Fractional order sliding mode control for circulating current suppressing of MMC

The bridge current distortion and system power loss of modular multilevel converter (MMC) are increased by the circulating current. In this paper, combined with fractional order theory, a circulating current controller based on fractional order differential sign function sliding mode control (FO-SMC) is proposed. By adding fractional order calculus into the sign function, the response speed is improved on the premise of reducing the chattering of the system. Under the same conditions, the system with PI, SMC and FO-SMC circulating current controller is simulated under AC power disturbance, DC side voltage step and received power step conditions, respectively. The simulation results show that the FO-SMC circulating current controller is better than the other in both steady and dynamic conditions and is more suitable for nonlinear systems as MMC.


Introduction
The topology of the modular multilevel converter (MMC) is cascades by multiple identical sub-modules.Due to the above circuit structure, it is convenient to improve the reliability of the converter operation by installing redundancy [1][2][3].Moreover, the cascading of several sub-modules can improve the working voltage level of the converter and reduce the harmonic content of the output waveform, etc. [4].Therefore, the MMC topology has become the preferred solution for high voltage and high power transmission.However, due to the limited capacity of submodule capacitors, capacitor voltage fluctuations are inevitable in the process of power transmission.As a result, there is circulating current inside the converter [5,6].
The circulating current not only increases the current amplitude of the bridge but also improve the rated capacity of the device [7].Moreover, it will increase the system loss and shorten the life of the equipment [8,9].Therefore, a series of advanced control methods have been used for circulating current suppressing in recent years.The PI circulating current controller was designed in [10], but its control performance is greatly affected by system parameters.The result is that the system has poor robustness and limited control accuracy.Further, proportional resonant (PR) circulating current controller is designed in [11,12], which only suppresses the double frequency circulating current.On this basis, multiple PR controllers are connected in parallel in [13] to control the 2, 4, 6 and high-order harmonic circulating currents, respectively.Although the PR circulating current controller can directly track the AC component without static error, but it is sensitive to the frequency fluctuation of grid.Moreover, the parallel connection of multiple PR controllers will greatly increase the complexity of the control system.For other controllers, the virtual impedance sliding mode circulating current controller is proposed in [14] to improve the system robustness and enhance the circulating current suppressing.However, chattering caused by sliding mode should not be ignored in practical applications, which will further increase loss and affect the stability of device.
Based on the above problems of traditional controllers, more and more researchers have introduced fractional order theory into control systems in recent years.Aiming to improve control system performance through fractional order controllers.The fractional order PI λ D μ controller is presented and applied to the BUCK converter control system in [15].Compared with the conventional controller PID, two free variables λ and μ are added and the range of values is real.Finally, the simulation proves that the fractional order PI λ D μ controller can improve the dynamic and steady performance of the system.Based on [15], the fractional order PI λ D μ controller is fuzzed and used in the DC motor speed control process in [16].The simulation results show that its dynamic response and steady-state error performance are better than the traditional integer order PID controller.Based on the fractional order characteristics of DC-DC converter, a fractional order terminal sliding mode controller is proposed in [17].Theoretical analysis and simulation results show that the controller can better cope with sudden load changes.With the further study of fractional order, the fractional order calculus is used to construct sliding mode surface and apply it to the speed control system of permanent magnet synchronous motor in [18].Compared with the conventional integer order sliding mode surface, it has better performance in speed tracking and interference immunity.Based on the above analysis, the application of fractional order theory to control systems has advantages that traditional integer order cannot achieve.
Based on the above analysis, this paper proposes a fractional order differential sign function sliding mode circulating current controller (FO-SMC).By adding fractional order sign function to the approach rate, the response speed of the system can be improved and the chattering of the sliding mode can be reduced.
The rest of this paper are organized as follows.The working principle and circulating current mechanism of the converter are analyzed in Sect. 2.Then, the fractional order differential sign function sliding mode circulating current controller is designed in Sect.3.And in Sect.4, its approach rate stability and sliding mode chattering are analyzed.In Sect.5, the PI, traditional sliding mode (SMC) and FO-SMC circulating current controllers are simulated under three different working conditions.Finally, the concluding remarks are presented in Sect.6.

MMC topology
The topology of the MMC is shown in Fig. 1.Each phaseleg is composed of upper and lower symmetrical arm, and each arm contains N identical SMs and a smoothing inductor L m .Each SM is composed of two IGBTs with anti-parallel diodes and a capacitor C. When the SM is in the input state, the output capacitor voltage u c ; when the SM is in the removal state, the output voltage is 0. At any time, the total number of SMs with upper and lower arm of each phase-leg in the input state should be N. Therefore, N + 1 level waveforms can be output on the AC side by rationally distributing the number of SMs in the input state of the upper and lower arms, respectively.
As shown in Fig. 1, u sj andi sj (j = a, b, c, the same below) are the voltage and current on the AC side of the converter; L s and R s are the inductors and resistors on the AC side; u pj and u nj are the equivalent voltage of the upper and lower arm; R m is the equivalent resistors of arm; U dc andI dc are voltage and current of DC side, respectively.From KCL, the upper and lower arm current i pj and i nj can be expressed as where i cirj is the circulating current of phase j.Which can be expressed by the upper and lower arm current as If e j is defined as the equivalent AC output voltage of phase j without arm reactance, then the single-phase equivalent circuit of MMC is shown in Fig. 2.
From Fig. 2, it can be seen that by writing the KVL equation for the AC and DC side of the equivalent circuit According to (3), the circulating current i cirj is determined by the voltage difference between the upper and lower arm and the DC bus.Where the unbalanced voltage drop u cirj can be written as

MMC circulating current mechanism
Considering the converter operating characteristics, the DC output voltages of the three phases connected in parallel are equal at all times in the ideal case.At this time, i cirj of each phase circulating current only contains the DC component I dc /3.However, in actual operation, due to the limited capacity of capacitors, the voltage of capacitors will deviate from the rated value U dc /N in the process of SMs input and removal.Therefore, the circulating current of AC component is generated between the three phases.It is proved that the AC component in the three-phases circulating current presents the characteristics of double line-frequency negative sequence from the perspective of the instantaneous energy balance of the bridge arm and the switching function in [9,10].Neglecting the higher harmonic components, the threephases circulating current can be expressed as where ϕ 0 is the initial phase angle of the double linefrequency circulating current, and I 2f is the peak value of the double line-frequency circulating current.Introducing the negative sequence coordinate transformation, the circulating current under the three-phase stationary frame is transformed to the double line-frequency negative sequence rotational reference frame, and the circulating current voltage equation is obtained as where u cird and u cirq are the d and q components of the inner unbalanced voltage u cirj ,i cird and i cirq denote the d and q components of the circulating current.
According to (6), the mathematical model of circulating current in the negative sequence rotating reference frame is shown in Fig. 3.
According to (3), ( 4) and ( 6), it can be seen that the ultimate purpose of circulating current suppressing is to adjust the output voltages of the upper and lower arm to compensate for the three-phase unbalanced voltage drop.Where, the reference signal u cirj_pref of unbalanced voltage drop of each phase is obtained from the circulating current controller.Therefore, on the premise that the output of the AC and DC side of the converter is not affected, the output voltage reference signals u pj_pref and u nj_pref of the upper and lower arm of each phase is expressed as 123

FO-SMC circulating current suppressing strategy
MMC circulating current shows strong interconnection and nonlinear characteristics, which is associated with grid voltage fluctuations, SMs capacitors voltage and loads.The change of any parameter will affect the circulating current model, making it difficult for the traditional linear controller to ensure the system dynamic performance.Therefore, to enhance the circulating current suppressing effect, improve the dynamic response performance of the system, and reduce the dependence of the circulating current controller on the model parameters, a fractional order differential sign function sliding mode controller is proposed in this paper.

Decoupling linearization of MMC circulating current systems
According to the mathematical model of the circulating current in Fig. 3, there is a strong cross-coupling between the d and q axis model of circulating current, and it should be accurately decoupled before designing the controller to improve the system control performance.Therefore, the input-output feedback linearization method is introduced to eliminate the coupling existing in the circulating current model.Rewriting (6) as an affine nonlinear model with two inputs and outputs as where state variablex According to the exact linearization condition of the multiinput-output affine nonlinear system, solving for the Lie derivative of the output variable h(x 0 ) at any point x 0 of the state variable x yields the matrix E(x 0 ) as According to (9), the relative order vector of the system r = r 1 + r 2 = 2, which is equal to the state variable dimension.Therefore, the system output variable to time derivative relationship is expressed as where By introducing a new control vector v = [v 1 v 2 ] T , the linear relationship between y and v is satisfied According to ( 9), ( 10), ( 11) and ( 12), the nonlinear state feedback control rate of the system can be obtained Then, the nonlinear relationship between the original output variable y and the input variable u is converted into a linear relationship with v, realizing the precise decoupling between the d and q axis circulating current.
According to (13), the decoupled circulating current model is shown in Fig. 4.Where the blue part represents the decoupling link, and then the input-output linear relationship is satisfied.

Fractional order sliding mode controller design
From the analysis in Sect.3.1, the linear model of circulating current is obtained.However, in practical engineering, the disturbance of external factors can seriously affect the accuracy of the circulating current model.Therefore, the nonlinear fractional order sliding mode controller is designed.
The integral sliding mode surface S is selected as Fig. 5 Fractional-order differential sign function wheree d = i cird_ref -i cird , e q = i cirq_ref -i cirq .k 11 , k 12 , k 21 , k 22 are the control parameters of integral sliding mode surface.
The introduction of integration eliminates the steady-state current error and allows the system to converge in a finite time.traditional sliding mode control does not move smoothly along the sliding mode surface after reaching the mode surface but keeps crossing on both sides of the sliding mode surface, resulting in chattering.In this paper, in order to weaken the chattering phenomenon and improve the system response speed.Combining fractional order theory and sign function, the sliding mode approach rate of fractional order differential sign function is designed.
Traditional sign function can be expressed as The fractional order differential sign function can be expressed as According to (16), the fractional order differential sign function can also extract the sign of the function f (t).When the same signal is input, the differential sign function of different orders is compared with the traditional sign function as shown in Fig. 5.
According to Fig. 5, the fractional order differential sign function 0 D t λ sign(t) > > 1 when the system state crosses the sliding mode surface moment, and the value of the function varies with the change of order.The value of the differential sign function gradually decays between 1 and 2 with time, until the system state returns to the sliding mode surface.The gain of the differential sign function is correlated with the deviation, which can improve the system response speed on the one hand, and effectively weaken the chattering of the sliding mode controller on the other.Therefore, the sliding mode approach rate of the fractional order differential sign function can be expressed as where the differential order 0 < λ < 1, β andε are constants.Selecting 0 < ε < 1 ensures that the differential sign function has a large gain when the system state cross the sliding mode surface, and the gain of the differential sign function is less than 1 when the system state is close to the sliding mode surface, which further weakens the sliding mode chattering.According to (12), ( 14) and (17), the system control rate can be expressed as The exact feedback linearization yields the system output control quantity as In summary, taking the d axis as an example, the control block diagram based on the fractional order sliding mode circulating current controller is shown in Fig. 6 (q-axis is similar).
Combined with ( 7) and ( 19), the overall control block diagram of the system can be drawn in Fig. 7.Where the red part represents the MMC control link, and the blue part represents the circulating current suppressing link.

Stability analysis of sliding mode approach rate of fractional differential sign function
To verify the stability of the sliding mode approach rate of fractional order differential sign function, the Lyapunov function is selected Both sides of Eq. ( 20) are derived, which derives Eq. ( 21) considering (17).
From the analysis in Sect.3.2, 0 < ε < 1.When the parameter β > 0, it is guaranteed that for any S = 0, V <0 is always established, which satisfies the Lyapunov stability condition and ensures that the system reaches the sliding mode surface in a finite time.

Chattering analysis
In practical control systems, the sliding mode approach rate is often analyzed and applied in a discrete form, and the ε 0 D t λ sign (t) plays a major role when approaching the sliding mode surface.Therefore, the discrete form of the approach rate of the fractional differential sign function is expressed as where T is the sample period.
When the system state starts to move to the sliding surface on the side of S > 0, assuming S(n) = 0 + , it can be obtained where 23), the chattering width of the sliding mode controller in the fractional order differential sign function is expressed as The chattering width of the traditional sliding mode controller is expressed as Comparing ( 24) and ( 25), it can be seen that the chattering width 1 under the traditional sliding mode approach rate is a fixed value that does not change with time.However, the fractional order differential sign function uses the memory effect of fractional order and releases energy slowly with time.By adjusting the differential order λ, the chattering width can be changed, so that the system can reach a stable point.

Simulation verification
In order to verify the effectiveness of the proposed FO-SMC circulating current controller and to compare its advantages and disadvantages with the conventional PI circulating current controller and sliding mode (SMC) circulating current controller, a MMC-HVDC model with a voltage level of 20 kV is built in this paper in the MATLAB.Where the modulation strategy adopts carrier stacking, and the sorting method is used to balance the voltage of capacitor.The dynamic and steady-state performances of three circulating current controllers are studied separately for comparison under operating conditions of power side disturbance, DC side voltage step and received power step change.The simulation parameters are shown in Table 1.

Anti-source disturbance performance comparison
Under the simulation parameters in Table 1, phase-A is analyzed as an example, and the PI, SMC and FO-SMC circulating current controllers are activated at 0.05 s for steady-state comparison simulation.At 0.2 s, the phase-A Fig. 8 Phase-A circulating current waveform voltage amplitude increases by 10%, and the disturbance is cleared at 0.22 s, to verify the performance of different circulating current controllers anti-source disturbance.The simulation results are shown in Fig. 8. Figure 8a and b shows the A-phase circulating current waveforms without and with three different circulating current controllers, respectively.Activating the circulating current controller at 0.05 s, three circulating current controllers are fast and effective in reducing circulating current fluctuations.When there is disturbance in the source, the circulating current fluctuation under the PI controller is the largest, and the circulating current suppressing effect is Fig. 9 FFT analysis spectrum different control methods the best with the SMC and FO-SMC controllers.In order to compare the anti-source disturbance performance of the circulating current controllers in detail, FFT analysis was performed on the circulating current before and after the disturbance.The circulating current analysis spectrum are shown in Fig. 9, and the data are shown in Table 2.
When the load is 20 resistance, the DC component of each phase circulating current should be 333.33 A. From the DC component values in Fig. 9, it can be seen that the FO-SMC circulating current controller has the least impact on the DC component and is closer to the ideal DC component of the circulating current; SMC is the second.The PI circulating current controller has the greatest impact on the DC component.It can be seen from the proportion of the double-line-frequency component of the circulating current in the two steady-state conditions before and after the disturbance removal, circulating current suppressing capability of the three controllers in the steady-state conditions can be obtained: FO-SMC > SMC > PI.Secondly, through the comparative analysis of the change of the double line-frequency component of the circulating current, the anti-source disturbance performance of the three controllers can be obtained: FO-SMC > SMC > PI.It can be verified that the performance of the FO-SMC circulating current controller is better than the other in both steady-state and source disturbance conditions.However, the influence of the three circulating current suppressors on the circulating current high-frequency component can be ignored, so the circulation high-frequency component is not analyzed in this paper.

Performance comparison of DC link voltage step
The converter is operated in the closed-loop rectification state with the system DC link output voltage of 20 kV.The system DC link voltage steps by 20% at 0.05 s to verify the dynamic response performance of the three circulating current controllers under DC link voltage step conditions.Figure 10 shows the three-phase circulating current waveforms with three different circulating current controllers, respectively.Figure 10a shows that the DC link output voltage is 20 kV at 0 ~0.05 s, and the DC output voltage steps to 24 kV at 0.05 s.Compare Fig. 10b, c and d, when the DC link voltage step, the original steady state of the system is broken and transited to a new steady state.Comparing the time to reach the new steady state with the three circulating current controllers, the system transition time is the shortest with the FO-SMC controller, while the time is longer with the SMC and PI controllers.The largest fluctuations of the circulating current with the PI controller and the smallest fluctuations with the SMC and FO-SMC controllers during the transition.To clearly compare the circulating current suppressing performance of the three controllers under the DC link voltage step, FFT analysis of circulating current i cira of the system is performed, and the data are shown in Table 3.
It can be seen from the data in Table 3 that comparing the proportion of the total harmonics and the proportion of the double line-frequency of circulating currenti cira , the three circulating current controllers are ranked as follows: FO-SMC > SMC > PI.When the DC link voltage is 24 kV, the ideal value of DC component in each phase of the circulating current should be 400 A. Comparing the value of phase-A DC component, it can be seen that the value of DC component is equal with the FO-SMC and SMC controllers, which is also

Performance comparison under received power step
The system operates at the power receiving side with constant active-reactive power control.At 0.2 s, the received power of the converter increases from 20 to 24 MW, to simulate the performance of different circulating current controllers when the received power step.Since the three circulating current controllers have similar effects on the transmission power, the power change with the FO-SMC circulating current controller is used as an example for illustration.Figure 11b, c and d shows that the three circulating current controllers have good dynamic response performance when power step at 0.2 s, respectively.However, during the step change in power, the circulating current transits from the original steady state to the new steady state.Overshoot exists in both PI and SMC controllers, while the circulating current smoothly transits to the new steady state with the FO-SMC controller.To clearly compare the circulating current suppressing performance of the three controllers during the received power step, FFT analysis of the circulating current i cira of the system is performed, and the data are shown in Table 4.
Comparing the data in Table 4, it can be seen that the DC component with the three different controllers are basically the same when the received power step.Comparing the total harmonic percentage of the circulating current, the circulating current i cira with the SMC controller is the highest.Comparing the percentage of the double line-frequency component of the circulating current, the three circulating current controllers are ranked as follows: FO-SMC > SMC > PI.It is proved that when the converter operates at the power receiving side, the FO-SMC circulating current controller not only has the best double line-frequency circulating current suppressing performance but also has no overshoot during the transition to the new steady state during received power step.

Conclusion
In this paper, the internal circulating current of MMC is analyzed.The input-output feedback linearization method is introduced to achieve linear decoupling of the circulating current model.And a circulating current control method based on FO-SMC is proposed.The simulation results show that the FO-SMC circulating current controller has better performance in suppressing the double line-frequency circulating current and improves the dynamic response performance and control effect of the system.

Fig. 2
Fig. 2 Equivalent circuit of one phase of MMC

Fig. 3
Fig. 3 Circulating current mathematical model of MMC

Fig. 4
Fig. 4 Circulating current decoupling model of MMC

Fig. 6 d
Fig. 6 d-Axis control block diagram with FO−SMC circulating current controller

Fig. 7
Fig. 7 Overall control diagram of MMC

Fig. 10
Fig. 10 Three-phase circulating current waveforms under DC link voltage step

Fig. 11
Fig. 11 Three-phase circulating current waveforms under received power step

Table 1
Simulation parameters of the MMC

Table 3
Phase-A circulating current under DC link voltage step

Table 4
Phase-A circulating current under received power step