamics. Applying (9) into (10), G idc (s) can be derived as follows, where k p and k i are the PI parameters of the dc current controller in (11), as shown at the bottom of the next page. The location of poles of G idc (s) mainly determines its performances. To tune the controller gains, a second-order system whose denominator is s 2 + 2ξω n s + ω 2 n is designed. ξ and ω n are damping ratio and natural frequency, respectively. G idc (s) will be used in Section III for analyzing the stability of the HVdc grid.
Therefore, the controller parameters can be written as follows: The values of natural frequency and damping ratio are presented in Appendix I. It can be seen from (12) that at different operating points (i.e., with differentv dc ,ī qL , andū q ), the PI parameters can be tuned adaptively to achieve a consistent performance.
Based on the dc current dynamics given in (11), a thirdorder dc voltage transfer function can be derived as follows: Details of a 2 , a 1 , a 0 , b 2 , b 1 , and b 0 can be found in (11).
The magnitude of G vdc (s) is plotted at various load powers as shown in Fig. 6. Parameters of the electrical machine and operating points are presented in Appendix I. Following information can be obtained from Fig. 6.
1) As the increase of load power from 5 to 25 kW, the magnitude becomes smaller. This is because as the increase of power, the dc current also increases. According to (1), this means a larger deviation between the actual dc bus voltage and reference. Therefore, the magnitude in Bode diagram will decline. 2) At 100 Hz, the magnitude damps around 3 dB compared with the original value in the low-frequency region. This is consistent with the designed bandwidth. It means the adaptive parameters in (12) can ensure a constant bandwidth for controlling the dc voltage.

III. DC GRID STABILITY ANALYSIS WITH DIFFERENT POWER SHARING RATIOS BETWEEN THE HPR AND LPR
To analyze the HVdc grid stability with different power sharing ratios between the HPR and the LPR, a dc side equivalent circuit is built as shown in Fig. 7, including the droop-controlled rectifiers, cables, dc capacitors, and loads. The superscripts HP and LP represent the variables of the HP and LP power generation channel, respectively. Since a current mode droop control is implemented [see (1)], the HP and LP channel can be regarded as current sources. In Fig. 7, i HP dc and i LP dc are current sources controlled by the local dc voltages v HP dc and v LP dc , and the droop gains k HP D and k LP D . The dc current dynamics G idc (s) derived in (11) is also considered. The dc currents can be expressed as follows: In Fig. 7, C HP and C LP are the capacitances of the local dc capacitors. The transmission cables are represented by RL branches and their admittances are denoted as X HP and X LP , respectively. v mb and C mb are the voltage and capacitor of the global main HVdc bus.

A. Impedance-Based Stability Analysis Using Middlebrook Criterion
Middlebrook [18] proposed an impedance-based approach to analyze stability, which allows definition of stability criterion for every individual subsystem through convenient impedance specifications. Considering the magnitude of impedances of source and load subsystems as ||Z s || and ||Z L ||, respectively, the Middlebrook criterion gives a sufficient stability condition, which is From (15) it shows that the system stability can be assessed by ||Z s ||. To obtain ||Z s ||, the open-circuit voltage v oc (s) and short-circuit current i sc (s) of the source subsystem in Fig. 7 in Laplace domain are derived as follows. The derivation process is given in Appendix II Therefore, the output impedance of the source subsystem can be derived as follows:  Table IV. It can be seen that as the proportion of the LPR's power increases, the peak magnitude of the source impedance at the low-frequency region will decrease. According to the Middlebrook criterion, reduction in the magnitude of the source subsystem impedance indicates that the system tends to be more stable [19], [20].

B. Time Domain Analysis Using State-Space Model
In order to verify the impedance analysis results in Section III-A, in this section the equivalent circuit shown in Fig. 7 is further analyzed using a state-space model. Then the outcomes of the state-space model and the impedance model are compared to see whether they are consistent.
The currents of inductors and voltages across capacitors are selected as state variables. Then the state-space model can be expressed as follows: Considering the droop control effect in (1), v LP dc can be written as where G LP vdc (s) is the dc voltage transfer function of the LPR control system which has been expressed in (13).
The original G LP vdc (s) as shown in (13) is a third-order transfer function, which will complicate the state-space model. Properly order reduction is required. Substitute the s with j ω, G LP vdc (s) can be transformed from the Laplace domain to the frequency domain, shown as follows: As shown in Appendix III, the control bandwidth for the dc voltage is designed as 100 Hz. Hence, the frequency range from 0 (ω = 0 rad/s) to 150 Hz (ω = 942.5 rad/s) which covers the control bandwidth is focused to investigate the characteristic of G LP vdc ( j ω) in (20). For the studied system whose parameters are given in Appendixes I and III, when the frequency component ω is within the range of interest [0, 942.5 rad/s], the following relations are met: (20) can be simplified as follows: To validate the order reduction, the magnitude and phase characteristics of the original G LP vdc ( j ω) in (20) and the simplified G LP vdc simp ( j ω) in (22) are demonstrated in Fig. 9. It can be seen that within the control bandwidth, the magnitude and phase of the original G LP vdc ( j ω) and the simplified G LP vdc simp ( j ω) are very similar. Therefore, in the frequency region that we are interested, i.e., within the bandwidth, the G LP vdc ( j ω) in (20) can be replaced by the G LP vdc simp ( j ω) in (22) with a sufficient accuracy and a much lower complexity.
Substitute the j ω in (22) with s, a simplified first-order transfer function G LP vdc simp (s) can be obtained Apply the simplified transfer function G LP vdc simp (s) in (23) to (19) and transform (19) into time domain, the following relation can be derived in the small-signal manner: where the term v ref dc is omitted since v ref dc is set as a fixed 270VDC as per the MILSTD-704F standard [21].
The voltage balance equation across the transmission cable can be expressed as follows: where R 2 and L 2 are the equivalent resistance and inductance of the LP channel's cable, i.e., Hence, the state-space equation of i c2 in the small signal manner can be derived as follows: The voltage equation of the main dc capacitor is In MEA many energy consuming loads, such as the thermal mats for wing ice protection system and the power converter driven compressors for environment control system, can be regarded as the constant impedance load (CIL) and the constant power load (CPL), respectively [22]. Hence, in this article the system is configured with a CPL, whose power is P CPL , and a CIL, whose impedance is R CIL . The current of load i Load can be derived as follows: Combing (27) and (28), the state-space equation of v mb in the small signal manner can be written as follows: wherev mb is the steady operating point of the main dc bus voltage.
With (24), (26), and (29), and considering the LP and HP channels are identical as shown in Fig. 7, the system matrix A in (18) can be derived as follows: The matrix (30) shows the system matrix A of the fifthorder state-space model in (18). Stability of the system can be assessed using the eigenvalues of A. Fig. 10 shows the eigenvalues loci when the power ratio between the LPR and the HPR changes from 0.5:1 to 2:1. It can be seen that as the LPR accounts for more power, the dominant eigenvalues will move far away from the right half-plane. It reveals that the system tends to be more stable.
The results in Fig. 8 derived from the Middlebrook approach and Fig. 10 derived from the state-space model are aligned, both indicating that increasing the proportion of the LPR's output power will contribute to a more stable HVdc gird. This crucial conclusion will be further validated in Sections IV and V through simulation and experiments. It can also be used as a criterion for researchers and engineers when designing onboard power generation centers with a sufficient stability margin for MEA application.

C. Influence of the Power Sharing Ratio on the Line Losses
The above stability analysis reveals the relation between the LPR's power and the HVdc grid stability, but it does not specify the HPR/LPR power ratio. To find the power sharing ratio range quantitatively, other approaches or optimization objectives should be considered. In this section, the impact of power sharing ratio on the transmission line losses is analyzed, and the optimal power ratio is derived for minimizing the line losses.
Assume that n 1 and n 2 are the ratios of the dc side current of the HPR and the LPR to the total load current. The current sharing ratio can be expressed as Typically, the geometry of power system onboard MEA is symmetrical. Hence, the cable length can be assumed to be identical for HP and LP channels. Denoting the cable resistance R 1 and R 2 in Fig. 7 both as R c , the minimization problem of line losses (P Loss ) can be formulated as min(P Loss ) = min n 2 1 + n 2 The Lagrange multiplier is applied to obtain the solution for (32), yielding The partial derivative of (33) is expressed as follows: Eliminating the parameter λ in (34), the optimal power sharing ratio aiming for line losses minimization can be obtained as follows:

D. Power Decoupling Characteristic of the APGC
The relation in (35) presents that the LPR and the HPR share equivalent power that can minimize the line losses. The findings in Figs. 8 and 10 reveal that increasing the proportion of the LPR's power is beneficial to the HVdc grid stability. Summarizing the above findings to achieve a tradeoff, it can be concluded that the LPR should deliver more power than the HPR to enhance the HVdc grid stability and reduce the line losses.
However, given the stability of engine, in MTO and top of climb modes, more MP is extracted from the HPS than the LPS to avoid the overspeed of the HPS and potential instability [13], [14]. For the PGC in Fig. 1, the PCE makes it infeasible to extract more power from the HPS and meanwhile delivering more power to the dc bus through the LPR. However, with the APGC, the BTB converter can transfer power from the HP to LP channel, allowing the HPS shares more power than the LPS, and meanwhile the LPR feeds more power than the HPR. Hence, the APGC presents a power decoupling characteristic.

IV. SIMULATION RESULTS
To verify the analytical findings in Section III, an equivalent nonlinear MATLAB/Simulink model of the PGC shown in Fig. 1 is built to test the HVdc grid stability under different power sharing ratios between the HPR and the LPR. The system parameters are presented in Appendix III. As stated in Section III, the impedance of transmission cable of the HP and LP channels is assumed to be identical. The control scheme for both rectifiers has been presented in Fig. 3, where the control bandwidths for the voltage loop and current loop are 100 Hz and 1 kHz, respectively. The associated control parameters are also given in Appendix III. Decoupling terms and antiwindup scheme presented in [23] are used for current controller to achieve a better dynamic performance.
The dc currents and voltage under different power sharing ratios between the HPR and the LPR are presented in Fig. 11. Fig. 11(a) and (c) present the dc currents and voltage where the power of the LPR (P LPR ) is triple of that of the HPR (P HPR ), i.e., P HPR : P LPR = 1:3. It can be seen that as the increase of load power from 10 to 30 kW, the dc voltage decreases due the droop control effect. The i LP dc is tightly controlled to be triple of i HP dc in the whole process, showing smooth performance in transient and steady-states.
Swapping the power sharing ratio between the HPR and the LPR, i.e., P HPR : P LPR = 3:1, and keeping the rest of configurations unchanged, the system responses are exhibited in Fig. 11(b) and (d). With 10-and 20-kW load power, the system remains stable, same as that in Fig. 11(a) and (c). However, as the load power increases to 30 kW, the system moves to an unstable state with severe oscillation in the dc currents and dc voltage. The comparative results in Fig. 11 show that increasing the power ratio of the LPR in the heavy load condition will contribute to a more stable HVdc grid. The simulation results are in accordance with the analytical findings in Sections III-A and III-B.
According to the operating points in Fig. 11 and the system parameters in Appendix III, Bode plot of source and load impedances under different load powers and power sharing ratios between the HPR and the LPR are presented in Fig. 12. The expression of source impedance Z s (s) has been formulated in (17). A dc/dc buck converter is regulated as a CPL, whose circuitry model is presented in Appendix III. The load impedance Z L (s) of the buck converter can be expressed as follows [24]: where the definitions and values of R ic , ω o , ω p , and Q 0 are given in Table V. It can be seen from Fig. 12(a) that under 20-kW load condition, the magnitude of Z s (s) at various power sharing ratios is much smaller than the magnitude of Z L (s). As shown in (15), according to the Middlebrook criterion, this indicates a sufficient system stability margin. This result is consistent with the simulation results in Fig. 11, where the system remains stable under 20-kW load condition for both P HPR : P LPR = 1:3 and 3:1. However, Fig. 12(b) shows that as the load power increases to 30 kW, the magnitude of Z s (s) with P HPR : P LPR = 3:1 and 1:1 is very close to the magnitude of Z L (s), showing no stability margin. This is consistent with the simulation results in Fig. 11(b) and (d), where the  system becomes unstable under 30-kW load power condition. By decreasing the ratio P HPR : P LPR to 1:3, the magnitude of Z s (s) decreases, meeting the requirement in (15). This is why in Fig. 11(a) and (c) where P HPR : P LPR = 1:3, the system operates smoothly even though under 30-kW load power.
The line losses under 20-and 30-kW load conditions with different power sharing ratios are presented in Fig. 13. The value of cable impedance is demonstrated in Appendix III. It can be seen that a minimal line losses is achieved at P HPR : P LPR = 1:1. This is consistent with the analysis in Section III-C. The increase of line losses caused by increased power ratio of the LPR is less than 5 W, which is trivial compared with the total load power. Hence, the HVdc grid stability should be the main consideration for designing the power ratio between the HPR and the LPR.

V. VALIDATIONS ON A DOWNSCALED LAB PROTOTYPE
According to the above theoretical analysis and simulation results, it can be concluded that the LPR is expected to share more power than the HPR to enhance the stability of the HVdc grid. However, in the high-power settings of engine, the HPS of engine needs to share more power than the LPS of engine to avoid overspeed of the HPS. It has been pointed out that the PGC in Fig. 1 cannot fulfill the two goals due to the PCE. While the APGC in Fig. 2 contains a BTB converter as a power decoupling bridge, which allows bidirectional power flow between the HP and the LP channels. Hence, the PCE is removed and the stability of HVdc grid and engine can be enhanced. To investigate this point, a downscaled lab prototype of the APGC consisting of two rectifiers and one BTB converter has been built, as shown in Fig. 14.
In Fig. 14, TMDSCNCD28379D is used as the digital control platform. A resistor bank and a dc electronic load are parallel connected to the dc bus, behaving as CIL and CPL, respectively. An autotransformer whose primary side is connected with the utility grid is used to emulate the LPG. Its electrical frequency is 50 Hz. A Chroma 31120 programmable ac source is used to emulate the HPG, whose voltage frequency is set as 80 Hz. The switching frequency is 5 kHz. Current loop and voltage loop execution frequencies are 5 and 1 kHz, respectively. The current and voltage control bandwidth is designed as 1 kHz and 100 Hz, respectively. The filter inductance is 2.5 mH. LV25-P and LA200-P are used as voltage and current transducers.

A. Control Performances of the APGC
The control scheme for rectifiers has been presented in Fig. 3. Details of control design for the BTB converter can be found in [11]. As revealed in Sections III and IV, the LPR is suggested to deliver more power than the HPR. Here the power sharing ratio between the HPR and the LPR is set as 1:2, i.e., P HPR : P LPR = 1:2. To simulate the situation that extracting more MP from the HPS than LPS, the ratio between the HPG's power (P HPG ) and LPG's power (P LPG ) is set as 2:1. Thus the BTB converter transfers power from the HP channel to the LP channel.
Control performance of the APGC under various load power conditions is demonstrated in Figs. 15 and 16. The whole process is divided into five stages. At stage 1 and 5, the CPL is deactivated and only a 40-CIL consumes 1.8-kW power.  At stage 2 and 4, the CPL is activated and the total load power increases to 2.5 kW. At stage 3, the CPL consumes more power, resulting in a 3.0-kW load power.
The results of v dc , i HP dc and i LP dc are presented in Fig. 15. It can be seen that as the increase of load power, v dc deviates from the rated value 270 to 263.0, 260.0, and 258.0 V due to the droop control effect. i LP dc is around double of i HP dc in the whole process (4.6 A/2.3 A, 6.6 A/3.3 A, 8.0 A/4.0 A), which means the output power of the LPR is successfully controlled to twice that of the HPR. The phase currents of the HPG, LPG and BTB H converter are exhibited in Fig. 16. It can be seen that as the increase of load power, the phase currents of the HPG and the LPG increase as they need to supply more power. The phase current of the BTB H converter also increase because the BTB converter transfers more power from the HP to the LP channel. It should be noted that the phase current of the LPG distorts, this is due to the inherent distortion of the utility grid voltage. Based on the above-mentioned power sharing ratios, the relationships of power flow at different stages are summarized in Table I.
To verify the values of power flow presented in Table I, the operating status of the HPG simulator is recorded as shown in Fig. 17. As given in Table I, ideally the power of the HPG emulator (P HPG ) should be 1.67 and 2.00 kW at stages 2 and 3, respectively. The actual results in Fig. 17 are 1.78 and 2.10 kW, which are well matched with the ideal values considering the inevitable power losses. This confirms the power control performance.
From the results from Figs. 15 to 17, it can be seen that using the APGC, the LPR can feed more power than the HPR to the dc bus, meanwhile more MP can be extracted from the HPS than the LPS by controlling their associated generators. Hence, the inherent power coupling defect of the PGC can be

B. Investigation of HVdc Grid Stability With Different Power Sharing Ratios Between the HPR and LPR
In this section, the impact of power sharing ratio between the HPR and LPR on the HVdc grid stability is examined. First, an equivalent simulation model as the lab prototype in Fig. 15 is built. The simulation results are exhibited in Fig. 18, where the dc bus voltage v dc , dc currents of the HPR and LPR (i HP dc and i LP dc ) are shown. From 1 to 2 s, the power ratio between the LPR and HPR is 2:1, i.e., P LPR : P HPR = 2 : 1. Subsequently, the power ratio between the LPR and HPR gradually decreases to 1:2, i.e., P LPR : P HPR = 1:2. It can be seen that in this case, the system tends to be unstable with severe oscillation in the dc currents and dc voltage. Then from 4 to 5 s, the power sharing ratio between the LPR and HPR restores to 2:1, the system becomes stable again.
The operating points shown in Fig. 18 are used to feed the source and load impedances in (17) and (36), and the magnitudes of their impedances are shown in Fig. 19. It can  be seen that when P LPR : P HPR = 2:1, the source impedance ||Z s (s)|| is covered by the load impedance ||Z L (s)|| with around 6-dB stability margin in the low frequency region. This indicates that the system can operate in a stable state, which matches with the results in Fig. 18 from 1 to 2 and 4 to 5 s. However, when the power ratio changes to P LPR : P HPR = 1:2, in the low frequency area shown by the red dashed rectangular, ||Z s (s)|| ≈ ||Z L (s)||, showing no stability margin. Therefore, this explains why the dc voltage and currents become distorted in Fig. 18 when P LPR : P HPR = 1:2.
To further verify the simulation and analytical results in Figs. 18 and 19, experimental validations are conducted and results are shown in Fig. 20, where the dc bus voltage v dc , dc currents of the HPR and LPR (i HP dc and i LP dc ), and the phase current of the HPG (i HP a ) are presented. Load power is also set to be 3.0 kW with the CPL.
Initially, the power ratio between the LPR and HPR is 2:1, i.e., P LPR : P HPR = 2:1. It can be seen that the system is stable. Subsequently, the power ratio between the LPR and HPR gradually decreases to 1:2, i.e., P LPR : P HPR = 1:2. It can be seen that in this case, the system loses stability with noticeable oscillation in the dc currents. Then the power sharing ratio between the LPR and HPR gradually restores to 2:1, the system becomes stable again. To quantitatively analyze the system performance with different power sharing ratios, harmonic analysis of the a-phase current of the HPG (i HP a ) is shown in Fig. 21. From  Fig. 21(c) and (d) it can be seen that the total harmonic distortion (THD) of i HP a is 4.15% when P LPR : P HPR = 2:1, while it increases to 8.23% when P LPR : P HPR = 1:2. It shows that the LPR shares more power than the HPR can improve system performance with a reduced current ripple.
To conclude this section, the experimental results in Fig. 20 are perfectly consistent with the simulation results in Fig. 18 and the impedance analysis in Fig. 19, confirming that increasing the proportion of the LPR's power, and make the LPR share more power than the HPR, can contribute to a more stable HVdc grid.

C. Analysis of Engine Performance With Different Power Sharing Ratios
To study the engine performance with different power sharing ratios between the HPS and LPS, a multispool turbofan model is developed using the intercomponent volume method and CFM56-3 engine generic maps [13], [14]. The MTO mode is focused as an example of high-power settings of engine. Results are exhibited in Fig. 22. Fig. 22(a) presents the HPC map and operating points with different power sharing ratios. It can be seen that when the power extracted from the LPS (P LPS ) is double than that of the HPS (P HPS ), i.e., P LPS = 2P HPS , the speed of the HPS exceeds the maximum allowable speed (N 2 = 1, where N 2 is an indicator in a cockpit gauge which presents the rotational speed of the HPS). Conventionally, to decrease the speed of the HPS, pilot needs to decrease the engine throttle. While with modern multigenerator topology within a more electric engine, overspeed of the HPS can be addressed by changing the power sharing ratios between the HPS and the LPS. In Fig. 22(a), when P HPS = 2P LPS , i.e., extracting more MP from the HPS than the LPS, air mass flow, and pressure ratio decrease back to the allowable region within N 2 = 1. And the operating point moves closer to the central efficiency contour with a higher operation efficiency.
From the viewpoint of the LPC map, as shown in Fig. 22(b), increasing the proportion of the LPS's power will lead to the decrease of the LPS's speed. Since the thrust is proportional to the speed of the LPS, this will decrease the thrust. Moreover, the operating point moves to a lower efficiency contour.
The results in Fig. 22 show that in the high-power settings of engine such as MTO, extracting more MP from the HPS than the LPS can avoid the overspeed and potential instability of the HPS and maintain thrust, which is beneficial for engine stability and efficiency.

D. Discussion
In Section V-B, it has been proven that the LPR delivers more EP than the HPR to the dc bus benefits the HVdc grid stability. In Section V-C, it has been validated that extracting more MP from the HPS than the LPS can improve engine stability and efficiency. The PGC shown in Fig. 1 cannot fulfill the two goals at the same time due to the PCE. While the APGC shown in Fig. 2 can eliminate this effect by transferring power from the HP channel to the LP channel through the BTB converter. This characteristic and relevant control performance has been demonstrated in Section V-A.

VI. CONCLUSION
In this article, the PCE of the state-of-the-art PGC is identified. Due to this effect, it is infeasible to make the LPR output more power to the dc bus and meanwhile extract more power from the HPS. As a result, it is difficult to enhance the stability of HVdc grid and engine simultaneously in the high-power settings of engine. The APGC removes the PCE, enabling the HPS to output more power than the LPS, and the LPR delivers more power than the HPR to the dc bus. Hence the stability of both the HVdc grid and engine can be enhanced. This article also derives the source and load impedances and provides detailed stability analysis for the HVdc grid. The derived impedance model and analytical findings may be of interest for other researchers who are interested in the stability issue of the onboard dc microgrid. Key performances of the APGC have been verified through simulation and experimental results. See Tables II and III.  (17) According to the Kirchhoff current law, the current flowing through the HP channel cable can be derived as follows: The current can also be derived from the voltage across the cable and the cable admittances, given as follows: Linking (A1) and (A2), the following relation can be derived: Due to the LP channel circuit is identical to that of the HP channel, the local dc voltage of the LP channel can also be written as follows: Under the short circuit condition, the main dc bus voltage is zero, i.e., v mb = 0. Hence, based on (A3) and (A4), the short circuit current of source subsystem can be expressed as follows: In the open circuit condition, the open circuit voltage can be expressed as follows: v  [25]. Based on a business jet aircraft platform, considering the distance from the engine-driven generator-rectifier unit to the main dc bus (5 m) and the auxiliary power unit-driven generator-rectifier unit to the main dc bus (20 m), the cable impedance is set as 3.2 m and 2.2 μH in this study.
The circuitry of the buck converter load used as a CPL in the simulation and analysis in Section IV is shown in Fig. 23.

APPENDIX IV CONSIDERATION OF THE INDUCTOR INSTALLATION
IN THE APGC It can be seen from Fig. 2 that the four power converters, i.e., HPR, LPR, BTB L converter, and BTB H converter, are all voltage source converters (VSCs). To make them operate compatibly, inductors should be deployed to separate these VSCs. Moreover, inductors can filter high-frequency pulsewidth modulation harmonics generated by the switching actions of power devices. There are four possible configurations with different locations of inductors, as shown in Fig. 24.
In Fig. 24(a) and (b), an inductor, denoting as L 2 , is placed at the front end of the BTB H converter. In this case, the terminals voltages of the HPG are limited within (v dc / √ 3) using the typical space vector pulsewidth modulation (SVPWM) [26], where v dc is the main dc bus voltage. The value of v dc is considered as 270 V to follow the MIF-STD-704F standard. Hence, for some permanent magnet-based generators, such as the electrical machine [27] developed under the frame of the Clean Sky project, it means that the field-weakening operation is still needed for the HPG with the architecture shown in Fig. 24(a) and (b).
In Fig. 24(c) and (d), L 2 is deployed at the front of the HPR. In this case, terminals voltages of the HPG are limited within (v BTB / √ 3) using the SVPWM, where v BTB is the dc-link voltage within the BTB converter. By increasing the voltage of v BTB , the HPG can operate at a high speed without field-weakening control. Hence the power losses within the HPG and HPR can be reduced [11]. The difference between Fig. 24(c) and (d) is the location of inductor L 1 . Since in most cases, most of the LP channel' power goes through the LPR to feed the onboard loads, and a relatively small proportion of power is transferred through the BTB converter, phase current of the LPR is larger than that of the BTB L converter. Placing L 1 at the front end of the BTB L converter instead of the LPR can reduce the power losses in the inductor L 1 . To conclude above, the configuration of inductors shown in Fig. 24(d) are chosen to build the APGC as shown in Fig. 2.