Control-Based Stabilization of DC Microgrid for More Electric Aircraft

，，*，，，

1.Key Laboratory of More Electric Aircraft Technology of Zhejiang Province，University of Nottingham Ningbo China，Ningbo 315000，P.R.China；

2.Power Electronics，Machines and Control Group，University of Nottingham，Nottingham NG7 2RD，U.K.

Abstract:Electrifying the on-board subsystems of aircraft becomes an inevitable process as being faced with the environmental pollution，along with the proposed concept called more electric aircraft（MEA）.With the increasing number of on-board power electronic based devices，the distribution system of the aircraft can be regarded as an onboard microgrid.As it is known that the load power electronic converters can exhibit constant power load（CPL）characteristics and reduce the system stability，it is necessary to accurately predict and enhance the system stability in designing process.This paper firstly analyzes the stability of an on-board DC microgrid with the presence of CPL.Then，discusses the reasons behind instability and proposes a control strategy to enhance system stability.Finally，the simulation results are worked out to validate the analysis and the effect of the proposed control strategy.

Key words：DC microgrid；stability analysis；impedance model；constant power load；more electric aircraft；dual active bridge converter；permanent magnet synchronous generator

0 Introduction

Regarding to the deterioration of environment，it was proposed for aircraft that the on-board hydraulic，pneumatic，and mechanical subsystems need to be replaced with electrical subsystems.With these replacements，the proportion of electrical power increases，so called more electric aircraft（MEA）.Compared with conventional aircraft，the MEA takes advantages of lower environmental impact，lower weight，and lower maintenance cost［1-3］.The process of electrifying on-board subsystems has been started recently.For example in Boeing 787，the wing de-icing system，the environmental control system and the engine starting system are powered electrically rather than powered pneumatically［3］.In addition，the starter/generator system is improved from constant speed drive（CSD）system to variable speed constant frequency（VSCF）system［4］.To be more detailed，the integrated drive generator（IDG）is removed，which was used to produce constant speed based on the variable speed of the jet engine［5］.As a solution，the generator is directly coupled to the jet engine through a gearbox.Besides，it was also proposed that the on-board AC microgrid can be replaced with DC microgrid.Compared with AC microgrid，the DC microgrid takes advantages of lower power losses on components，fewer conversion stages and lower bus current［6］.Fig.1 shows a DC microgrid containing 540 V high voltage（HV）DC buses and 28 V low voltage（LV）DC buses as potential for MEA.

As shown in Fig.1，the power electronic converters become the key element to achieve the conversion between AC/DC voltage and different voltage levels，and there are varieties of loads interfaced with the converters，such as the loads insensitive to frequency，loads for energy storage，motors and pulsed loads.To be more detailed，the facilities for heating such as wing de-icing system，galley ovens and cargo heaters can be regarded as resistive loads，while the batteries，fuel cells and super capacitors are the loads for energy storage.Since these loads are tightly controlled by converters，the combination of the converters and the loads can be regarded as load subsystems，and the characteristics（such as power and impedance）of the load subsystems are determined by the characteristic of loads and the control strategy of converters.Unlike the resistive loads and energy storage units which have slow variation behavior during switching periods，the pulsed loads have fast and significant variation because they can instantly absorb large power from system，such as the radars and the electromagnetic launch and recovery systems.With this characteristic，the general averaging techniques such as switching averaging and state-space averaging cannot be applied directly.In this paper，the pulsed loads are not considered so that the later analysis is only valid if the amount of pulsed loads is a very minor part of the overall power requirements.As for the loads which have slow variations and require to be supplied with constant power，the subsystems to which the loads belong can be regarded as constant power load（CPL）.CPL has characteristic of negative incremental impedance at operating point［7］.TheI-Vcurve of CPL is shown in Fig.2.

Fig.1 An example of DC microgrid for MEA

Fig.2 I-V curve of CPL

With such a behavior，the system stability can be a main concern.Hence，it is necessary to predict the system stability during design.So far，several approaches were proposed by researchers to assess the stability during system design.One main approach is based on the impedance.To implement this，the system needs to be decomposed into a source subsystem and a load subsystem.Fig.3 shows an equivalent circuit of a DC system，whereVSis an ideal voltage source，VS，OUTthe output voltage of source subsystem，VL，INthe input voltage of load subsystem，ZSthe source impedance andZLthe load impedance.Hence，the load voltage can be defined as

Fig.3 An equivalent circuit of a DC system

whereGMLG=.Here a concept called minor loop gainGMLGis introduced，which is defined as the ratio of source impedance and load impedance.It is worth mentioning that if the system is supplied by a current sourceIS，theGMLGwill be reciprocal［8］.The load current can be given as

whereYSis the source admittance，YLthe load admittance，andGMLG=.With the minor loop gain，several impedance-based stability criteria were developed.The most basic one is the Nyquist Criterion that provides the sufficient and necessary condition for system stability.It depicts that for a system of which the source subsystem and load subsystem are individually stable，the whole system will be stable if the Nyquist contour ofGMLGdoes not encircle（-1，0）.Based on that，other criteria were proposed.The most conservative is Middlebrook Criterion since only the gain margin is considered［9］.The constraint is given as

whereGMis the desired gain margin.Although the middlebrook criterion ensures the system stability，it sometimes results in overdesign since the size of input filter could be large.The gain margin phase margin（GMPM）criterion as a compromise mean，loose the requirement of system design by liberating the phase margin［10］.The constraint of GMPM criterion is given as

whereGMis the desired gain margin andPMthe desired phase margin.With such a constraint，the system can be optimized more while the system stability is guaranteed.Recently，there have been several research outcomes of the system stability analysis based on the impedance.The stability of a multisource multi-load system with single DC bus is analyzed based on the impedance model［11］.The accurate impedance models of permanent magnet synchronous generator（PMSG）and dual active bridge（DAB）converter are validated and the relevant stability analysis is carried out［12］，moreover，the individual stability of PMSG is analyzed based on its impedance model［13］.The instability of DC bus voltage of grid-connected voltage source converter（VSC）is analyzed based on the impedance model［14］.The stability of a three-phase AC system with CPL is analyzed in terms ofd-qimpedance［15］.In addition，the minor loop gain is modified when it comes to a single-bus system including converters with different types of control［16］.

In this paper，the stability of a DC microgrid consisting of a PMSG as source，and a DAB converter and a permanent magnet synchronous motor（PMSM）as load，is analyzed in terms of their impedance models，where the impedance of bus cable is taken into consideration.Section 1 presents the impedance model of subsystems，and the system instability is pointed out by Bode diagram.Section 2 gives the explanations of system instability and proposes a method to mitigate the resonance resulted from the CLC circuit of DC system.Section 3 gives the simulation results of a switching model in PLECS.Section 4 draws the conclusion.

1 Impedance Model and System Instability

In this section，the output impedance of PMSG，input impedance of DAB converter and PMSM are given first，then the parameters of them and the DC bus cable are given.Next，the system instability is investigated.

1.1 Impedance model

The output impedance of PMSG has been derived［12-13］and given as

whereVdcis output DC voltage，CPMSGthe output capacitance，V qthe output voltage ofq-axis current controller which will be transformed and applied to the rectifier，Iqthe equivalent current onq-axis goes through the stator，Iothe output current of rectifier，Gi（s）the transfer function of current controller andGv（s）the transfer function of DC link voltage controller，andGd，PMSG（s）the first-order delay function with time constant of one switching period.RsandLsare the stator resistance and inductance.

The input impedance of DAB converter has been derived［12，17］and given as

whereCiis the input capacitance，Cothe output capacitance，Rloadthe load resistance，andGv，DABthe transfer function of voltage controller.G1，G2，G3andG4are the small signal gains.

The control of PMSM is similar to the control of PMSG，except that the voltage controller is changed to be speed controller.In this case，the mechanical load torque is assumed to be constant.By considering that the bandwidth of current controller is large，it can quickly compensate the dynamics on DC link voltage by adjusting the output duty cycle.Hence，the PMSM is tightly controlled and the input impedance of PMSM can be given as

whereViis the input DC voltage andIithe input DC current.

1.2 Determination of DC bus cable parameters

According to the specifications of Airbus A 380，the capacity of electrical power reaches 500 kW［3］.Since the DC bus voltage is chosen as 540 V，the rated current of bus cable must be larger than 1 000 A.Hence，the type of wire ASNE0438-YV AWG 4/0［18］is selected and seven of them are bundled up to be the cable to maximize the utilization of space.The cross-section is shown as Fig.4.

Fig.4 Cross-section of seven-wire bundled DC bus cable

Considering that the engines are located at the end of wings，it can be assumed that the length of bus cable is equal to the wingspan.With the known length and cross-section diameter of wire，the inductance of wire can be calculated using the formula［19］，given as

wherelis the length，dthe cross-section diameter，andμthe relative permeability.

1.3 Specifications

Before analyzing the system stability，the parameters of PMSG，DAB converter and DC bus cable need to be given.To facilitate the process of selecting parameters，the parameters in Ref.［12］are used.

1.4 System instability

With the known impedance models and cable impedance，the system stability can be predicted using GMPM Criterion.As a criterion based on Bode diagram，GMPM Criterion requires proper grouping of subsystems since different groupings of subsystems can have different gain margin and phase margin for system design［20］.Hence，for converteroriented design，the subsystems are regrouped and shown in Fig.5.With the known transfer functions of source impedanceZSand load impedanceZL，the Bode diagram of them at system power of 500 kW is obtained and shown in Fig.6.It can be seen that the Bode curves are intersected at frequency of about 570 Hz.At the frequency of intersection，the gain margin is minus and the phase margin is larger than 180°.This means that the system will be unstable at system power of 500 kW，and the instability harmonics on DC bus has a main component with frequency of 570 Hz.

Fig.5 Grouping of subsystems

Fig.6 Bode diagrams of ZS and ZL at system power of 500 kW

2 System Stabilization

2.1 System r esonance

It is known that for a three-phase AC system，the widely used LCL filter could cause instability.However，for a DC system，the CLC circuit could be formed by the filter capacitance of converters and inductance of DC bus cable.In Fig.6，it also can be observed that the Bode curve ofZSbehaves as CLC circuit at high frequencies.The existence of CLC circuit in system can make resonance by charging and discharging process among filter capacitance and cable inductance，which can be a contribution to system instability.In this case，the output capacitanceCPMSGof PMSG，the inductanceLwireof DC bus cable and the input capacitorCiof DAB converter form a CLC circuit（the resistance of bus cable is neglected here to simplify the computation process）.The impedance of CLC circuit seen from the load side is given as

By re-arranging the terms in Eq.（8），it has

Hence，the resonant frequency in radian of the CLC circuit can be obtained as

With the known values ofCPMSG，LwireandCi，the resonant frequencyωrcan be calculated.The calculated result shows a good match with the frequency labeled in Fig.6.

2.2 Proposed strategy for r esonance mitigation

As discussed above，the resonance caused by the CLC circuit could make system tend to instability.If the resonance can be mitigated，the system stability can be enhanced.One main approach to suppress the resonance is proposing new control strategies.Considering that for PMSG the current control is directly applied to plant，and the bandwidth of inner current loop should be much larger than the outer voltage loop，the influence of current loop on performance of PMSG is dominated.Hence，to reshape the source impedanceZS，the current control of PMSG is modified.Generally，the type of current control is proportional-integral（PI）control，which has good behavior of eliminating the DC error.The resonant control proposed in Ref.［21］，which has good behavior of eliminating the AC error，is widely used in the stationary frame control of machine and grid.Taking account of the advantages of PI control and proportionnal resonant（PR）control，an ideal proportional-integral-resonant（PIR）control for DAB converter was proposed to eliminate the second-order harmonic component of the grid frequency waveforms in a DC grid in Ref.［22］.Here，the idea is to add resonant control into the current control.Hence，the type of current control becomes PIR control and can take care of both DC and AC errors.The transfer function of a non-ideal resonant controller is given as

wherekris the resonant coefficient，ωcthe cut-off frequency in radian，andωrthe desired resonant frequency in radian.

Before applying the controller，the selection of parameters becomes important.To investigate the impact of parameters on the performance of controller，different parameters are set and the corresponding Bode curves are checked.Fig.7 shows the Bode curves of PI controller and PIR controller with different parameters.Compared with PI controller，the PIR controller has the large gain at the resonant frequency.As for the PIR controller，increasing the cut-off frequencyωcwill slightly increase the gain and bandwidth of sideband centered on resonant frequency，while the gain at the resonant frequency keeps the same.In addition，increasing the resonant coefficientkrwill obviously increase the gain and the bandwidth of sideband centered on resonant frequency，and the gain at the resonant frequency is also increased.In this paper，the PI parameters of current controller are determined using pole-zero cancellation method［12-13］.As for the resonant controller，kris set as 1，ωcis set as 50 Hz andωris set as 570 Hz.Fig.8 shows the Bode diagram of open loop gain of current control loop using PI controller and PIR controller.It can be seen that the addition of resonant controller increases the gain at resonant frequency.Fig.9 shows the Bode diagram ofZLandZSwith PI and PIR current control of PMSG at system power of 500 kW.It can be seen that with PIR controller the gain ofZSat 570 Hz is attenuated，avoiding the intersection withZL.Fig.10 shows the Nyquist contour of minor loop gainGMLG.It can be seen that the contour with PI control encircles（-1，0），indicating that the system will be unstable，while the system becomes stable when PIR control is applied.

Fig.7 Bode diagrams of PI controller and PIR controller with different parameters

Fig.8 Bode diagrams of open loop gain of current control loop using PIcontroller and PIR controller

Fig.9 Bode diagrams of ZL and ZS with PI and PIR current control of PMSG at system power of 500 kW

Fig.10 Nyquist contour of minor loop gain with PI and PIR control of PMSG at system power of 500 kW

3 Simulation Results

In this section，a switching model is built in PLECS to check the system instability caused by the resonance of CLC circuit and the effect of PIR controller on system stabilization.Fig.11 shows the simulation waveforms of output voltage andq-axis current of PMSG using PI current control.In Fig.11，the power of system is set as 300 kW initially and then increases to 500 kW att=0.4 s.It can be seen from Fig.11（a）that the system becomes unstable aftert=0.4 s，and the frequency of instability harmonic component can be observed in Fig.11（b），which is 570 Hz.Hence，the simulation results are consistent with the prediction results of Bode diagram in Fig.6 and the Nyquist contour in Fig.10.Fig.12 shows the simulation waveforms of output voltage andq-axis current of PMSG using PIR controller.It can be seen from Fig.12（a）that with PIR controller the system keeps stable after the power stepped to 500 k W att=0.4 s.And it is shown in Fig.12（b）that the 570 Hz harmonic component is eliminated.These simulation results also show a good match with the prediction results of Bode diagram in Fig.9 and the Nyquist contour in Fig.10.

Fig.11 Simulation waveforms of output voltage and q-axis current of PMSG using PIcontrol at t=0.6 s

Fig.12 Simulation waveforms of output voltage and q-axis current of PMSG using PIR control at t=0.6 s

The simulation results presented above shows that the PIR current control has effect on suppressing the resonance of DC bus which is introduced by the capacitance and inductance on bus.With this feature，the stable region of system is extended，and the system power can go higher with the same parameters.In addition，the inductance of bus cable can be increased，and the input capacitance of load subsystems can be reduced meanwhile the size of capacitors is also optimized.

4 Conclusions

This paper analyzes the stability of an on-board DC microgrid based on the impedance models of subsystems.The system instability is pointed out from the Bode diagram，and the reasons behind it is revealed，that is，the instability is mainly due to the resonance of the CLC type circuit.To suppress the resonance，the resonant control is introduced into the current control of PMSG，to reshape the source impedance.It can be seen from the Bode diagram that the gain at the frequency of instability harmonic component is significantly attenuated.As for the validation，the simulation of a switching model is carried out.The simulation results are consistent with the predicted results by analysis.