Steady‐state power operation region of a modular multilevel converter connecting to an AC grid

State Grid Corporation of China Science and Technology Research Project, Grant/Award Number: 5105‐202018129A‐0‐0‐00; State Grid Corporation of China, Grant/Award Number: Unassigned China; China Scholarship Council, Grant/Award Number: Unassigned China Abstract The modular multilevel converter (MMC) has become a widely used topology for voltage source converter–based high‐voltage direct current transmission projects. To calculate the power operation region of an MMC directly and efficiently, we propose a novel steady‐state phasor model of the MMC. The model is expressed as an equivalent capacitance in series with a voltage source, where parameters are independent of alternating current (AC) electrical quantity. Based on the model, we propose an open‐ loop approach to determine the operation region boundary of MMC by calculating the envelopes of the P–Q curves. The effect of MMC parameters, the circulating current control, and the short‐circuit ratio of the AC grid to the power operation region can be determined by the size and position of the power operation region. The proposed approach and analysis results were verified by an MMC/AC simulation model built in PSCAD/EMTDC.


| INTRODUCTION
High-voltage direct current (HVDC) transmission systems and technologies continue to advance as they make their way to commercial applications [1][2][3]. It was based initially on thyristor technology and, more recently, on fully controlled semiconductors and voltage-source converter (VSC) topologies [3]. The modular multilevel converter (MMC) is suitable for high-voltage and large-capacity power transmission due to its modular structure and has become a widely used topology for VSC-HVDC transmission projects [4][5][6]. The MMC and the two-level VSC have apparent differences in steady-state operating characteristics, where all the electrical quantities and control variables are coupled in an MMC due to the effects of capacitor voltage ripples in each submodule (SM) [7,8].
Studying the operation region of an MMC plays an essential role in the parameter design of an MMC-HVDC system. The VSC's active and reactive power output is determined by its virtual electromotive force (EMF). Reference [9] determined the operation region of the MMC by borrowing from the two-level VSC model and mapping the MMC's power outputs to its EMF. This method can determine the boundary of the operation region. However, there is a specific deviation in the calculation result without considering the difference between MMC and two-level VSC.So far, most current methods to determine the operation region of MMC are 'seeking-and-checking' or 'point-scanning' analysis methods instead of determining the boundary of the operation region. In [10], the constraints of the voltage stability, the voltage, and the current are studied. Reference [11] examined the limitations of internal dynamics. In [12], researchers studied the operation region of an MMC under unbalanced power grid conditions. The influence of circulating current control on the power operation region is studied in [13,14]; however, the methods to determine the operation region are proposed for the circulating current suppression and injection controls, not universal. Hence, a general approach to determine the operating region boundary of the MMC directly is still needed.
The key to accurately determine the operation region boundary of MMC is to establish a steady-state MMC with an open-loop solution method. Much research has been conducted on the steady-state model of MMC. A steadystate time-domain analysis model of an MMC was proposed in [15], ignoring the circulating current and its suppression control. Reference [16] proposed a simplified phasor model of the MMC, ignoring the circulating current. The accurate models presented in [8,17] can solve the circulating current, where the accuracy is relatively high. In [8], a closed-loop time-domain analysis model of MMC was established by defining an accurate switching function. The authors of [17] proposed a general MMC steady-state analysis model for MMC's optimization considering various circulating current control methods [13,[18][19][20][21][22]; that is, the circulating current injection control, the third-harmonic signal injection control, and the average capacitor voltage control. The existing steady-state models can accurately describe the coupling characteristics of an MMC. Still, an iterative solution had to be adopted because they failed to decouple the control variables and electrical quantities. There is a problem in and around the critical point that the iteration may not converge.
In this paper, we study the open-loop method to determine the power operation region boundary of an MMC. We first proposed a novel steady-state phasor model of MMC by deriving the function relation between the output voltage and current of MMC. Based on the model, determining the steady-state power operation region is transformed to solve the envelopes of a family of P-Q curves. The rest of this paper is organised as follows: In Section 2, the time-domain model of MMC is described to state the principles and assumptions of the MMC steady-state model. In Section 3, the complete form of the novel steady-state phasor MMC model is proposed, and its derivation is presented in detail. Based on the proposed model, a method to determine the steady-state power operation region of an MMC is proposed in Section 4. Section 5 describes the simulation validation in PSCAD/ EMTDC that was done to verify the research in this study. In Section 6, the conclusions are summarised.

| Equivalent circuit and mathematical model of MMC
The single-phase equivalent circuit of an MMC [8] is shown in Figure 1. In each arm, there are N series-connected SMs and an arm inductor L 0 . The arm voltages across the N seriesconnected SMs in the upper and lower arms of phase i are u pi and u ni , respectively, and the arm currents are i pi and i ni , respectively. The MMC is connected to the alternating current (AC) grid at the point of common coupling (PCC) through a three-phase converter, where L T is the leakage reactance, u ti and u vi are the single-phase AC voltage on the grid side and valve side, respectively, and i ti and i vi are the single-phase AC currents on the grid side and valve side, respectively. Figure 1b shows the equivalent model of an MMC from the AC side, including u ci , the inner EMF of phase i, which is also called the output voltage of the MMC, and i ci , which is the output current of the MMC. Figure 1c shows the equivalent model of the MMC from the direct current (DC) side, where U dc is the DC line voltage. u diff and u com , defined by Equation (1), are the differential-mode and common-mode components of the arm voltages, respectively. The definitions of i diff and i com , the differential-mode and common-mode components of arm current, are shown by Equation (2).
where u L0 is the voltage across the arm inductor.
where i dc is the DC current and i ciri is the circulating current. Assuming capacitor voltages are well balanced, and the switching frequency is very high, all SMs in one arm can be considered identical [8]. Therefore, all capacitors in the N series SMs of one arm can be equivalent to an equivalent arm capacitor C eq , where C eq = C 0 /N, and C 0 is the SM capacitance. The current flow across the equivalent arm capacitor is produced by the corresponding switching function and arm current. By integrating it, the corresponding voltage of the equivalent arm capacitor is obtained. The upper and lower arm voltages are produced by multiplying the corresponding equivalent arm capacitor's voltage with the switching function. The mathematical expression can be written as Equation (3).

| Steady-state equivalent and coordinate transformation
In steady-state, ignoring the third-and higher-order components, the arm current of the upper and lower arms of phase i can be written as Equation (4), and the accurate switching functions [8] of the upper and lower arms can be defined as Equation (5). In [8], by substituting Equations (4) and (5) into Equations (1)-(3), the steady-state time-domain model of an MMC was obtained.
where I diff0 is the DC component of the differential-mode arm current, I com and φ comi are the magnitude and angle of the fundamental component of the common-mode arm current of phase i, and I diff2 and φ diff2i are the magnitude and angle of the second-order component of the differential-mode arm current.

> > > <
> > > : where M dc is the DC modulation index; M e and θ ei are the magnitude and the phase angle of the fundamental modulation components, which come from the inner-loop current controller's outputs; and M 2 and θ 2i are the magnitude and the phase angle of the second-order modulation components, which come from the circulating current controller's outputs. The Park transformation matrix from the abc reference frame to the dq0 rotating frame is defined as Equation (6), where the superscript + represents the positive sequence, and the superscript -represents the negative sequence.
Thus, Equations (4) and (5) can be rewritten in the form of d and q components, as shown by Equations (7) and (8), respectively. The subscripts d and q are the corresponding d and q components, respectively.
where θ+ a = 0, currents; M ed and M eq are the dand q-fundamental modulation components; and M 2d and M 2q are the dand q-secondorder modulation components. In a three-phase symmetrical condition, there are a total of five control components: M dc , M ed , M eq , M 2d , and M 2q , which controls the five arm current components, including I diff0 , I comd , I comq , I diff2d , and I diff2q . Nevertheless, the control process is not decoupled due to the effects of capacitor voltage ripples in each submodule [7,8].

| Derivation of MMC steady-state phasor model
To derive the function relationship between the MMC's output voltage U c and the output current I v in phasor form, let us define the current vector as Define the switching function transformation matrix S M , the equivalent capacitance voltage transformation matrix S Ucap , and the voltage output transformation matrix S UMMC , as shown by Equation (9). Thus, according to the symmetry of the upper and lower bridge arms, Equation (3) can be rewritten in the matrix form as shown by Equation (10).
8 > > > < > > > : Note that the DC component of capacitance current equals 0 in the steady state, which is expressed as Equation (11): According to Equation (1), the equivalent of the output voltage vector U MMC is obtained as Substituting Equation (10) into Equations (11) and (12) and combining them, the steady-state operation equation of an MMC can be obtained as Equation (13). The expressions of the matrices in Equation (13) Thus, the functional relationship between the AC voltage and current outputs of an MMC can be derived, as Equation (14) shows. It is the steady-state phasor model of MMC at the AC side.
The impedance matrix Z MMC can be written in the form of ; which means that the MMC from the AC side can be described as the equivalent model with the reactance X MMC and the voltage source E c in series. Let us take X L0 = ωL 0 and X Ceq = 1/(ωC eq ), and normalise them. The rated MMC DC voltage U dcN and the rated DC power P N are selected as the base values of the DC voltage and power, respectively. The selection method of U vN , the rated voltage on the valve side of the converter transformer, is shown by Equation (15), and it is chosen as the base value of the MMC's AC output voltage.
where m k is the no-load transformation ratio of the converter transformer; usually 0.85. Thus, the parameters of the equivalent model in a per-unit system will be obtained, as shown in Equation (16).
where 8 > > > > > > > > > > < > > > > > > > > > > : : It should be noticed from Equation (14) that even when the DC voltage is specified, the EMF of the MMC U c cannot be completely determined by the controller outputs, but is related to the AC currents. That is why [8] was required to adopt a closed-loop analysis method to calculate the unknown variables of an MMC.

| Parameters analysis of MMC phasor model
In the MMC-HVDC system of half-bridge SMs, the DC modulation index M dc is generally constant at 1. In a full-bridge or a hybrid system, M dc is approximately equal to 1 [23][24][25][26].
Equation (17) shows the parameter selection methods of the SM capacitor and the arm inductor [27]: where U capN is the rated capacitor voltage, ε cap is the designed ripple percentage of capacitor voltages, I km is the designed peak value of the circulating circulation, ω 0 is the fundamental angular frequency, and cosφ is the power factor, taken as 1.
If we define ε Ikm = I km /(√2I vN ), which is the designed percentage of the circulating circulation peak value, there is the function relation shown by Equation (18) between X L0 and X Ceq : Ignoring the second-order modulation components of the switching function, because its index M 2 is usually small, the equivalent reactance X MMC can be simplified to Equation (19). It illustrates that X MMC < 0 when the fundamental modulation index M e varies between 0 and 1, which means the reactance in the MMC steady-state model is capacitive.

| Model of an MMC connecting to an AC grid
The equivalent model of an MMC connecting to an AC grid is shown in Figure 2. The MMC is described by an equivalent capacitor X MMC in series with the equivalent voltage source E c , as shown inside the blue box. The AC grid is equivalent to a Thevenin model of the equivalent voltage source E s in series with the equivalent resistance Z s ∠ θ s . The value of Z s equals the reciprocal of the AC grid's short-circuit ratio (SCR). X T and k T are the reactance and the tap of the converter transformer, respectively. U t and I t are the voltage phasor and WANG ET AL.
-5 current phasor on the transformer's grid side, and U v and I v are the voltage phasor and current phasor on the transformer's valve side, respectively. P and Q are the active and reactive power outputs of MMC to the grid. After obtaining the X MMC and E c by Equation (16), we can solve the AC current of MMC as follows: where According to Figure 2, the control process of an MMC-HVDC system can be described as follows: The controller controls the output voltage and current by changing the d and q components of the equivalent voltage source E c and the value of the equivalent capacitor X MMC , to track the control target of output power. The explicit expressions of P and Q to the control components M dc , M ed , M eq , M 2d , and M 2q can be obtained by substituting Equations (16) and (20) into Equation (22).

> > > > > > > > <
> > > > > > > > : Thus, the open-loop analysis method can still be adopted to analyse the effect of control variables on the power outputs, by directly mapping the constraint condition of the controller outputs to the constraint on the output power.

| Operation region boundary of converter constraint
Due to MMC's topology, the number of SMs limits the available range of the arm voltage, which is the most dominating constraint on the power outputs of an MMC. It is impossible to achieve the operating points beyond this limit. Therefore, the number of SMs required to be in the 'ON' state should not be more than the total number of SMs in a bridge arm, as shown in Equation (23).
Thus, M e should satisfy the constraints shown in Equation (24), while considering the component for circulating current control. If the envelope exists, the points on the envelope follow that: The power flow equation at the PCC is P = P(U td , U tq ) and Q = Q(U td , U tq ), where the Jacobian is defined by Equation (26) : ð26Þ It is derived that the determinant of the Jacobian equals 0. The proof is shown in Appendix A. Thus, the envelope of the family of P-Q curves is equivalent to the curve corresponding to the condition that the Jacobian of the power flow equation is singular.
Plot P-Q curves with various M e in its defined domain, as shown in Figure 3. The case shown in Figure 3 is of SCR = 1.5 and θ s = 80°. In the P-Q chart, P > 0 means that the MMC operates in the inverter state, and P < 0 reflects that the MMC operates as a rectifier; Q > 0 illustrates that the reactive power output of MMC is inductive, while Q < 0 means MMC outputs capacitive reactive power.
The lower boundary, as shown by the solid red line, is determined by the singularity condition of the Jacobian of the power flow equation, on which the points follow as Equation (27).
A P-Q curve is divided into two parts by its tangent points with the lower boundary if the tangent points exist, where the saddle-node bifurcation occurs. Considering the control process of MMC and the voltage stability at the PCC, the points in the solid lines are high voltage equilibrium and are stable; conversely, the points located in the dashed lines are unstable [28]. Some P-Q curves with small M e have no tangent point with the lower boundary, where all the operating points are unstable. MMC can operate at the dashed line points by directly controlling the fundamental modulation indices M ed and M eq , but cannot adjust the power outputs at their target value. Therefore, without considering the security constraints of the voltage offset and the current overload, the upper boundary of the power operation region of an MMC is determined by M e = M e _ max , as shown by the solid blue line.
The upper boundary is related to the MMC's parameters, while the lower boundary is determined only by the AC grid's parameters. The region enclosed by the two boundaries, shown as the light grey area, is the feasible operation region of an MMC connecting to an AC grid constrained by the converter. The dark-filled area consists only of the dashed line part of the P-Q curves, where the MMC could operate by directly generating the controller outputs; that is, M dc , M ed , M eq , M 2d , and M 2q , but this condition is abnormal. Points A and B are the tangent points of the upper and lower boundaries, where the MMC delivers and receives the maximum active power, respectively.

| Effect of the second-order modulation components on the operation region boundary
into Equation (16), note that both the derivatives of X MMC and ‖E c ‖ with respect to θ 2 can factor out the item of sin(θ 2 -2θ e ). The detailed expressions of X MMC , ‖E c ‖ and their derivatives are shown by Equations (B-9) and (B-10) in Appendix B. The derivates of P and Q with respect to θ 2 can factor out the item of sin(θ 2 -2θ e ) as well, according to Equation (28), which means that when θ 2 = 2θ e or θ 2 = 2θ e + π, the upper boundary is the highest and the size of the operation region is the largest, if M 2 is specified.
Thus, substituting Equation (29) into Equation (22) and calculating it with θ e changing from 0 to 2π, the upper boundary can be determined: Figure 4 shows the boundaries of the power region with various second-order modulation indices M 2 when the SCR is 1.5 and θ s is 80°. The solid line represents the case of θ 2 = 2θ e , and the dashed line corresponds to θ 2 = 2θ e + π. Under this F I G U R E 3 P-Q curves and operation boundaries WANG ET AL. condition, the upper boundary is the highest when θ 2 = 2θ e . However, the second-order modulation index M 2 occupies the range of the fundamental modulation index M e , reducing the size of the operation region. Therefore, the operation region is maximised when M 2 = 0; that is, when there is no circulating current control.

| Effect of the converter parameters on the operation region boundary
It is noted from Figure 2 that the equivalent reactance X MMC could affect the power outputs of MMC, which means both the arm inductor L 0 and the capacitance C 0 affect the operation region of MMC. According to Equation (19), reducing the submodule capacitance C 0 can reduce X MMC . However, the arm inductor L 0 should be increased with the reduction of C 0 according to Equation (17) to guarantee the peak value of the circulating current. Therefore, we analyse the influence of the designed parameters ε cap and ε Ikm on the maximum active power transmission. Figure 5 shows the results when the SCR is 1.5 and θ s is 80°.
As shown in Figure 4, when ε Ikm is large (that is, the arm inductor L 0 is small), reducing the submodule capacitance C 0 can increase the power transmission of the MMC. When ε Ikm is small, reducing C 0 will reduce the MMC power transmission. When ε Ikm is very small and ε cap is very large, the maximum power transmission will be significantly reduced, which may not satisfy power transmission requirements. Therefore, when designing the converter parameters, the requirements for inductance and capacitance parameters by the maximum power transmission should also be checked.

| Operation region considering the current constraint
To protect the semiconductors in an MMC from overload, an additional current limiter is included in the current controller to keep the current I v within the rated current of the MMC [8]. Similar to the analysis of converter constraint, according to Equation (22), the equation of P-Q curves can also be written as P = P(I v , φ v ) and Q = Q(I v , φ v ), where I v and φ v are the magnitude and the angel of the current I v , respectively. Therefore, the region satisfying the current constraint comprises a family of P-Q curves of various I v within the range of [0, I v _ max ], where I v_max is the current limit. The boundaries of the current constraint consist of the P-Q curve with I v = I v _ max and the curve determined by the condition that the power flow Jacobian is singular.
The operation region considering the security constraints is shown in Figure 6, where SCR is 1.5, θ s is 80°, and I v_max is 1.18. The red curve is the lower boundary of |J| = 0, the blue curve is the upper boundary of the converter constraint, and the green curve is the boundary of the current constraint. Region 1, shown in light grey, is where the operating states meet both the converter constraint and the current constraint. Region 2, located by both the left and right sides of region 1, is where only the current constraint is violated. MMC could operate in region 2 by using oversized insulated-gate bipolar transistors to provide system support features during emergency conditions [29]. However, an MMC connecting to an AC grid cannot operate stably in the region outside the union of regions 1 and 2.

| Effect of the SCR on the operation region
The power operation regions under various SCR conditions are shown in Figure 7. The SCRs are set to 1, 2, 3, 6.68, 10, and 25, and θ s is set as 80°. In this example, I v_max is set as 1.18, M e_max as 1, and M 2 as 0, and the |Z ∑ | is approximately equal to the inverse of 6.68. It could be noted that both region 1 and region 2 expand with the increase in the SCR. In Figure 7a,b, where the SCR is relatively small, the lower boundaries of region 1 are determined by |J| = 0, and both the P-Q curves of M e = M e_max and of I v = I v_max have tangent points with the lower boundary. If the SCR is quite small, as shown in Figure 7a, the dominating constraint is only the converter constraint, and the MMC could always transmit power without violating the current constraint. After the SCR increases, as shown in Figure 7c

| SIMULATION VALIDATION
To verify the proposed model and power region calculation method, we established a single-terminal half-bridge MMC-HVDC simulation system in PSCAD/EMTDC; the system parameters are shown in Table 1.

| Verification of the steady-state phasor model
We calculated the proposed model in this paper, the models in [9,10,13], and the accurate models in [8,17] using Matlab2016, and compared their results with the simulation results in PSCAD/ EMTDC. In this example, the DC modulation index is set as 1, the fundamental modulation index M e as 0.95, and the secondorder modulation index M 2 as 0. Changing θ e from 0 to 2π by the step ∆θ e as π/1800, the CPU time of the three models in Matlab2016 is 0.1716, 0.1035, and 110.7943 s, respectively. The comparison between the calculation results and the simulation results in PSCAD/EMTDC is shown in Figure 8.
There are 12 sets of comparison curves, including the active power P and reactive power Q, the dand q-components of voltage U t , the dand q-components of the common voltage U v , the dand q-components of the current I v , the DC component of the equivalent arm capacitor voltage U∑ cap0, the DC  [9,10,13] are denoted by the dashed red lines, which do not fit well with the simulation results because the models cannot describe the coupling characteristic inside the converter. The results of models [8,17] are denoted by the solid red lines, which is accurate. Thus, it is verified that the proposed model in this paper has high accuracy and has apparent advantages in computational efficiency due to its open-loop calculation process.

| Verification of determination method of the operation region
To verify the determination method of the upper boundary, we set M e_max as 0.95 and calculate three examples of θ 2 = 2θ e , θ 2 = 2θ e + π, and θ 2 = 2θ e + π/2. The calculation results denoted by the solid lines and the simulation results denoted by the black balls are shown in Figure 9. The average relative errors are 1.0827%, 1.1573%, and 1.0318%, respectively. The upper boundary of θ 2 = 2θ e is at the highest position, which matches the analysis results in Section 4.1.2. Thus, it is verified that the proposed determination method for the upper boundary of the power operation region is correct and can be used to analyse the influence of the second-order modulation components on the operation region.
To verify the calculation method of the lower boundary and the current constraint boundary, two typical operating points were simulated in PSCAD/EMTDC, where the first point was in region 2 and the second was outside the sum of regions 1 and 2. Calculating by Equation (27), the active power was 1.006 when setting the reactive power to 0, which is on the lower boundary. Figure 10 shows the response to the change in active power from 1 to 1.01 at t = 20 s. We found that the system is stable at the first point although the current I t exceeds its rated value (1.375 kA), but the system would be unstable at the second point, which confirms the theoretical calculation of the power operation region.

| CONCLUSIONS
Considering the different operating characteristics of the MMC and the two-level VSC, we propose a novel steady-state phasor model of an MMC done by deriving the function relation between the voltage and current outputs in the d-q frame. We also propose an open-loop calculation method for the steadystate power operation region of MMC based on the model and study its influencing factors.
The main conclusions are as follows: 1. The MMC is equivalent to an equivalent capacitor in series with an equivalent voltage source. In contrast, there is no equivalent capacitor at the AC-side equivalent model of a two-level VSC. The equivalent capacitor in the MMC model reflects the influence of converter internal coupling characteristics on the AC outputs; that is, the difference between an MMC and a two-level VSC. 2. The limit power operation region is constrained by the converter, which is the most fundamental constraint. The upper boundary is determined by M e = M e_max and the lower boundary is determined by the condition that the Jacobian of the power flow equation is singular. The lower boundary is determined only by the AC grid parameters, but the upper boundary can be affected by the MMC parameters and the circulating current control mode. 3. Considering the current constraint of MMC, the operation region of MMC is divided into two parts: region 1 is where the operating states meet both the converter constraint and the current constraint, and region 2 is where only the current constraint is violated. The SCR has an essential influence on the size and the dominating constraints of the MMC operation region.