Global optimization of high-power modular multilevel

Electr Eng DOI 10.1007/s00202-017-0523-5

ORIGINAL PAPER

Global optimization of high-power modular multilevel active-front-end converter using analytical model Amin Zabihinejad1 · Philippe Viarouge1

Received: 7 January 2017 / Accepted: 21 March 2017 © Springer-Verlag Berlin Heidelberg 2017

Abstract Modular multilevel converters (MMCs) have emerged in response to increasing the demands of converter in high-power applications. Because of the modularity, MMC structures are ideal in order to use in high voltage and current applications. Increasing the number of semiconductors and passive components made it so bulky and expensive. On the other hand, high number of variables and the circular interaction between the components values and electrical quantities of the MMCs make it difficult to analyze and design. In this paper, an accurate steady-state analytical model of modular multilevel active front end has been presented and developed. The proposed steady-state model provides more precise analytical expressions for capacitor voltage ripple and circulating current based on the input and output specifications. Then, a novel optimization approach has been proposed and integrated to determine the passive component values in order to maximize the performance and minimize the total volume using analytical model with respect to the technical and mechanical constraints. The optimization procedure uses a nonlinear numerical solver to calculate the optimal values of sub-module capacitor and arm inductance in order to minimize the total energy stored in the converter which is directly related to the total converter mass. Keywords Modular multilevel converter · Analytical model · Steady state · Global optimization · Active front end

List of symbols Csm

B 1

Sub-module capacitor value

Amin Zabihinejad [email protected] LEEPCI Laboratory, Universite Laval, Quebec, Canada

Ia Icirc hn Icu,cl Iu,l L M N m R Sm Sau,al θ1 θ2 ω1

AC line current (phase a) Circulation current nth Harmonic of capacitor current Upper/lower arm current Arm inductance value Mutual inductance value Number of sub-modules per arm Number of parallel arms Inductor resistance Modulation index Upper/lower switching function Phase of AC current Phase of second-order current Main angular frequency

1 Introduction Due to increasing rating of electrical equipment, demands to medium and high-power converters have been accelerated. Modular multilevel converters (MMCs) have emerged in response to demands of high-power converters with lower THD and higher efficiency [10,21,22,28]. MMC was proposed to reduce the switching losses, increase the output power quality and distribute the voltage across the switches [22,28,31]. Modularity and functionality of MMC made it very compatible to a wide range of applications. MMC converters generate high-quality voltage waveform with lower switching frequency which is ideal for many applications such as high-power transmission in HVDC and renewable energy [2,15,20,23], high-power motor drives such as railway traction motor drives [1,26] and high-power accelerators [4] (Fig. 1). In the literature, MMC converters were investigated in different ways. Most of attends have been focused on different

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Fig. 1 The proposed multilevel topology

MMC topologies and their specifications [6,9]. Researchers classified the different MMC topologies based on their applications and nominal power [6,9]. Also, MMC converters provide more switching states in comparison with conventional converters. As a result, various control strategies have been proposed and developed by researchers [9,18,19]. On the other hand, converter losses are known as an important issue in MMC design. Increasing the number of IGBTs leads to increase the semiconductor losses and decrease the converter efficiency. Various MMC models have been proposed in the literature in order to estimate the semiconductor and inductor losses [17,18,24]. These models are known as appropriate tools to analyze the MMC losses and used as analytical tool to calculate the capacitor and inductor values [17,18,24]. An important subject which has been rarely investigated is to propose an integrated model to make a relation between the converter losses and total converter mass. Due to increasing rating of converters, researchers investigated the converter losses and efficiency which affects the final converter volume and price [11,29]. Global optimization algorithm has been proposed to find minimum converter size considering the technical and manufacturing constraints. The increase in the number of modules leads to fast growing of the converter size and price. Due to the limitations of IGBT switches in high-power applications, various optimization procedures have been proposed in order to find the value and size of passive components which provides the best performance [5,17,18,25,30]. The first approach is to put the simulation in the optimization loop. In the case of power electronic converters, it will be complicated and time-consuming

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and sometime impossible to converge. The second approach is to extract and utilize the analytical model. Easy to analyze, access to all important parameters, choosing the targeted initial values and fast convergence are some advantages of using analytic model in optimization procedure [3]. Also, the lower accuracy in comparison with the simulation is the weakness of analytic models [3]. The accuracy of the analytic model is the most important point to success of optimization algorithms. Researchers proposed various steady-state analytic model of MMC [7,12–14,16,21,31]. The circular interaction among the passive components and electrical quantities made it complicated to analyze and optimize. Most of steadystate models are based on average quantities [2,21,31]. These types of models generally neglect the effect of switching frequency. The main difference is to utilize the method which is utilized in order to determine the circulating current, capacitor and arm inductance selection. Also, generally, the effect of coupled inductors which is important to decrease the total volume and losses of the converter is ignored. [32]. In this paper, a precise analytic model of an active-front-end (AFE) converter has been proposed and developed. Utilizing the actual switching function, measuring the precise THD and considering the coupling between arm inductances are the most important advantages of this model in comparison with the previous works. To validate the model accuracy, the outputs are verified using MATLAB/Simulink model. A novel nonlinear constrained optimization algorithm has been presented and integrated to determine the optimal values of passive components and the number of sub-modules per arm in order to achieve the best performance considering the

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of this model is to calculate the precise harmonic value and power quality of the converter. Figure 2 shows the average equivalent circuit of MMC converter in the phase A. The total sub-module voltage is dependent on the modulation index (Sau , Sal ) and total capacitor voltage [12]. ⎧ N  ⎪ i ⎪ Vcu ⎨ Vus = Sau i=1

(1)

N  ⎪ ⎪ ⎩ Vls = Sal Vcli i=1

where Vcu and Vcl are the total capacitor voltage of upper and lower arm, respectively. The state equations of the leg will be easily written by considering KCL. The current of upper and lower branches is the state variables. The state equations are written as below: 

p Iu p Il

Fig. 2 Average equivalent circuit of phase A

technical and manufacturing constraints. Optimization procedure utilizes the proposed analytic steady-state model of MMC. The generalized reduced gradient method (GRG) was employed in optimization [8]. Using multi-start technique, the solver search an area in order to find the global minimum value. Figure 1 shows the high-power converter of Compact Linear Collider at the CERN Company. The Compact Linear Collider (CLIC) study is an international collaboration working on a concept for a machine to collide electrons and positrons (antielectrons) headon at energies up to several tera-electronvolts (TeV). As shown in Fig. 1, a high-power multilevel active-front-end converter is connected to the grid via DY three-phase transformer. The multilevel rectifier supplies a time variable load, while H-bridge converter provides the current of the load which is adjusted using a current controlled controller.

2 State equations of active-front-end converter The three-phase AFE converter consisted of three legs that each leg is made of at least two branches. To suffer the high voltage, several sub-modules are connected in series as well as the coupled inductors. To analyze the MMC converter, it is considered that the capacitor voltages are the same. Therefore, the average model of the branch will be the voltage source in series with a resistor and inductance. Unlike the conventional steady-state model which employs the average quantities, the state equation model calculates the timedomain waveform of each variables. On the other words, the state equation model does not neglect the actual switching function and switching frequency. The major advantage



⎡



  RL Iu − RαM α = Il − RαM RαL     L M Vu − Vus α −α + L Vl − Vls −M α α

(2)

⎤ N  i    p V cu ⎢ i=1 ⎥ Sau 0 Iu ⎢ ⎥= N N ⎣  ⎦ Csm 0 Sal Il p Vcli

(3)

i=1

where α = L 2 − M 2 . There is a circular interaction between the current and capacitor voltages. This makes a little difficult to analyze the steady-state condition [12,33].

3 Steady-state average model of MMC In the case of MMC converters, the steady-state model is important to explain the converter behavior and determine the appropriate components value. There is a circular interaction between the capacitor and inductance value and the electrical quantities of the converter. Therefore, we need to start by estimation of circulating current. The existing literature has proven that the most significant component of the circulating current is the second-order harmonic [27]. Thus, the upper arm current for a three-phase converter is defined as below: ⎧ ⎪ ⎨ Iau (t) = ⎪ ⎩

Ial (t) =

dc Icirc 3

+

Ia 2 cos(ω1 t

+ θ1 ) + Icirc cos(2ω1 + θ2 )

dc Icirc 3

−

Ia 2 cos(ω1 t

+ θ1 ) + Icirc cos(2ω1 + θ2 ) (4)

The upper and lower arm currents consisted of DC source current divided by three, phase current divided by two and the

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circulating current. Generally, the switching function is not a pure sinusoidal waveform. If the number of sub-modules is high enough or the switching frequency is high enough, the harmonic components in the switching function can be ignored. To estimate the magnitude of circulating current, it will be considered that the switching function is sinusoidal. Therefore, the switching function for the first leg (phase A) will be written as below: 

Sau = 21 (1 − SM cos ω1 t ) Sal = 21 (1 + SM cos ω1 t )

(5)

Considering to the sub-module configuration, the current passes through the capacitor when the lower switch is ON and the upper is off. The capacitor current waveform is calculated as below: 

Icu (t) = Sau (t)Iau (t) Icl (t) = Sal (t)Ial (t)

(6)

The sub-module capacitor current waveform consisted of DC component, main frequency component, and second-order and third-order harmonics. 1 1 Idc − SM Ia cosω1 (7) 6 8 1 Ia h1 (t) = − SM Idc cos(ω1 t) + cos (ω1 t + θ1 ) Icu 6 4 Icirc − SM cos (ω1 t + θ2 ) (8) 2 Icirc Ia h2 cos (2ω1 t + θ2 ) − SM cos (2ω1 t + θ1 ) (9) (t) = Icu 2 8 Icirc h3 (10) Icu (t) = − SM cos (3ω1 t + θ2 ) 4 dc (t) = Icldc (t) = Icu

To calculate the capacitor voltage, it is sufficient to multiply the capacitor current by the capacitor impedance. By eliminating the DC section of the result, the capacitor voltage ripple is determined. The capacitor voltage ripple consisted of main frequency, second-order and third-order harmonics. By adding three terms of ripple, the total capacitor voltage ripple is calculated for each sub-module. The importance of capacitor voltage ripple comes from the limitation of semiconductor rating. Finding the maximum capacitor voltage is a criterion to select the semiconductor switch. On the other hand, to find the precise losses function, the capacitor voltage ripple plays an important role [12,33]. h1 (t) = Vcu

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−SM Idc sin(ω1 t) 6ω1 CSM Ia + sin(ω1 t + θ1 ) 4ω1 CSM SM Icirc − sin(ω1 t + θ2 ) 2ω1 CSM

Icirc sin(2ω1 t + θ2 ) 4ω1 CSM Ia SM − sin(2ω1 t + θ1 ) 16ω1 CSM −Icirc SM h3 Vcu (t) = sin(3ω1 t + θ2 ) 12ω1 CSM

h2 (t) = Vcu

(13)

The phase voltage ripple is the summation of all submodule capacitors voltage ripple multiplied by the switching function. The phase voltage ripple is composed of the pair harmonics and the odd harmonics were eliminated. The output voltage ripple consisted of DC component, second-order and fourth-order harmonics [24].    N N h1 h2 h3 − SM cos ω1 t Vcu + Vcu + Vcu Va = 2 2    N N Vclh1 +Vclh2 +Vclh3 + + SM cos ω1 t 2 2 (14) The voltage ripple function is a valuable analytic function which is important to estimate the grid-side THD value. This function is used to estimate the THD constraint in the optimization procedure, or it could be an appropriate criterion to design the external filter. 2 I N SM N SM Ia circ sin θ1 + sin θ2 8ω1 Csm 4ω1 Csm 2 I N SM dc = sin(2ω1 t) 12ω1 Csm 3N SM Ia − sin (2ω1 t + θ1 ) 16ω1 Csm   2 N Icirc 1 + 76 SM + sin (2ω1 t + θ2 ) 4ω1 Csm 2 I N SM circ = sin (4ω1 t + θ2 ) 24ω1 Csm

Vadc = − Vah2

Vah4

(15)

(16) (17)

It is noted that the circulating current consisted of secondorder harmonic. Thus to estimate the I circ , the second-order harmonic of the voltage is utilized: Icirc cos (2ω1 t + θ2 ) =

Vah2 − j4ω1 (L + M)

(18)

Thus, the value of Icirc and θ2 are calculated as below:     2 N 1 + 76 SM Icirc cos (2ω1 t + θ2 ) 1 − 16ω12 (L + M)Csm =

(11)

(12)

2 I N SM dc

cos(2ω1 t) + M)Csm 3N SM Ia − cos (2ω1 t + θ1 ) 64ω12 (L + M)Csm 96ω12 (L

(19)

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Fig. 3 The flowchart of proposed analytic model

Icirc =

 2 (B cos ω1 +C)2 +(B sin ω1 ) 1−A

< θ2 = tan−1

B sin ω1 B cos ω1 +C

2 ) N (1+ 67 SM 16ω12 (L+M)Csm 2 I N SM dc 96ω12 (L+M)Csm

where A = C=



Fig. 4 Circuit diagram of half-bridge sub-module

(20)



,B=

−3N SM Ia , 64ω12 (L+M)Csm

4 Precise analytic model using average model To calculate of steady-state average model, average switching function is utilized. It leads to eliminate the effect of switching frequency. Generally, high-power MMC converters utilize low switching frequency. Therefore, the results of average model are different from the reality. To resolve this issue, a novel analytical model is proposed. The new analytical model utilizes the SPWM switching function which is generated by comparing the sinusoidal and triangular waveforms. Figure 3 shows the flowchart of proposed analytical model for the MMC converter. The analytic model takes the value of passive components and the input/output specifications of the converter in order to estimate the AC line current which consisted of dc current, main frequency and circulating current. The analytical model calculates all sub-module capacitor currents and voltages. Then, the summation of sub-module terminal voltages is calculated to determine the upper and lower arm voltage sources. In the final step, the current waveform in the upper and lower arm considers the inductance matrix. It is possible to calculate all other parameters such as line current, THD, switch losses from the previous parameters.

5 Steady-state losses analysis The important part of losses in MMC structures is the switch losses. Figure 4 shows the circuit diagram of a halfbridge sub-module. The capacitor current passes through S1 , whereas another switch (S2 ) passes the rest of upper branch current.  Ps1 =

2 Vd0 Icu,avg + Rd0 Icu,rms 2 VT 0 Icu,avg + RT 0 Icu,rms

Icu > 0 Icu < 0

(21)

Also, the losses function of S2 will be as below:  Ps2 =

2 Vd0 Is2,avg + Rd0 Is2,rms 2 VT 0 Is2,avg + RT 0 Is2,rms

Icu > 0 Icu < 0

(22)

Considering to KCL, the current which is passed S2 is: Is2 = (1 − Sau ) Iu = Sal Iu

(23)

To estimate the total sub-module losses, it is sufficient to multiply (Ps1 + Ps2 ) by 6N . The last part of MMC losses is the inductor losses. Inductor losses consisted of copper losses and core losses. From Fig. 2, it is clear that Iu passes through the inductor. Thus, the copper losses will be estimated as below:

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Electr Eng Table 1 Load specifications of MMC converter Parameter

Value

Nominal power

2.5 MW

AC line voltage

2000 V

DC link voltage

5000 V

Power factor

1

Nominal frequency

50 Hz

Number of sub-module per arm

6

Sub-module capacitor

11.1 mF

Arm inductance

4.22 mH

Switching frequency

755 Hz

(a) 2 Pl,cu = Rl,ac Iu,rms

(24)

To achieve a precise estimation about core losses, we have to consider about the harmonics of the current. The inductor current consisted of the main frequency and the second-order harmonic. Therefore, Pl,core = K 1 f 1m (Bh1 )n + K 2 f 2m (Bh2 )n

(25)

(b)

6 Analytical model verification using Simulink It is necessary to evaluate the precision of the proposed model before utilizing in optimization algorithm. In this research, the results of analytical model are verified using MATLAB/Simulink. The analytical model was implemented by Excel macro platform in visual basic. To compare the results, a case study has been defined to make an equal condition in two simulation approaches. Table 1 shows the case study parameters. Figures 5 and 6 show the comparison among the waveforms of capacitor current and voltage which are the results of Simulink and Excel in steady-state condition. Figure 7 shows the arm currents in the upper and lower side. The accuracy of arm current is important, because there are many parameters which are dependent on the arm current. Comparing the output of Simulink and analytical model proves the preciseness of the model. Figure 8 shows the AC line current and its THD value. The current waveform of the analytical model is very close to the Simulink result. Also, the THD measurement has done using two different approaches. The result confirms the accuracy of the model.

Fig. 5 Capacitor current waveform, Simulink and analytic model. a Simulink. b Analytic model

In this issue, the goal function is to minimize the total converter mass with respect to the technical constraints. The optimization variables are sub-module capacitor value, arm inductance value, modulation index and switching frequency. The nonlinear constrained optimization algorithm which used the GRG Nonlinear Solving method is employed to optimize the converter components. The flowchart of proposed optimization algorithm is shown in Fig. 9. The main constraints are the capacitor voltage ripple Vc , DC link voltage value and AC line current THD, total converter efficiency and the maximum switch losses. The goal function is the total energy stored in the converter that is the summation of electric energy of capacitors and the magnetic energy of inductors. mN    3 2 2 Csm Vcu + V cli i 2 i=1   1 2 1 2 =3 L Iu + L Il + M Iu Il 2 2 = E cap + E ind

E cap =

(26)

E ind

(27)

7 Global optimization and result discussion

E total

Each optimization procedure consisted of optimization variables, mathematical model, constraints and goal function.

In this optimization, the THD must be less than 2 percent and capacitor voltage ripple must be less than 20 percent of

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(28)

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(a)

(a)

(b)

(b)

Fig. 6 Capacitor voltage waveform, Simulink and analytic model. a Simulink. b Analytic model

Fig. 8 AC line current and THD, Simulink and analytic model. a Simulink. b Analytic model

(a)

Fig. 9 Proposed global optimization algorithm

(b) Fig. 7 Upper and lower arm currents, Simulink and analytic model. a Simulink. b Analytic model

dc value. Also, the number of sub-module is between 3 and 8 and the number of parallel arm is between 1 and 3. In order to calculate the size of converter, we need to introduce an appropriate criterion which demonstrate the converter size. The sub-module capacitors and arm inductance are bulky components. Therefore, The energy stored in

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Electr Eng Table 2 Operating point and constraints of optimization Parameter

Value

Nominal power

2.5 MW

DC link voltage

5000 V

Power factor

1

Sub-module voltage ripple

0.2*Vdc/m

Line current THD

95%

Fig. 12 Total converter efficiency

Fig. 10 Capacitor energy stored versus number of module

Fig. 13 Capacitor voltage ripple versus capacitor energy stored and number of modules

Fig. 11 Inductor energy stored versus number of module

the capacitors and inductor is the best parameter that present the size of the converter. The summation of energy stored in the capacitors and the inductor is known total energy stored in the converter which is a criterion to the converter size. Table 2 shows the specification of the operating point that optimization procedure tries to find the minimum converter size in this point. Figures 10, 11 and 12 show the optimization results. The optimization procedure was repeated for each number of module and parallel arm. The optimized capacitor energy stored is shown in Fig. 10. The capacitor energy stored is the representative of the total capacitor size which should be installed. The result shows that the total capac-

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itor size is almost constant and is not dependent on the number of sub-modules and parallel arms. It seems that the capacitor size is dependent on the voltage ripple. Figure 11 shows the magnetic energy stored in the inductor which represents the total size of the inductor. The magnetic energy stored is decreased by increasing the number of modules. Figure 12 shows the total converter efficiency. The total converter efficiency is reduced by increasing the number of sub-modules and parallel arm. By increasing the number of modules, optimization procedure decreases the switching frequency to reduce the losses and keep the efficiency bigger that 95%. Therefore, by increasing the number of submodules per arm, it is possible to reduce the total converter size with respect to the efficiency and THD constraints. Figure 13 presents a better recognition of the effect of capacitor voltage variation on the capacitor voltage ripple. The contour of capacitor voltage ripple versus capacitor energy stored and number of sub-module is shown in Fig. 13, while the arm inductance is 2.3 mH, and switching frequency is 1000 Hz. It shows that the rate of ripple reduction is decreased by increasing the capacitor energy stored. It means that to achieve lower ripple, more capacitor volume is needed.

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Fig. 14 THD and total efficiency versus coupling factor

Also, it is possible to achieve lower ripple by increasing the number of sub-modules. Figure 14 shows the effect of coupling factor among the upper and lower arm inductances on the line current THD and total converter efficiency. It shows that the negative coupling factor enhances the converter efficiency while reduces the power quality. On the other side, the positive coupling factor leads to lower THD and efficiency.

8 Conclusion In this paper, a novel and precise analytic steady-state model of high-power MMC converter has been proposed and developed. This model proposes the analytical expressions of converter electrical quantities to represent the behavior of arm current and sub-module capacitor voltage in the steadystate condition. Due to increasing the number of components in MMC converter, the analytical model is utilized to find the optimal value of sub-module capacitor and arm inductance. The optimal selection of MMC components leads to minimize the total converter mass and price. To verify the proposed analytic model, the load specification of an AFE converter which is used in the high-power accelerator has simulated using MATLAB/Simulink and compared to the analytic model outputs. Finally, using the proposed model, an integrated optimization plan was proposed to minimize the capacitor and inductor energy stored which represents the capacitor and inductor mass. The nonlinear constrained optimization method was employed to optimal selection of converter components. The optimization procedure was repeated for each number of sub-modules and parallel arm. Finally, using the proposed model, the effect of increasing the capacitor volume and coupling factor among the upper and lower arms has been investigated.

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