Multi-objective Model Predictive Control of Grid-connected Three-level Inverter Based on Hierarchical Optimization*

,* , , , ,

(1.College of Electrical and Information Engineering,Hunan University,Changsha 410082,China;2.School of Automation,Central South University,Changsha 410083,China;3.College of Electronic Information and Electrical Engineering,Changsha University,Changsha 410022,China)

Abstract:In order to solve the problem of weighting factors selection in the conventional finite-control-set model predictive control for a grid-connected three-level inverter,an improved multi-objective model predictive control without weighting factors based on hierarchical optimization is proposed.Four control objectives are considered in this strategy.The grid current and neutral-point voltage of the DC-link are taken as the objectives in the first optimization hierarchy,and by using fuzzy satisfaction decision,several feasible candidates of voltage vectors are determined.Then,the average switching frequency and common-mode voltage are optimized in the second hierarchy.The average ranking criterion is introduced to sort the objective functions,and the best voltage vector is obtained to realize the coordinated control of multiple objectives.At last,the effectiveness of the proposed strategy is verified by simulation results.

Keywords:Multi-objective model predictive control,grid-connected three-level inverter,hierarchical optimization,fuzzy satisfaction decision,average ranking criterion

1 Introduction1

The grid-connected inverter is widely used as an important link between renewable energy and the power grid[1-2].Compared with two-level inverters,the three-level inverters have several advantages,including low voltage stress of switching devices and low total harmonic distortion (THD) of the grid-connected current[3-4].

At present,the finite-control-set model predictive control (FCS-MPC) is one of the main control technologies for grid-connected inverters[5-6].In this technique,based on the discrete prediction model and the cost function of the control objective,an optimal operation can be obtained for the next sampling period from a limited number of voltage vectors[7-8].The obvious advantages of this technique are that it does not require modulation modules and avoids the complicated process of controller parameter adjustment in the traditional method[9-10].In addition,FCS-MPC has ability to control several objectives simultaneously,which are transformed into a single cost function with different weights[11].However,each control objective has a different dimension and control standard,which leads to incommensurability.Moreover,the objectives may conflict with each other,and choosing the appropriate weighting factors between multiple objectives becomes difficult.In Ref.[12],the guidelines to select suitable weighting factors is presented,but the method is empirical.In Ref.[13],the weighting factors are clearly derived based on the analytical expression of torque ripple.However,this method relies heavily on model accuracy and cannot optimize the switching frequency simultaneously.In Ref.[14],by training an artificial neural network to replace the power model,the weighting factors can be selected.However,it takes a long time to repeat the simulations to obtain the data needed for the algorithm.

In this paper,a multi-objective model predictive control for grid-connected three-level inverter based on hierarchical optimization is proposed.As the first hierarchy,the grid current and neutral-point voltage of the DC-link are optimized by fuzzy satisfaction decision to obtain several feasible candidates of voltage vectors.The average switching frequency and common-mode voltage are taken as the control objectives in the second hierarchy.Their cost functions under candidate voltage vectors obtained from the first hierarchy are calculated,and the optimal voltage vector is selected according to the average ranking criterion of the objectives.Compared to the conventional FCS-MPC method,the proposed scheme solves the difficulty of designing the weighting factors of multi-objective predictive control.

The other parts of this paper are organized as follows.The FCS-MPC method of the grid-connected three-level inverter is introduced in Section 2.Section 3 presents the proposed multi-objective model predictive control scheme based on hierarchical optimization.In Section 4,the simulation results are provided,and Section 5 is the conclusion of this paper.

2 FCS-MPC of grid-connected three-level inverter

2.1 Discrete prediction model

The power circuit of grid-connected three-level inverter is depicted in Fig.1.Each leg of the inverter consists of four power electronic switches and two diodes,resulting three switching states,as follows

wherex=a,b,c.

According to switching combinations,there are 33=27 switching states corresponding to the output voltages in the three-level inverter,and the output voltage of inverter can be expressed as follows

whereVdcis the DC source voltage.

Fig.1 Grid-connected 3-level inverter

The voltage space vector is usually used to describe the output voltage of the inverter,which is defined by Eq.(3).The voltage space vectors associated with switching states are shown in Fig.2.

Fig.2 Voltage space vectors

From Fig.1,the mathematical model of the inverter in the stationarya-b-cframe is obtained by applying Kirchhoff’s law.

whereLis the filter inductance;Ris line resistance;ea,eb,andecare the grid voltages;ia,ib,andicare the grid-connected currents;andvNois common-mode voltage,which can be calculated as follows

Combined with Eq.(3) and Eq.(4),the output voltage space vectors can be written as follows

Considering the definition of space vector,the grid-connected current space vector and grid voltage space vector can be defined as follows

Then,considering thatvNo+εvNo+ε2vNo=0,by combining Eq.(6),Eq.(7),and Eq.(8),the output voltage space vector can be rewritten as follows

Further,whenC1=C2=C,the neutral-point voltage of the DC-link can be calculated by using the grid-connected current and switching status

When the sampling period is small enough,the forward Euler formula can be used to approximate the differential term[15],that is

whereTsis the sampling period.

Finally,combined with the Clarke transformation,the discrete time model in the stationaryα-βframe is obtained as

2.2 Cost function

In the conventional FCS-MPC scheme,the cost function is first created according to the control objectives of the system.Then,by rolling all candidate voltage vectors,the best one that minimizes the value of the cost function is selected.If multiple control objectives are required,by linear weighting,a single cost function that includes the multi-objectives is used[16].In the grid-connected three-level inverter,a tracking reference current with minimum error is expected to improve the performance of grid connection and reduce the harmonic component;the imbalance of neutral-point voltage of the DC-link will increase the maximum inverse voltage of the switch tube,which will damage the device in severe cases;the switching loss of switch tube is closely related to the switching frequency,and it is necessary to reduce the switching frequency to improve the system efficiency;and the common-mode voltage will generate high-frequency leakage current to the ground through parasitic capacitance,which forms electromagnetic interference,shortens the life of load equipment,and even endangers personal safety.Hence,the control objectives are controlling grid-connected current to track reference current,balancing the neutralpoint voltage of the DC-link,reducing the common-mode voltage,and decreasing the average switching frequency.The cost function can be expressed as follows

whereλ1,λ2,λ3,λ4,andλ5are the weighting factors;iαandiβare the predicted grid current in the stationaryα-βframe,which can be obtained by Eq.(12);vcmis the common-mode voltage,which can be obtained by Eq.(5);vois neutral-point voltage of the DC-link,which can be obtained by Eq.(12);andNSis the switching number,which can be calculated as follows

2.3 Time delay compensation

Based on discrete model and cost function,the switching state of the system in the next sampling period can be predicted by FCS-MPC.However,a time delay always exists due to the calculation time of predictive control.The method of one-step-ahead prediction for grid current can be used to compensate the time delay.The formula for one-step-ahead prediction can be written as follows

wherei′αβis calculated by combining the switching state predicted at thek-1 sampling period and Eq.(12).

The time delay compensation can be realized by replacingi(k+1) andi*(k+1) in Eq.(13) withi(k+2)andi*(k+2),respectively.i*(k+2) is obtained by further updating Eq.(15).

Once the cost function and time delay compensation are completed,combined with the measured values of current and voltage at the current moment,the state at the future moment of system can be predicted.

However,there are some problems in the conventional FCS-MPC scheme.It is hard to choose the appropriate weighting factors,because the dimension of each control object is inconsistent,which leads to the problem of incommensurability.Further,the objectives interact with each other.When an objective is over-optimized,the control quality of the remaining objectives may decline.At present,there is no scientific calculation method to design the weighting factors.The common approach is to use a large number of experimental and simulated data for offline analysis,but the debugging process is complicated.Hence,in order to solve the above problems,the model predictive control based on hierarchical optimization (HO-MPC) is proposed below.

3 Proposed HO-MPC strategy

Hierarchical optimization is a common strategy to solve multi-objective optimization problems[18].It is suitable for situations where multiple objectives evidently have different priorities or importance.When the four control objectives are considered simultaneously,tracking the reference current and reducing the neutral-point voltage of the DC-link have significantly higher priorities.Owing to their mutual influence,separate analysis cannot obtain good control performance,so the two are divided into the first hierarchy for combined analysis.Decreasing the switching number and common-mode voltage have lower priorities.Thus,they are put into the second optimization hierarchy.The complete flow chart of the proposed HO-MPC strategy is shown in Fig.3.

Fig.3 The flow chart of the HO-MPC scheme

3.1 Control design of the first hierarchy

In order to optimize the performance of the grid current and neutral-point voltage of the DC-link,the fuzzy satisfaction decision is applied in the first optimization hierarchy.This is one of the most popular multi-attribute decision-making methods and works well in the control areas of power system and power electronic,such as predictive torque control for induction motor drives[19].In this chapter,the satisfactory errors of grid current and neutral-point voltage are set to control these two objectives in the satisfactory interval.Then,the global membership function is used to judge whether to enter the second optimization hierarchy for further optimization or directly select the optimal voltage vector in the first optimization hierarchy.

The absolute tracking error of grid current is first defined as follows

Then,the fuzzy satisfaction membership function of grid current (μα,μβ) and neutral-point voltage of the DC-link (μv) are calculated as follows

whereIlimandVlimare the satisfactory errors of the grid current and neutral-point voltage,respectively.

Thus,the global membership function can be written as follows

Next,the values ofμare calculated by an exhaustive search of 27 voltage vectors.

Whenμ=1 exists,it indicates that the grid current and neutral-point voltage can be satisfactorily controlled.At this point,the second optimization hierarchy is entered and the optimal voltage vector is selected according to the method in Section 3.3.The setJincluding feasible voltage vectors can be defined by Eq.(20),where the number of elements in setJis denoted byn.

Whenμ=1 does not exist,it means that the satisfactory control of grid current and neutral-point voltage cannot be achieved.At this time,the switching number and common-mode voltage of the second optimization hierarchy cannot be taken into account,and the objectives of first optimization hierarchy need to be optimized.Thus,the best voltage vector corresponding to the switching state to be applied at the next sampling time is selected by minimizing the value of global membership function,that is

3.2 Control design of the second hierarchy

The second optimization hierarchy is based on satisfying the first optimization hierarchy.In this case,the candidate voltage vectors obtained from setJare judged to obtain the best one.The switching number and common-mode voltage are the optimization objectives.The average ranking method,involving ranking the value of the individual evaluation function,is used[20].The detailed steps are given below.

The evaluation functions of the two objectives are defined as follows

wheref1is the absolute value of common-mode voltage andf2is the switching number,which both depend on the voltage vectors procured from setJ.

Then,each evaluation function associated with feasible voltage vectors is sorted,and a ranking value is allocated to each of them,that is

wherer1andr2are the ranking values corresponding tof1andf2,respectively.

The smaller the value of evaluation function,the lower the ranking of the voltage vector that will be assigned.Thus,the best voltage vector is selected by selecting the smallest ranking value of evaluation function.On the one hand,the voltage vector corresponding to the lowest value ofr1can be selected to reduce common-mode voltage.On the other hand,by selecting the lowest ranking value ofr2,it is possible to select an optimal voltage vector from the point of the decreasing switching number.However,in the second hierarchy,the switching number and common-mode voltage are of equal importance.It is necessary to choose a voltage vector among these alternatives to achieve the best overall control objectives.That is,an average sorting criterion is used by calculating the average ranking value of two evaluation functions,and the voltage vector with the lowest average ranking value can be selected.Thus,the optimal voltage vector based on the average sorting criterion can be expressed as follows

It is worth noting that the different voltage vectors may lead to the same value of the evaluation function.For example,the candidate voltage vectors contained in setJarev0(000),v1(100),v2(110),v7(1-1-1),v9(11-1),andv19(111).Thus,the values off1associated with the voltages obtained by Eq.(5) and Eq.(23) are [0,Vdc/6,Vdc/3,Vdc/6,Vdc/6,Vdc/2].Among these,the second,the fourth and the sixth are equal.In this case,the ranking value of voltage inr1can be written as [1,2,5,2,2,6],and the optimal voltage vector is obtained by Eq.(25).

4 Simulation results

In order to verify the effectiveness of the proposed strategy,the conventional FCS-MPC method based on weights and the HO-MPC method based on hierarchical optimization are studied through the simulation results.The simulations are carried out in Matlab/Simulink,and the parameters of the two simulations are presented in Tab.1.

Tab.1 Parameters of the system

For the conventional FCS-MPC strategy,the weighting factors of the cost function are chosen by empirical,thus the values areλ1=λ2=1,λ3=0.9,λ4=0.000 5,andλ5=0.06.

In HO-MPC,the satisfactory errors of grid current and the neutral-point voltage are set to 1.2 A and 1.5 V,respectively.In addition,the amplitude of reference current is set to 16 A in both simulations.

Furthermore,the root-mean-square error (RMSE)is used to judge the tracking error of current,neutral-point voltage of the DC-link and common-mode voltage,that is

whereMis the number of sampling points considering the 33 μs duration of the sampling step.

The average switching frequency can be calculated as follows

The percentage error between the grid current and the reference current is calculated as follows

The percentage of tracking error of current exceeding the satisfactory range is defined as

The percentage of neutral-point voltage exceeding the satisfactory range is defined as follows

The simulation results of grid-connected current are shown in Fig.4.It can be seen that,compared with the conventional FCS-MPC scheme,the proposed HO-MPC strategy controls the grid current within a satisfactory range.The maximum error of current is 1.219 A,which is only 1.58% greater than the satisfactory range,whereas that of conventional FCS-MPC strategy is 1.625 A,which is 35.42%greater than the satisfactory range.The maximum value ofipeis 7.62% in HO-MPC strategy,which is 2.54% less than that of the FCS-MPC strategy.Besides,the RMSE between grid current and reference current in HO-MPC is 0.501 A,which is higher than the 0.499 A of the traditional strategy.In addition,the THD of grid current in the conventional FCS-MPC and HO-MPC strategy are 3.52% and 3.84%,which are both controlled under the IEEE standard.

Fig.5 shows the waveforms of neutral-point voltage.Compared with the traditional strategy,the ability to reduce the neutral-point voltage is significantly improved in the HO-MPC strategy.The maximum value of neutral-point voltage in HO-MPC is 1.591 V,which is only 6.07% more than the satisfactory range,while that of the FCS-MPC strategy is 2.249 V,which is 49.93% more than the satisfactory range.Meanwhile,the RMSE of neutral-point voltage in the proposed strategy is 0.911 V,which is lower than 0.984 V that of the traditional strategy.

Fig.4 Simulation results of grid-connected current

The waveforms of common-mode voltage are shown in Fig.6.The maximum common-mode voltages of the two methods are the same.The RMSE of the common-mode voltage in the proposed strategy is 101.7 V,whereas that of the traditional strategy is 102.7 V,which is 1 V higher.

Fig.5 Simulation results of neutral-point voltage for two strategies

Fig.6 Simulation results of common-mode voltage for two strategies

Meanwhile,by using Eq.(27),the average switching frequency can be derived.Thus,the value of the proposed strategy is 5 831 Hz,whereas the average switching frequency of the traditional method is 5 727 Hz,which is lower than that of the proposed strategy.This condition is inevitable,because in a multi-objective control strategy,it is impossible to optimize all control objectives.The results of the comparative simulations are presented in Tab.2.

Tab.2 Results of the comparative simulation

The above analysis shows that the HO-MPC strategy can eclectically optimize the common-mode voltage and switching frequency while satisfying the tracking errors of the current and neutral-point voltage.Thus,the coordinated optimization of multiple objectives is achieved without the need for a complex selection process of weighting factors.

In addition,Fig.7 shows the dynamic simulation results when the neutral-point voltage of the DC-link is deviated at the start to further confirm the effectiveness of the HO-MPC strategy.At this point,the reference current is still set to 16 A,and the satisfaction error of current is still set to 1.2 A.During simulation,the initial voltage deviation of the DC-link is set to ±50 V.

Fig.7 Dynamic responses of the system

As can be seen from Fig.7,the neutral-point voltage is large at the beginning.Owing to the large voltage deviation,the tracking error between the grid current and reference current is distorted,as well as the common-mode voltage.Meanwhile,the average switching frequency increases to 6 636 Hz by Eq.(27).In the role of the proposed HO-MPC strategy,the neutral-point voltage decreases rapidly.After 0.02 s,the grid current,common-mode voltage,and neutral-point voltage are controlled stably,and the average switching frequency is reduced to 5 597 Hz.

5 Conclusions

In this paper,a multi-objective model predictive control method based on hierarchical optimization is presented to solve the complex adjustment process of weighting factors in FCS-MPC.The tracking error of the current and the offset of the neutral-point voltage can be controlled in a satisfactory range,the common-mode voltage and average switching frequency can also be controlled while avoiding the design of weighting factors.The simulation results verify the effectiveness of the proposed strategy.Although the average switching frequency of the proposed strategy is higher than that of the FCS-MPC strategy,the common-mode voltage of the system is decreased.In addition,the maximum tracking error of the current is reduced by 2.54% and the maximum value of the neutral-point voltage exceeding the satisfactory range is reduced by 43.86%.