An enhanced second carrier harmonic cancellation method for multi-source DC electric power systems

Multi-source DC power systems have been widely used in electric transportations, including more-electric aircraft, electric ship and electric vehicles. These systems normally involve in power electronic converters whose switching actions may cause current fluctuation on DC-bus capacitors. Eliminating certain order harmonics can help the system potentially reduce the volume and weight of the capacitor. In this paper, a simplified model to estimate 2nd carrier harmonic of de current in two-level converters is proposed. In derived model, magnitude of the harmonic is only determined by the DC-bus current and modulation index of the converter. Meanwhile, the phase angle of it is determined by angle of carrier signals. Based on this model, a new harmonic cancellation method is proposed. The method has high robustness and can work under any fundamental frequency and power sharing ratio. Simulation results are presented in this paper to verify the validation of proposed harmonic model and the enhanced cancellation method.


INTRODUCTION
With the development of transportation electrification, multi-source DC power systems start to penetrate in hybrid vehicle [1], shipboard [2], and more electric aircraft [3], [4]. The multi-source systems give system advantages such as high redundancy, high flexibility of power management and so on. Fig 1 shows a simple multi-source power system. In the system, there are starter-generators (SG) supplying one common DC bus and give power to load together. Using different control algorithm in controller, optimization targets can be achieved in terms of efficiency, power quality, and reliability.
In the system, switching actions of converters cause current pulses injected into DC-bus capacitor. These pulses will significantly determine the weight, volume, and cost of capacitor. To overcome this problem, research on reducing current pulses on capacitor should be investigated.
In recent years, researchers have published several papers about capacitor reduction for single converter. In [5], mathematical model of DC-link current was developed using double Fourier solution. It gave researchers a potential to analyse harmonics and minimize it. In [6], lower capacitor harmonics are achieved by applying high modulation on DC-AC converter using an additional DC-DC converter. More switching devices are used in the system and the control becomes more complicated. In [7], capacitor ripple current was reduced by applying nonadjacent switching vectors.
For multi-source DC power system, there are few paper published about reducing capacitor. Fortunately, back-to-back converter is similar to multi-source power system. One rectifier and one inverter share same DC-bus capacitor. Meanwhile, fundamental frequencies of inverter and rectifier are not same. Methods of eliminating the switching frequency components of the first order and the second order on DC-bus capacitor were investigated in [8] and [9]. However, the performance of reducing harmonic highly relies on performance of band-pass filter, which limits the application of this method.
In this paper, the research is about cancellation method for the second carrier harmonics on capacitor. Firstly, a simplified mathematical model of second carrier harmonics on DC-bus is investigated. Based on it, a new method on minimizing 2nd carrier harmonic is investigated. This approach can significantly minimize harmonics under different fundamental frequencies and power sharing ratio. Compared with method proposed in [9], measurement on phase angle of harmonic on DC bus is not needed, which makes this method easily applied on low communication bandwidth system. At last, the proposed method has been validated by simulation, and the results are given in Section IV.  To eliminate 2nd carrier harmonics, analytical solution for harmonics generated from single converter should be investigated firstly. The 3-phase 2-level converters have been widely used in AC-DC conversion, as shown in Fig. 2. Analysis of the DC-bus current (idc) before capacitor will be introduced in this section.

A. Mathematical analysis on DC-bus 2nd carrier harmonic
Ignoring switching-frequency fluctuation on AC side current, for a two-level converter, AC side current is as follows Where Iac is the amplitude of fundamental component of AC current, f0 is the fundamental frequency of iabc, β is the angle between phase current and its back-EMF, k=0, 1 and 2 is to represent phase A, B and C.
The current on DC bus is flow from AC side through power converter. Using illustration in [9], for asymmetrical regular sampling PWM, the switching function for one phase leg can be expressed by [ ] ( ) = , 2 + + + 2 3 Where fc is the switching frequency, [ ] is PWM carrier angle for each leg, α is the phase angle between AC fundamental current and AC-side converter voltage (power factor angle), Km,n is the harmonic amplitude using Bessel function of first kind. Based on double Fourier analysis [10], Km,n can be expressed by Here, function Jn() is Bessel function of the first kind. In (3) and (4), m and n are orders of main and band side harmonic. For instance, when m=1 and n=3, Km,n means the magnitude of harmonic component on frequency fc+3f0. Using (1) -(4), the DC-bus harmonic currents generated from one phase leg can be derived as , Current harmonics on DC-bus should be expressed as sum of 3 legs' harmonics, which is , , ( ) = 2 , 2 ( + ) Where i and j mean orders of main and band side harmonic of current harmonics. This is different from m and n which are orders for switching function.
Phase angle of components in (8) (15), there are two components with same frequency but different magnitude and phase angle. Using magnitude-angle form, these two components can be expressed as , ∠(2 − ) and , ∠(2 + ) (16) Look into parameter K2,-1 and K2,1 firstly, In general, fc is much larger than f0. Hence term f0/ fc is approximately equal to 0. Substituting f0/ fc =0 in to (17) and (18) makes For given output converter power P, the AC-side terminal real power of the two-level converter can be formulated as Where Idc is the value of DC component current on DCbus. In (22), the simplified model also eliminates the influence from power factor angle α, which is always determined by many elements such as rotor speed, output power, and so on. Magnitude of 2nd carrier harmonic current is only influenced by Idc and M. Meanwhile, phase angle of the component is only determined by θc.  (19) to (22), term 1 and 2 reform to same magnitude which is shown in red. The simplified synthesized vector also has a fixed phase angle which is 2θc. In Fig 3, the difference between synthesized vectors in blue and red is obviously visible. However, this is drawn for clear illustration. In practical, the difference is quite small due to f0/fc is approximately equal to 0. This will also be proved in next subsection.

C. Comparison between simplified and original model
The comparison between original (15) and simplified (22) model is given in Fig 4. Fig 4a and b show the accurate magnitude and angle under different rotor speed and output power together with simplified model respectively. The calculation is based on a PMSG power generation system. Modulation index is set as 0.95. Parameter of PMSG is given in Table I. These two figures shows a higher error when PMSG works under higher speed. This is caused by assuming f0/fc=0 in (19). Higher rotor speed means larger f0 which makes f0/ fc not near 0 as it when under low speed.
Meanwhile, both magnitude and angle show larger error under low power output. Actually, this is a little complicated to analyse the reason because it is caused by power factor angle and magnitudes of two terms in (16) together. There are too many interaction relations from machine parameter and control scheme.
However, it is obvious that magnitude under low power is not significant. That means it will affect less on DC-bus current harmonics compared with that under higher power. Hence, this error can be ignored in practical, because the control system only care about the worst case which is high power operation.

Number of poles 6
Machine resistance 1.058mΩ

Machine d-axis inductance 99µH
Machine q-axis inductance 99µH Flux linkage of permanent magnet 0.03644Vs/rad III. PROPOSED METHOD Using simplified model illustrated in previews section, a new 2nd carrier harmonic cancellation method for multisource system was proposed in this section.
A. Method for 1:1 power sharing ratio A system only contains 2 sources and one resistance load was researched in this section, which is shown in Fig 5. The purpose of controller is to adjust DC bus voltage constant at reference value Vref. A centralized controller is applied to realize power sharing control and PWM generation. In practical, controller can also be decentralized controller with information communication. Using centralized controller here is just for simple explanation.
The main concept of proposed method is 2nd carrier harmonics generated from converters interactively cancel each other. It means two components has same magnitude but 180-degree phase shift. When two sources work in average, Idc1=Idc2. Seen from (22) and {2} mean the index of SG.
The control diagram was shown in Fig 5. A centralized voltage controller works for making DC bus voltage following reference. DC current reference is generated and broadcast to each PWM generator, hence power sharing between two converters is in average. Then, by a 90-degree phase shift on carrier signal, harmonic cancellation can be achieved. B. Method for other power sharing ratio In practical system, non-average power sharing ratio was always applied according to different optimizing objects. Only applying 90-degree phase shift cannot eliminate each other totally because the magnitudes of 2 components are different. To overcome this problem, a new control scheme was proposed and shown in Fig 6. Between the local controllers and voltage controller, an additional 2nd carrier harmonic controller block was applied. This block is to calculate the individual current references and modulation index based on power sharing ratio and maximum modulation index. Firstly, define K as power sharing ratio between two sources which is Hence, the exact value of iref1 and iref2 can be expressed as Then, same magnitudes of 2nd carrier harmonics should be achieved. Seen from (22), the relation should be Modulation index of each converter can be manipulated to achieve same 2nd carrier harmonic magnitude between two sources. Smaller modulation index can increase the magnitude which is shown in (22). Assuming K<1, which means SG2 produce more power than SG1, and it should operate under maximum modulation index Mmax. M1 is manipulated for 2nd carrier harmonic to reach the same magnitude value, which is Define function Manipulated M1 can be derived using inverse function of (27) which is However this inverse function is hard to calculate. Hence, a look-up table is applied in practical. Fig 7 shows the relation between M and K when Mmax=0.9 and 0.95 respectively. A lower limitation 0.5 was set to prevent high negative d-axis current, which will generate higher loss.
When K>1, same solution can be implemented. Modulation index for both sources are Using manipulated modulation index together with 90 degree phase shift on carrier signal, 2nd carrier harmonics on DC-bus capacitor can be supressed under non-average power sharing ratio.   C. Method for system with more than two sources This paper focuses on dual-channel system. The simplified can also be utilized for system with more than two converters with same basic concept. However, this is not the important part of this paper, and will be presented in future work.

IV. SIMULATON RESULTS
Simulations based on MATLAB/Simulink and PLECS was implemented to evaluate the performance of proposed harmonic model and cancellation method. Some basic control parameters for multi-source power system was shown in Table  II.

V. CONCLUSION
In this paper, a simplified mathematical model on 2 nd carrier current harmonic was investigated firstly. The results show the magnitude of component was only determined by the value of DC current and modulation index, while phase angle of it is caused by carrier phase angle. Based on simplified model, a new 2nd harmonic cancellation method was proposed. By manipulating modulation index together with 90-degree phase shift on carrier signal, cancellation can be realized in a 2-source power system. The method can work under any machine speed and need no communication of phase angle among sources.
Finally, simulation has been done to verify the validation of the proposed 2nd harmonic model and cancellation method.