Control Design and Voltage Stability Analysis of a Droop-Controlled Electrical Power System for More Electric Aircraft

This paper focuses on the analysis of a single dc bus multigenerator electrical power system (EPS) for future more electric aircraft. Within such a single bus paradigm, the paper proposes a detailed control design procedure and provides a stability analysis based on the derivation of the output impedance of the source subsystem and input impedance of the load subsystem, including control dynamics. The single bus characteristic is analyzed and the stability properties of the EPS are investigated when supplying constant power loads. In addition, the paper highlights the impact on stability of the number of parallel sources and of the power sharing ratio. The theoretical analysis is instrumental in designing an optimally stable single dc bus EPS. The key findings are validated by experimental results.


I. INTRODUCTION
HE more-electric aircraft (MEA) concept is one of the major trends in modern aerospace engineering aiming for reduction of the overall aircraft weight, operation cost and environmental impact. Electrical systems are employed to replace existing hydraulic, pneumatic and mechanical actuators. As a consequence, the onboard installed electrical power increases significantly and this results in challenges in the design of the aircraft electrical power systems (EPS). The tendency is to replace traditional AC distribution with highvoltage DC distribution. This can increase efficiency, reduce weight and remove the need for reactive power compensation devices [1], [2].
In literature, the primary power distribution in aircrafts has been traditionally based on the single-generator-per-bus paradigm with switched distribution providing the connectivity Manuscript  and system integrity. Instead, the proposed "single-bus" concept uses the micro-grid approach in which all the generators and loads are connected to a single distribution bus. This single bus configuration has been widely used in other applications such as residential microgrids [3]. Such a system has the potential to considerably reduce the EPS weight since bus mass is reduced and load and generator fault isolation function can be integrated in power converters; in addition the controlled power sharing between generators has the potential to reduce generator capacity and operate at maximum efficiency levels.
should be carefully examined for stability in order to guarantee safe and predictable EPS operation for a wide range of operation scenarios.
To evaluate voltage stability properties and design a robust EPS, two approaches can be highlighted here. The first option is to consider the stability problem with a general state space model, design the control and evaluate the impact of different parameters by studying the eigenvalues [14]- [16]. A more practical approach is the impedance method, which is based on the ratio of the output impedance of the source subsystem and input impedance of the load system [17]- [20]. Thus, the contribution of each component to the system stability can be assessed in the frequency domain. In addition, the output impedance of the components or subsystems can be reshaped to stabilize the overall power system and the impact of the connection of new converters can be assessed.
In the specific context of aircraft power systems, so far, several publications have discussed the system stability in MEA EPS. The stability of a switched reluctance motor-based 270V DC EPS has been analyzed in [21]. A permanent magnet synchronous generator (PMSG)-based hybrid AC-DC MEA EPS is investigated in [22], [23] and the influence of parameter variation on system stability is presented. In addition, several publications report the stabilization methods which can be generally divided into two categories: passive damping and active damping [24]- [31]. Passive damping methods incorporate a passive damping network (add a resistor in series with the dc-link inductor or capacitor) to stabilize the system [24], [25]. However, the required extra hardware and space can be expensive and system efficiency is reduced. Alternatively active damping methods can stabilize the system by implementing some modifications on the control. So far, the active damping approaches using a high pass filter [26], nonlinear feedback control [27], an observer based stabilization [29], virtual capacitor [28], a constraint optimization algorithm [30], a negative input resistance compensator [31] are reported in the literature. Nevertheless, the published works on the context of MEA EPS are mainly focused on the single source system, droop control is not used and the dynamics of the generator is not taken into account. In [32], stability of a single DC bus MEA EPS with multiple generators is investigated. However, the interaction of the parallel sources is not validated experimentally, and the impact on stability of different operating frequencies of the different sources is not addressed.
In alignment with the single bus concept, this paper presents a design oriented analysis of the single DC bus EPS with multiple sources based on Permanent Magnet Synchronous Generators (PMSGs) for future civil aircrafts. As illustrated by Fig. 1, two PMSGs are connected directly to the turbine shafts through active rectifiers (ARs) and can operate in variable speed mode. The two generators take power from the main engine through high pressure (HP) shaft and low pressure (LP) shaft, guaranteeing an efficient exploitation of the power generated by the engine, and operate in parallel to transfer the power to the DC bus. The rated voltage of the main bus in aerospace applications is 270 V, as described in the standard MIL-STD-704F [33]. In this configuration, the two generators can operate at different speeds, typically during climb, cruise, and descent of the aircraft, and as a consequence, the corresponding connected converters operate at different AC input frequencies. As shown later in the analysis, this may lead to voltage instability if the load power is not properly distributed between the two sources. The main contributions of the paper can be highlighted as follows: (1) Detailed control design for the single bus EPS with multisource and active/passive loads has been performed.
(2) The steady-state power sharing performance and main bus characteristic are presented.
(3) The interaction between multiple sources (LP/HP generators) has been investigated in terms of voltage stability: the influence of the power sharing ratio between different sources and the increased number of parallel sources on stability are analyzed.
The paper is organized as follows: In Section II, the control strategies for the active rectifier and active load converter are presented and detailed control design procedure is shown. Section III derives the source and load impedances including the control dynamics. The interaction effect between parallel sources on voltage stability as well as on steady-state characteristic are also performed in this Section. Validation of the analysis with experimental results is presented in Section IV.

II. SYSTEM MODELING AND CONTROL DESIGN
The system shown in Fig. 1 can be divided into source and load subsystem with the break point chosen at the DC bus. The definition of the source/load subsystem is based on [34]. The converter which takes part in the bus voltage regulation is defined as the source converter whereas the converter consumes/absorbs the power from the bus is regarded as the load converter. This section will present the detailed control design process for the source and load converter. Fig. 2 presents the control block diagram of the single PMSG-AR system. By controlling the flux in the d-axis and the active power in the q-axis, the PMSG can operate in generation mode. Conventional PI controllers are used to deflux the machine (d-axis) and control the output DC current (q-axis). Below the base speed ("constant torque" region), the d-axis current component equal to zero, assuring the maximum torque-to-current ratio. Under high speed operation region, the back electromotive force of the PMSG, which is proportional to the speed, may become higher than the converter ac voltage. The limited value of the reference voltage of the feeding converter (Vc max ) defines the constant power region where flux weakening is implemented [35]. The stator current references in d and q axes are obtained from the output of the flux weakening controller and Idc controller respectively. The reference of the AC voltage (Vc max ) is dependent on the DC voltage. The DC current reference (idc * ) is determined by the desired droop characteristic. It is worth noting that instead of tightly regulating the DC bus voltage, DC current is the control target in this study in order to regulate the proper injected active power into the DC bus. When using this topology, this control structure should be implemented in all generatorconverter modules intended for power flow control.
Control scheme utilized in the single PMSG-AR system.

A. Source Subsystem-Inner Loop
Using the vector control for the PMSG-AR, one can obtain the voltage equations in dq frame as where R and L represents the stator resistance and inductance; ωe is the electrical rotating velocity in rad/s; ϕm is the flux linkage of the permanent magnet; idq is the stator current in dand q-axis respectively. The control plant of the inner current loop Gp_idq can be expressed as Assuming the inner current loop is designed to be a first-order system with the central frequency ωc, the zero of the PI compensator is set to cancel the pole of the plant. The proportional gain kpc and integral gain kic of the PI controller can be written as follows: ,

B. Source Subsystem-Outer Loop
As shown in Fig. 2, the DC current reference is obtained by the droop characteristic: where ki is the droop coefficient, vo is the nominal voltage (270 V in this study) and vdci is the local DC output voltage of the converter. Fig. 3 shows the control block diagram of the source converter. As discussed in Section II-A, the inner current loop is simplified by a first-order delay block. In order to simplify the control design, DC terminal voltage Vdc is assumed to be constant Vdco. Fig. 3. Control block diagram of the source converter.
As one can see from Fig. 3, the control-to-output (idc to iq * ) GP_C can be expressed as below Given a PI controller of the DC current Gdc, the expression for DC current dynamics GDy can be derived as follows: Substituting (5) into (6), the DC current dynamics can be derived as: where kpdc and kidc are the proportional gain and integral gain of the Idc controller. Given the damping ratio ξIdc and natural frequency ωIdc, the following equations can be obtained: From (8), the DC current controller parameters can be derived as follows: In order to keep a good performance of such a cascaded control loop structure, the bandwidth of the outer loop is chosen as 1/10 of the inner current loop.

C. Load Subsystem
In the load subsystem, apart from the passive loads like resistive loads (RLs), many active loads in EPS such as motor drives and power converters behave as CPLs. In this paper, a buck converter is selected as a representative of CPLs connected to the DC bus, as shown in Fig. 4. The load of the buck converter is a resistor Rbuck and the buck converter is controlled as a CPL. Different from the conventional control strategies for the buck converter, a current-mode control strategy is presented to achieve the CPL operation, as shown in Fig. 5. Vb represents the bus voltage. As one can see, the inductor current is regulated via a PI controller. The inductor current reference is given according to the load power demand as follows: where kpi and kii are the proportional gain and integral gain of the PI controller. Given the damping ratio ξL and natural frequency ωL, the PI control parameters can be derived as follows: To summarize, this section presented the control design for the source converter and load converter, which lays the foundation of the subsequent impedance analysis.

III. ANALYSIS OF THE SINGLE BUS MULTI-GENERATOR SYSTEM
In the EPS configuration shown in Fig. 1, multiple sources provide electrical power to feed the common DC bus. This section describes the steady-state performance and derives the output impedance of the source subsystem and input impedance of the load subsystem. The voltage stability is then investigated using the impedance-based approach.

A. Steady-State Analysis
The DC-side equivalent circuit of Fig. 1 is presented in Fig.  6. Droop characteristic is implemented by means of an additional current source which is controlled by the local voltage Vdci. Since the parasitic capacitance of the cable is much smaller than bus capacitor (Cb) and local capacitor (Ci), it can be included in the converter DC side capacitors and as a consequence, the cable is represented as the R-L branch.
Instead of changing the droop slope, the power sharing can be also achieved by adjusting the reference voltage. Under this circumstance, an additional control loop is required to change the reference voltage in order to obtain the desired sharing ratio. In this paper, power sharing control is still implemented by changing the droop gains.
If the system is loaded with CPLs and resistive loads (see Fig. 4), similar to the single source system, the bus voltage Vb and total output current Io match the following equation: Similar to the individual droop line in each parallel source, the steady state V-I characteristic at the DC bus can be expressed as a function of a global equivalent droop coefficient which can be written as below where Ri is the i th branch cable resistance and the droop coefficient is assumed to be much larger than the cable resistance (ki >>Ri). It can be inferred from (16) that the main bus voltagecurrent characteristic is still a droop curve which is stiffer than the individual droop characteristic. It indicates that with more number of parallel sources, the voltage deviation at the main bus will be reduced at the same load condition. It also be inferred from (16) that even if the individual droop coefficient settings are different, the bus voltage can be invariant as long as the individual droop coefficients yield the fixed global droop coefficient kt in (16).
By solving the above equations (14) and (15), the main bus voltage and total current can be derived as follows: To guarantee existence of a solution for DC voltage and current (17), global droop gain should satisfy Physically this can be explained as follows. A larger droop constant will cause larger voltage deviation at the main DC bus and can even result in no intersection point between the source curve and CPL curve. The upper limit shown in (18) is the critical condition to ensure existence of an equilibrium point.
If the global droop gain satisfies the abovementioned requirement, the next step is to decide the injecting current from each parallel source. As one can see from (13), the current sharing ratio is proportional to the inverse of individual droop gain. Combining (13) and (16), the individual droop gain can be designed as follows: where ni represents the desired current sharing proportion of the i th source and N is the total number of parallel sources.

B. Impedance Analysis
Using the ratio of open-circuit voltage and short-circuit current of the source subsystem, the output impedance of the droop-controlled source subsystem in Fig. 6 can be derived as where the cable admittance is Yib and GDyi is the DC current tracking performance given in (7). For the load impedance, the input impedance of the buck converter operated with the control strategy discussed above (see Fig. 5 After deriving the equivalent source and load impedance, the stability assessment will be evaluated in next subsection.

C. Voltage Stability Analysis
As one can infer from (20) and (21), operating parameters (load power, generator speed) and control parameters (droop gain, control bandwidth), which have a direct/indirect effect on GDy, will change the source impedance. It is well-known that an increased power absorbed by CPLs will degrade the system stability [22], [23]. The effect of the control parameters was also discussed in [32]. Thus, this paper focuses on the impact of the number of parallel sources and different power sharing ratios between HP and LP sources on the source/load impedance and voltage stability.

(1) Effect of number of parallel sources
If multiple sources work in parallel with the same individual droop coefficient (ki = 2), the global droop coefficient kt decreases with the increasing number of parallel sources as expected from (16), as listed in Table I. Fig. 7 shows the Bode plot of the source/load impedance of the parallel sources system. As one can see from Fig. 7(a), the source impedance is reduced with the increased number of parallel sources. Meanwhile, it is observable from Fig. 7(b) that the load impedance magnitude increases with the increasing number of parallel sources. This is because the global droop coefficient decreases with increasing the number of parallel sources, yielding less voltage drop at the main bus under the same load condition. As a consequence, the load impedance at the low frequency region will increase, as indicated from (23). Hence, the system stability is improved if parallel sources are working with the same individual droop coefficient (ki).  (2) Effect of load sharing ratio The power sharing among multiple sources can be adjusted by tuning the individual droop gains. This section will investigate the impact of different load sharing ratio on system stability. If the two source is operating at different frequencies (50 Hz for LP source and 400 Hz for HP source), Fig. 8 shows the equivalent source impedance under different power sharing ratios. It can be observed that the peak magnitude of the overall source impedance decreases if the LP system shares more power. Under the prerequisite that the load power does not exceed the ratings of each generation channel (LP or HP), the Bode plot demonstrates that the parallel system will be more stable if the LP system provides more power to feed the load. This finding would facilitate engineers in designing an appropriate power management scheme that would guarantee voltage stability in a sufficiently wide margin.

IV. EXPERIMENTAL VALIDATION
A test rig to validate the theoretical analysis developed for the multi-source MEA EPS architectures was designed and built, as shown in Fig. 9. Fig. 10 shows the schematic of the experimental system. The laboratory prototype contains two active front-end converters Conv 1, interfacing the HP generator, and Conv 2, interfacing the LP generator. For the experimental results shown in subsections IV-A and IV-B the two converters are connected to the same programmable AC source (Chroma QuadTech 31120) through two step-down transformers, to emulate two identical synchronous machines, thus focusing on a simplified scenario where HP and LP generators are identical. This choice has been made with the objective of proving the concept of global droop gain and the generic impact of parallel operation on stability. Instead, a different scenario is considered in subsection IV-C, where Conv 1 (HP) and Conv 2 (LP) are connected to two different generators. The control algorithm for both source converters (Conv 1 and 2) is implemented following the same scheme shown in Fig. 2. A DC/DC (buck) converter is tightly controlled as a CPL using the control strategy shown in Fig. 5. To maintain coherence with the theoretical analysis, the experimental system parameters are listed in Table II.

A. Case 1: Steady-State Investigation
Under a 2.6 kW CPL, the two converters are operating in parallel with the identical individual droop coefficient as listed in Table III. Fig. 11 shows the experimental result of the main DC bus voltage and the injecting DC current from each source. As one can see, the bus voltage decreases as the global droop coefficient kt increases and the current injected by Conv 1 and 2 are identical. The individual DC currents agree with the expectations in (4) and the main bus V-I characteristic (bus voltage and total load current) is consistent with the discussion in Section III-A.  Fig. 11

k1 k2
Global droop gain (kt) In the next experimental result depicted in Fig. 12, the global droop coefficient kt is fixed to 1, however the current sharing ratio between two sources is varied, as shown in Table  IV. The individual droop coefficients are set according to (19). As one can observe, the steady state bus voltage is not affected by the step changes in the sharing ratio, but the load current provided by Conv 1 and 2 varies according to the set ratio. As before, the results are consistent with the theoretical analysis in Section III-A.  Fig. 12 Power sharing ratio (I1 : I2) k1 k2 1:1 1/0.5 1/0.

B. Case 2: Stability Analysis-Effect of Parallel Generation
The effect of the number of parallel sources on voltage stability is examined in Case 2. Firstly a single converter (Conv 1) is operated with 200 Hz AC input from Chroma and the droop coefficient k1 is set to 1. It shows in Fig. 13(a) that the system consisting of a single converter is oscillating at high power load (6 kW). In contrast, Fig. 13(b) shows the experimental result when Conv 2 is connected to a 50 Hz AC source and Conv 1 is disconnected. It can be seen that the system with Conv 2 is still stable when the load power reaches 6 kW.
The effect of parallel operation with the same individual droop gain (ki = 1) has been also tested. As illustrated in Fig.  10, both converters (Conv 1 and 2) are connected to the system with 200 Hz AC supply and Fig. 14 shows the experimental result. At t = 0 s, the load is set at 3 kW and only Conv 1 is operating. At t = 1.1 s, Conv 2 is connected and begins to operate in parallel with Conv 1. As a result, the global droop gain reduces and the bus voltage increases. As discussed in Section III-C, the global droop gain reduces with the increased numbers of parallel modules with the same individual droop constant. The overall source impedance reduces and consequently, stability is improved by parallel operation. It can be seen that the two source system is stable below 7kW CPL. This result demonstrates that compared with single source operation, parallel operation with the same individual droop gains significantly improves the system stability which is in accordance with the discussion in Section III-C.

C. Case 3: Stability Analysis-Effect of Different Power Sharing Ratio
The effect of different power sharing ratio among the parallel sources in Case 3. Different from the previous cases, the Chroma and a Variable Autotransformer are connected to transformer TF1 and TF2 respectively. Under this circumstance, Chroma provides a 200 Hz AC input to Conv 1 and the Variable Autotransformer provides a 50 Hz AC input to Conv 2. Both sources are operated at different frequencies to emulate the HP and LP generators. Global droop gain kt is fixed at kt = 1. The influence of different power sharing ratio on the system stability has been tested and the result is shown in Fig. 15. When the ratio between HP and LP channel is 3:7, it can be seen from Fig. 15(a) that the system is stable over the load power ranging from 0 to 5 kW. When the power is equally shared between HP and LP channel, oscillation occurs in the presence of 5 kW CPL (see Fig. 15(b)). It can be seen from Fig. 15(c) that instability shows even earlier (4 kW CPL) when the current sharing ratio between HP and LP is set to 7:3. This indicates that the overall system stability will be degraded if the HP generation channel shares more power. Under the ratio HP: LP = 7:3, Fig. 15(d) shows the HP channel AC current (230 V/200 Hz) which is generated from CHROMA and LP channel AC current (230 V/50 Hz) which is generated from the Variable Autotransformer at 1 kW load condition.
The experimental result is consistent with the theoretical analysis in Section III-C, demonstrating that the system is more stable if the LP channel shares more power (see Fig. 8).

V. CONCLUSION AND FUTURE WORK
The paper presented a detailed control design and voltage stability analysis for one of the most promising power system architectures for the future MEA: a single bus based multisource DC EPS. This proposed architecture has the potential to considerably reduce the EPS weight in the MEA. Based on a developed mathematical model of the droop-controlled multisource system, a detailed control design of the source and load converter is presented. Further, the equivalent source and load impedance have been analytically derived and the stability assessment has been carried out to explore the impact of number of parallel sources and power sharing ratio. The main findings that have been experimentally validated can be highlighted as follows: (1) Compared to the single source operation, parallel sources operation will reduce the voltage deviation at the DC bus. As a result, the load impedance magnitude increases, the source impedance decreases and consequently the stability is improved.
(2) Different power sharing between HP and LP generators is emulated and it demonstrates that the system is more stable if the LP generator provides more power than the HP generator to feed the bus.
Overall, this paper provides a systematic design and analysis procedure for a future single DC bus EPS.
With the improvement of the system reliability and the potential to reduce the size of main generators, the installation of energy storage system (ESS) is essential in the MEA [36]. ESS can balance the power between sources and loads [37], improve the transient performance and stability [38], and provide the backup electrical power in case of a main power fault [39]. A corresponding coordinated control strategy needs to be developed dealing with the power balance between main generators and ESS. When the load is within the capacity of main generators, main generators will supply power to feed the load while the ESS can work in charging mode or "offline" mode. When the load exceeds the generator capacity under certain scenarios, together with main generators ESS works in discharging mode and provides power to the load. If more than one set of ESS is installed onboard, the ESS with higher state of charge (SoC) should provide more power to the load. In this sense, optimal coordinated control methods using dynamic consensus algorithm, fuzzy logic, or droop principle could be developed in the future.