Research on modular multi-level converter control based on passivity theory

All In view of the poor dynamic performance of the conventional PI control method in modular multilevel converter (MMC) systems, an up-to-date control strategy using passive theory is proposed. First, the topology of the system and its mathematical model are presented, then its Euler-Lagrange (EL) model is developed and its passivity is proved. Next, the dq-axis component of the current is chosen as the state variable to derive the passivity-based control (PBC) law and from this, the current passivity controller is designed. Simulation shows that compared with conventional PI control, the PBC in this paper can respond quickly to the change of current or power under different operating conditions, with good harmonic characteristics and better dynamic characteristics.


Introduction
Areas rich in clean energy, such as wind and photovoltaic, are usually far from electrical load centers, requiring the construction of high-capacity, inter-regional power transmission networks.Compared to AC transmission, high-voltage direct current (HVDC) transmission is characterized by flexible regulation, reactive power angle stability issues, and low power loss [1], which makes it an ideal transmission method for achieving green power transmission from clean energy bases to electrical load centers [2].Modular multilevel converters (MMC) are attracting more and more attention in the field of HVDC transmission due to their good technical performance, such as high voltage level, low harmonic content, and ease of expansion [3][4].
An appropriate control method can greatly affect the performance of the MMC.Typical MMC control adopts a double-closed-loop structure based on PI controllers, which utilizes the good tracking ability of PI controllers for direct flow to achieve the control of MMC, but it is difficult for PI controllers to achieve static-free regulation due to the influence of the circulating flow in MMC [5][6].Proportional resonance (PR) control has better suppression of frequency-specific harmonics and can directly control AC objects, but the PR controller is more complex to design [7].Both PI and PR are essentially linear controllers whose parameter selection is based on small-signal stability analysis, which makes it difficult to achieve satisfactory dynamic performance, and the complex control structure increases the difficulty of parameter tuning for linear controllers [8][9].Some researchers have studied the application of model predictive control (MPC) to MMC, where multi-objective control can be achieved by designing an appropriate value function, but the value function needs to be calculated in each adoption cycle, and optimization is too computationally intensive [10].
The essence of passivity-based control (PBC) is to establish the passivity of the controlled system by designing passivity controllers to make the closed-loop system meet the passivity conditions and achieve the closed-loop system asymptotic stabilizing near the equilibrium point [11], which is widely used in the field of electrical energy conversion [12].In [13], a robust PBC method for multi-terminal HVDC systems was presented based on voltage source converters, which improved the robustness of the system against power fluctuations.In [14], a passivity controller was designed to improve the dynamic characteristics of the VSC-HVDC by optimizing the interconnection and damping allocation scheme.In [15] and [16], passivity theory was applied to the control of DC-DC conversion and PV grid connection respectively.
This paper investigates the PBC problem of the MMC system.First, the MMC topology and its mathematical model are introduced, then the passivity of the system is discussed in terms of system energy dissipation, and the Euler-Lagrange (EL) function of the system is set up.Next, the passivity controller for the MMC is structured based on the energy function.Finally, based on the comparative analysis between the designed PBC and the conventional PI control, the simulation verifies that the MMC based on PBC has greater dynamic performance.

MMC main circuit topology
The MMC topology in this paper can be represented in Figure 1, in which the MMC comprises six bridge arms, each containing a number of sub-systems, a resistor, as well as an inductor.Every subsystem contains a half-bridge inverter unit connected in parallel with a capacitor.The sub-system has three states: latching state, input state, and removal state.If both V1 and V2 are off, the sub-system is latching; if V1 is conducting and V2 is off, the sub-system is in the connected, and capacitor C is charging or discharging; if V2 is conducting and V1 is off, the sub-system is in the disconnected.The insertion or removal of every sub-system is achieved through different combinations of the switching devices, thus outputting a multi-level modulated wave of three-phase voltage on the MMC AC side.

MMC mathematical modeling
Corresponding to the MMC topology represented in Figure 1, the MMC single-phase t circuit is shown in Figure 2. In Figure 2, o is the AC voltage reference point; m is the DC voltage reference point; u pj , as well as u nj , are the output voltages of the upper and lower bridge arm sub-systems; L s is the AC-side inductance; L p and L n are the series inductances of the bridge arms; R p and R n are the equivalent resistances of the upper and lower bridge arms; i pj and i nj are the currents flowing through the bridge arms; U dc is the DC system voltage; u sj is the AC voltages, and i j is the AC currents.
To simplify the discussion, it is assumed that the inductance and resistance of the upper and lower bridge arms are equal, i.e.L p =L n = L 0 and R p =R n = R 0 .Then Equation ( 1) can be rewritten as Equation (2).

EL model for MMC
The EL model for the MMC is the basis for proving system passivity and designing passive controllers, and the following is an example of deriving the EL model for the MMC inverter.The mathematical model for the MMC expressed with Equation ( 3) is rewritten as Equation ( 4).
Rewriting Equation (4) in the dq-axis coordinate system and taking into account that the u gotransformed fundamental wave component is zero, Equation (4) in the dq-axis coordinate system can be expressed with Equation (5), where i d is the d-axis output current, and i q is the q-axis output current; ω is the angular frequency of the base wave; u sd is the d-axis input voltage, and u sq is the q-axis input voltage; u diffd and u diffq are the differential mode voltages of the MMC in dq-axis coordinate system.
By selecting X = (x 1 x 2 ) T =(i d i q ) T as the state variables, Equation ( 5) can be rewritten as Equation ( 6), which is the EL model of the MMC inverter.In Equation ( 6), M = [L 0; 0 L] is the positive definite diagonal matrix; J = [0 -ωL; ωL 0] mirrors the system's inner connection structure; K=[R 0; 0 R] mirrors the system dissipation characteristics; U = [-u sd -u diffd ;-u sq -u diffq ] is the energy exchange matrix;

MMC system passivity
According to the PBC theory [17], a multi-input-output system is strictly passive if it satisfies the dissipation inequality shown in Equation (7), where H(x) is the system energy storage equation, Q(x) is a positive definite function, and u and y are the input and output of the system, respectively.
We let the MMC system storage energy equation be expressed as Equation ( 8).
We let y X = and T ( ) in Equation ( 10), then Equation ( 10) can be written as Equation (11).(11) and Equation (7) evidences that the MMC system is passive.

MMC passivity controller design
Assuming that the desired system equilibrium point is , where * d i and * q i are the reference values of the state variables, d i and q i , the error of the state variables can be shown as Equation (12).

U -(MX + JX + KX ) = MX + JX + KX
The energy error function of the system is T e e 0.5 e H

X MX =
, and its derivation leads to Equation (14).
) For the system to recover quickly to the desired equilibrium point after a perturbation, the energy error should converge to zero as fast as possible, and for this purpose, the energy dissipation is accelerated by injecting damping.The expression for the dissipation term after injection of damping is given by Equation.(16), where Combining Equations ( 15) and ( 16) leads to Equation (17).
(17) When the system has no steady-state error, the right term of Equation ( 17) is equal to zero.Thus, the PBC law of the system is obtained, as shown in Equation ( 18 From Equation ( 19), it is known that e 0 H <  , the PBC law shown in Equation ( 18) can effectively speed up the convergence of the error energy function.
Since the given expectation * X is constant and its reciprocal is zero, Equation ( 18) can be rewritten as follows: Using Equation ( 20), u diffd and u diffq can be expressed as shown in Equation ( 21).
From Equation (21), the injection of damping R h , and voltage coupling quantities ωLi d and ωLi q linearizes the state equations of the system, thus achieving decoupled control of the system current.The structure of the PBC used in the MMC system is shown in Figure 3.
The power outer loop offers the reference current for the current inner loop, and the commonly used PI controller is still used in this paper.The control frame diagram of the MMC inverter is illustrated in Figure 4, where P * and Q * are the active and reactive power setpoints, respectively, and k p and k i are the outer loop PI controller parameters.From Figure 4, the calculated power real-time value makes a difference to the power reference value, outputs the dq-axis current reference value after passing through the power outer-loop PI controller, then outputs the voltage reference value after passing through the current inner-loop passivity controller, and finally obtains the PWM control signals of every sub-system after the inverse Park transformation.

Simulation experiment
To check the feasibility of the PBC shown in Figure 4, the MMC inverter simulation system, in which the amount of sub-systems is 22, is established based on MATLAB (it is assumed that the power factor is 1), and the parameters of the simulation system are given in Table 1.Keeping the other modules of the system unchanged, this paper compares the current harmonic content and voltage waveforms of the grid side of the inverter and the power fluctuation under various scenarios when the MMC control is used with the PBC and conventional PI control respectively.It is supposed that the following two types of disturbances occur in the system: 1) the given active power jumps from 2.4 MW to 3 MW at 0.6 seconds; 2) the AC system voltage drops by 0.2 p. u. in 0.8 seconds with a duration of 0.05 seconds.Table 2 gives the harmonic percentage of the inverter gridside current under different operating conditions.As can be seen from Table 2, the harmonic content of the PBC is marginally lower than that of the common PI control, but overall the harmonic content values of the two control methods are very close to each other.Figure 5 and Figure 6 show the inverter grid-side voltage waveforms under different types of disturbances, and the corresponding inverter grid-side power is shown in Figure 7.In Figures 5, 6, and 7, the left graphic represents the simulation results using the PBC controller, and the right graphic represents the simulation results using the PI controller.According to Figure 5, we can observe that when the active power is given at 0.6 seconds to increase to 3 MW, the inverter grid-side voltage waveform remains as a step wave close to sinusoidal in both control modes, while the inverter grid-side current increases and its harmonic content slightly increases.As we can observe from Figure 6 when the AC system voltage drops during 0.2 p. u., the harmonic content of the inverter grid-side current is 8.68% and 8.93% under the two control modes, therefore, there is basically no difference in the current waveforms; while the harmonic content of the inverter grid-side voltage is 7.21% under the PBC and 17. 45% under the PI control, it is obvious that the PBC is much better than the PI control, which can also be proved by using the voltage waveform shown in Figure 6.IOP Publishing doi:10.1088/1742-6596/2703/1/0120828 disturbance type 2; the reactive power fluctuations of the two control modes do not differ much when the power factor is unity.From the above analysis, it can be seen that the PBC is more effective in suppressing system voltage harmonics and has more advantages over PI control.

Conclusions
This paper develops a PBC for the MMC system in order to improve the dynamic characteristics against system disturbances.The following main conclusions are obtained through theoretical calculations and simulation analyses: 1) Compared to conventional PI control, the PBC has the positives of simple structure, quick dynamic response, and convenient adjustment of control parameters, which has a promising application prospect in MMC.
2) When the system is in normal operation or when the power setpoint changes, the waveform and harmonic content of the inverter grid-side current are close to each other when PBC or PI control is used.When the grid voltage drops, the PBC has a smaller inverter grid-side voltage distortion rate compared to the PI control, and the active power fluctuation frequency and amplitude are also smaller.Therefore, the PBC has better anti-interference capability, which is conducive to the dynamic stability of the MMC system.

2 UFigure 2 .
Figure 2. MMC single-phase equivalent circuit.According to the reference direction shown in Figure2, using Kirchhoff's law, the single-phase voltage-current equations can be derived as shown in Equation (1).dc pj p pj p pj s j sj om dc nj n nj n nj s j sj om pj nj j can be expressed as Equation (3), and Equation (3) is the MMC mathematical model (j=a,b,c).

Figure 4 .
Figure 4. Frame diagram of dual closed-loop control of MMC inverter.

Figure 5 .
Figure 5. Current-voltage at the grid side of the inverter under disturbance type 1.

Figure 6 .
Figure 6.Current voltage at the grid side of the inverter under disturbance type 2.

Figure 7 .
Figure 7. Power at the grid side of the inverter under disturbed conditions.From Figure7we can observe that the active power fluctuations under the couple of control modes are close to each other during normal operation of the system and disturbance type 1, while the power fluctuation frequency and amplitude under the PBC is much smaller than under the PI control during ).

Table 2 .
Harmonic Content of Inverter Grid-side Current.