Due to the attractive features of high waveform quality, low loss and well expansibility compared with traditional voltage sourced converter (VSC) which only has two or three levels, the modular multi-level converter (MMC) is increasingly used in modern high voltage direct current (HVDC) power systems (Zou et al., 2018; Liu et al., 2021; Wang et al., 2021). Therefore, more rigorous requirements are posed on MMC, especially when the grid conditions are unbalanced, which is very common in actual MMC-HVDC power system (Guo et al., 2019).

Currently, the vector control (VC) has been widely used in MMC-based DC power system (Nami et al., 2015; Wang et al., 2015; Xia et al., 2021), which is a direct current control strategy characterized by fast current feedback. The specific forms of VC is double closed-loop control using PI control strategy (PIVC), which can obtain the high quality current response. Based on VC, MMC-HVDC can achieve great steady-state performance. However, PI controller cannot ensure a satisfying dynamic response and robustness due to the lag characteristic of integrator under complex work conditions, e.g., parameter perturbation.

Sliding-mode control (SMC), which is an effective nonlinear control method (Hu et al., 2010; Song et al., 2021), is extensively applied in doubly fed induction generator (DFIG), pulse width modulation-based rectifiers, AC/DC converters (Hemdani et al., 2015; Dan et al., 2018; Long et al., 2021) thanks to the well-known advantages of satisfactory steady-state precision, excellent dynamic control performance and disturbance rejection. Further researches apply SMC strategy in the VC system, and an improved SMC strategy is presented for MMC in Li et al. (2020), which has better dynamic response, great disturbance rejection and less chattering. However, the control strategies discussed earlier assume that the grid condition is balanced. In fact, the grid condition always becomes unbalanced in actual engineering due to grid harmonics, imbalance fault, etc. (Hao et al., 2020; Wang S. et al., 2020; Freytes et al., 2021), resulting in negative sequence components in the grid voltage and current, which seriously affects the operating performance of MMC-HVDC. Consequently, in order to improve the transient control performance of AC/DC converters under unbalanced grid conditions, a great deal of researches have been widely studied with two basic transient control objectives which are restraining negative-sequence current and eliminating the oscillating components at twice the fundamental grid frequency (2ω oscillations) in active power and reactive power simultaneously under unbalanced grid conditions (Kong et al., 2013).

Further studies are presented for the transient control of VSC/MMC under unbalanced grid condition. Classic VC strategy with two sets of PI controllers (PIVC) under unbalanced grid conditions is proposed in Chen and Xu (2007), the outer-loop controller calculates the positive- and negative-sequence reference current in dq reference frame while the inner-loop controller achieves current-tracking by PI control strategy. However, the active and reactive power fluctuations cannot be eliminated simultaneously and the robustness of PI control is deficient. Furthermore, proportional-resonant (PR) control and proportional-integral-resonant (PIR) control are also combined with VC strategy for grid-connected converters in Zhou et al. (2013) and Yang and Nian (2015), respectively, which are similar to PIVC.

As an alternative solution, resonant compensation control is widely studied for unbalanced conditions which adds compensations to corresponding control reference values to realize different transient objectives (Wei et al., 2011; Nian and Cheng, 2013). A resonant compensation control strategy is proposed in Wei et al. (2011), which combines the dual αβ-PR method with the power pulsation compensation to suppress the 2ω oscillations in active and reactive power to a small range. However, the control system cannot eliminate power oscillations completely. An improved direct power control strategy combining with a resonant regulator is presented in Nian and Cheng (2013), which can suppress the 2ω oscillations in active and reactive power at the same time by constructing a resonant closed-loop. However, the AC current would have non-sine aberration when the active and reactive power keep constant.

Conventionally, these control strategies above are all using the traditional instantaneous power definition (Akagi et al., 2008), which cannot ensure that the AC-side current remains sinusoidal while eliminating both active and reactive power fluctuations under unbalanced grid conditions. To cope with the problem, an extended reactive power (ERP) is proposed in Sun and Lipo (2006), which only modifies the definition of traditional instantaneous reactive power to be more suitable for AC/DC converters under unbalanced grid conditions. In Zhang and Qu (2015), Liang (2017), the ERP is used in different control systems such as predictive control and reduced-order vector resonance control for AC/DC converter, which shows a better performance when grid conditions are unbalanced. However, the ERP has not been combined with SMC for MMC to obtain a satisfactory control performance under unbalanced grid conditions.

The main contribution of this paper is to propose an improved sliding-mode vector control (ISMVC) strategy combined with ERP for MMC, which can eliminate the 2ω oscillations in the active power and ERP simultaneously under unbalanced grid conditions while obtaining sinusoidal AC-side currents. Compared with the PIVC, the proposed ISMVC has obvious control advantages in dynamic response and robustness. The rest of this paper is organized as follows. In Section 2, the mathematical of MMC and overall control scheme are simply presented. In Section 3, contrastive theoretical analysis of traditional reactive power (TRP) and ERP are conducted under unbalanced grid conditions while the formulas of the ERP-based outer-loop controllers are deduced. In addition, the ISMVC is proposed and applied to the positive- and negative-sequence inner-loop controllers with discussions on its stability. In Section 4, comparative simulations in PSCAD/EMTDC of the ISMVC and PIVC using different reactive power definition are carried out to validate the feasibility and superiority of the proposed methods under different grid conditions. In the end, conclusions are drawn in Section 5.

Figure 1 shows a lumped schematic of the MMC. Among this figure, Z and Z′ represent the neutral points (U _zz’ = 0). R _ac , L _ac , and R ₀, L ₀ indicate the equivalent resistance and inductance of AC grid and per bridge arm, respectively; u _pk , u _nk , and i _pk , i _nk (k = a, b, c), respectively, represent the SM voltages and bridge arm current in the high-side and low-side bridge arms; the AC and DC grid voltages are expressed as U _sk and U _dc , respectively; u _vk and i _vk are AC voltage and current output by MMC.z

By using Kirchhoff’s laws, the mathematical model for AC circuit and DC circuit of MMC in Figure 1 can be derived separately as L i ̇ v k + R i v k = U d i ff k − U s k L 0 i ̇ l k + R 0 i l k = U d c 2 − U c o m k (1) where U _diffk = (U _nk − U _pk )/2 and U _comk = (U _nk + U _pk )/2 denote the differential-mode voltage (DMV) and common-mode voltage (CMV) in phase k, respectively; i _lk = (i _pk + i _nk )/2 denotes the loop current of phase k, which consists of DC component and second harmonic components (Li et al., 2020); R = R _ac + R ₀/2; L = L _ac + L ₀/2.

Conducting Park Transformation on Eq. 1 for better control effect, Eq. 1 in dq reference frame can be transformed as L i ̇ v d i ̇ v q = − R ω L − ω L − R i v d i v q + U d i ff d U d i ff q − U s d U s q L 0 i ̇ l d i ̇ l q = − R 0 − 2 ω L 0 2 ω L 0 − R 0 i l d i l q − U c o m d U c o m q (2) where ω is the fundamental grid angular frequency; U _sd ,i _vd , U _diffd , U _comd , i _lq and U _sq , i _vq , U _diffq , U _comd , i _lq are d-axis and q-axis component of corresponding electrical variables in Eq. 1, respectively.

Figure 2 shows the common hierarchical control structure of the MMC-HVDC control system, consisting of three control levels which ensure the proper operation of MMC from different dimensions. Among them, the station-level control provides the DMV and CMV in Eq. 1 by controlling the inner-loop current i _vk precisely according to the reference value of DC voltage, active power and reactive power, which decides the control effects of MMC directly.

When the AC grid condition in Figure 1 become unbalanced, the grid voltage and current may contain the positive-sequence, negative-sequence and zero-sequence components, respectively. Since the transformer on the AC side often adopts the Y-Δ connection method in actual projects, the zero-sequence component of valve-side current is eliminated. Thus, the MMC controller is designed without considering the zero-sequence component in this paper. Due to the structure of the three-phase circuit on the AC side of MMC is approximately symmetrical in Figure 1, the mathematical model for AC circuit of MMC under unbalanced grid conditions can be rewritten from Eq. 2 as L i ̇ v d + i ̇ v q + = − R ω L − ω L − R i v d + i v q + + U d i ff d + U d i ff q + − U s d + U s q + L i ̇ v d − i ̇ v q − = − R ω L − ω L − R i v d − i v q − + U d i ff d − U d i ff q − − U s d − U s q − (3) where U s d + , U s q + , U d i ff d + , U d i ff q + , i v d + , i v q + , and U s d − , U s q − , U d i ff d − , U d i ff q − , i v d − , i v q − are positive-sequence components and negative-sequence components of corresponding electrical variables in Eq. 2, respectively.

According to Eq. 3, the inner-loop controller in Figure 2B can be further decomposed into two parts, namely, positive- and negative-sequence inner-loop controllers, which collaborate to achieve different control objectives by controlling the positive current i v d q + and negative current i v d q + in dqreference frame track their reference values accurately, respectively.

It should be noted that the reference value of loop current in the DC circuit of MMC is always set to zero during the whole control process in Figure 2, which means the loop current controller is independent of the unbalanced grid conditions. Therefore, this paper is focused on the improved double closed-loop control of MMC in Figure 2B which aims to obtain constant active and reactive power as well as distortionless grid current at any time especially when the grid condition is unbalanced.

The traditional AC-side complex power S can be expressed as S = 1.5 U α β i α β ∗ (4) where “*” denotes the conjugate of a complex vector; U _α , i _α and U _β , i _β are the α-axis and β-axis components of the voltage and current on the AC side, respectively.

Thus, the traditional power can be expressed as P = 3 2 R e U α β i α β ∗ Q = 3 2 I m U α β i α β ∗ (5)

When the traditional active and reactive power are chosen as the controlled variables of outer-loop controller in Figure 2 under unbalanced grid conditions, the active and reactive power will contain 2ω oscillations otherwise the grid current will have non-sine aberration, which is pernicious to the grid operation and the MMC itself. Consequently, the control method of MMC needs further improvement to be more applicable under unbalanced grid conditions.

In this paper, an ERP is introduced to improve the control performance of MMC under unbalanced grid conditions (Sun and Lipo, 2006), which can be expressed as Q n e w = 3 2 R e U α β l a g i α β ∗ (6) where, U α β l a g = U α β ( t − T / 4 ) lags U _αβ by a quarter of electrical period.

When the grid condition becomes unbalanced, the grid voltage and current can be calculated as U α β = U α β + + U α β − = U d q + e j ω t + U d q − e − j ω t i α β = i α β + + i α β − = i d q + e j ω t + i d q − e − j ω t (7) where U α β + , i α β + , and U α β − , i α β − , represent the positive- and negative-sequence grid voltage and current vectors in the stationary reference frame respectively, which rotate counterclockwise at ω and −ω respectively; U d q + , i d q + , and U d q − , i d q − are positive- and negative-sequence grid voltage and current vectors in the forward and backward dq reference frame respectively, which rotate counterclockwise at ω and −ω respectively.

According to the rotating direction of forward and backward dq reference frame, the delayed grid voltage vector under unbalanced grid condition can be expressed as U α β l a g = U d q + e j ω t − π 2 + U d q − e − j ω t − π 2 = − j U d q + e j ω t + j U d q − e − j ω t (8)

Substituting Eqs 7, 8 into Eqs 5, 6, the active power, TRP and ERP under unbalanced grid condition can be expressed as P = P 0 + P 2 c ⁡ c o s 2 w t + P 2 s ⁡ s i n 2 w t Q = Q 0 + Q 2 c ⁡ c o s 2 w t + Q 2 s ⁡ s i n 2 w t Q n e w = Q 0 n e w + Q 2 c n e w ⁡ c o s 2 w t + Q 2 s n e w ⁡ s i n 2 w t (9) where, P ₀, Q ₀, and Q 0 n e w denote the DC components of the active power, TRP, and ERP, respectively, which should be the reference value of the outer-loop controller; P _2c , Q _2c and Q 2 c n e w denote the coefficients of 2ω cosine oscillating component of the active power, TRP, and ERP, respectively; P _2s , Q _2s , and Q 2 s n e w denote the coefficients of 2ω sinusoidal oscillating components of the active power, TRP, and ERP, respectively.

For better illustration, the coefficients of 2ω cosine and sinusoidal components of the active power, TRP, and ERP can be expressed in the matrix form as P 0 P 2 c P 2 s Q 0 Q 2 c Q 2 s Q 0 n e w Q 2 c n e w Q 2 s n e w = 3 2 U d + U q + U d − U q − U d − U q − U d + U q + U q − − U d − − U q + U d + U q + − U d + U q − − U d − U q − − U d − U q + − U d + − U d − − U q − U d + U q + U q + − U d + − U q − U d − − U q − U d − U q + − U d + U d − U q − U d + U q + i d + i q + i d − i q − (10)

According to Eq. 10, the outer-loop controller in Figure 2 which select active power and reactive power as controlled variables (P/Q control) to obtain constant output power can be designed.

For the PIVC using TRP, the positive and negative current reference value in their corresponding synchronous frame which are the reference inputs of the positive- and negative-sequence inner-loop controllers can be derived as i v d r e f + i v q r e f + i v d r e f − i v q r e f − = 2 3 U s d + U s q + U s d − U s q − U s d − U s q − U s d + U s q + U s q − − U s d − − U s q + U s d + U s q + − U s d + U s q − − U s d − − 1 P 0 r e f P 2 c P 2 s Q 0 r e f Q 2 c Q 2 s (11)

Since the reference values of active and reactive power (P _0ref , Q _0ref ) are default setting received from system-level control in Figure 2, there are only two available controlled variables left (choose two controlled variables from P _2c , Q _2c , P _2s , Q _2s ), which should eliminate the 2ω oscillations in active power and reactive power simultaneously.

It can be concluded from Eq. 10 that P 2 c = 0 → U d − i d + + U q − i q + = − U d + i d − − U q + i q − P 2 s = 0 → U q − i d + − U d − i q + = U q + i d − − U d + i q − (12) Q 2 c = 0 → U q − i d + − U d − i q + = U d + i q − − U q + i d − Q 2 s = 0 → U d − i d + + U q − i q + = U d + i d − + U q + i q − (13)

Obviously, Eqs 12, 13 cannot hold at the same time which means the 2ω oscillations in TRP still exist when the 2ω oscillating component coefficients of active power in Eq. 11 are set to zero (P _2cref = P _2sref = 0) under unbalanced grid conditions. Consequently, they cannot be eliminated concurrently by adopting TRP under unbalanced grid conditions, otherwise, it will only lead to distortion of the AC current with high harmonics.

Similarly, for the ERP, it can be seen from Eq. 10 that P 2 c = Q 2 s n e w P 2 s = − Q 2 c n e w (14) which means the 2ω oscillations in ERP will be eliminated completely with grid current keeping sinusoidal when choose P _2c and P _2s as controlled variables and set them to zero under unbalanced grid conditions, which is why the ERP is more suitable than TRP under unbalanced conditions as well as reducing the control complexity to improve the control performance.

Therefore, choosing active power and the ERP as controlled variables under unbalanced conditions with P _2c = P _2s = 0 in Eq. 10, the positive- and negative-sequence current reference value in their corresponding synchronous frame can be derived as i v d r e f + i v q r e f + i v d r e f − i v q r e f − = 2 3 U s d + U s q + U s d − U s q − U s d − U s q − U s d + U s q + U s q − − U s d − − U s q + U s d + U s q + − U s d + − U s q − U s d − − 1 P 0 r e f 0 0 Q 0 r e f n e w = 2 P 0 r e f 3 A U s d + U s q + − U s d − − U s q − + 2 Q 0 r e f n e w 3 A U s q + − U s d + U s q − − U s d − (15) where A = [ ( U s d + ) 2 + ( U s q + ) 2 ] − [ ( U s d − ) 2 + ( U s q − ) 2 ] ≠ 0 ; P _0ref and Q 0 r e f n e w are the reference values of active power and ERP respectively.

Moreover, when grid condition keeps balanced, it can be seen from Eqs 5, 6 that Q = 1.5 I m U α β i α β * = U β i α − U α i β Q n e w = 1.5 R e U α β l a g i α β * = U β i α − U α i β (16)

It can be seen from Eq. 16 that the ERP can still remain effective under balanced grid condition which increases the applicability of the proposed control strategy.

After obtaining the reference values of positive- and negative-sequence dq form grid current in Eq. 15, the next-step work is to design a better inner-loop controller to track these reference value accurately. Thus, an ISMVC strategy is proposed in this paper to enhance the control performance of the inner-loop controller in response speed and robustness compared with the traditional PI control strategy. Since the control structure of the positive- and negative-sequence inner-loop controllers are identical, the rest of this section will focus on the positive-sequence inner-loop controller as an example.

The first stage in the design of SMC is to select an effective sliding surface. Due to the minimal steady-state error and great chattering rejection (Jiang and Gao, 2021), the integral sliding surface is adopted in this paper to improve the steady-state control performance, which can be designed as S = S 1 S 2 T (17) S 1 = e p + + k p + ∫ 0 t e p + τ d τ S 2 = e q + + k q + ∫ 0 t e q + τ d τ (18) where k p + and k q + denote the integral coefficients of the positive-sequence d-axis and q-axis grid current, respectively; e p + , e q + are the tracking errors of the positive-sequence d-axis and q-axis grid current, respectively, i.e. e p + = i v d r e f + − i v d + e q + = i v q r e f + − i v q + (19)

To guarantee that the reaching and sliding conditions of SMC which enables uncertain chaotic systems to reach sliding-mode surface within finite time are satisfied, a Lyapunov function (Wang et al., 2019; Wang R. et al., 2020) is defined as V = K S T S = K S 1 2 + S 2 2 ≥ 0 (20) where, K is a positive constant.

The derivative of V can be calculated as d V d t = S 1 d S 1 d t + S 2 d S 2 d t = S T d S d t (21)

From Eqs 18, 19, the derivative of S can be calculated as d S d t = d d t S 1 S 2 = d d t i v d r e f + − i v d + i v q r e f + − i v q + + k p + e p + k q + e q + (22)

Taking Eq. 3 into Eq. 22, the derivative of S can be rewritten as d S d t = J + D U (23) where J = 1 L R − ω L ω L R i v d + i v q + + U s d + U s q + + k p + e p + k q + e q + + d d t i v d r e f + i v q r e f +

D = − 1 L U = U d i ff d + U d i ff q + .

For the purpose of better dynamic performance and robustness with less chattering phenomena compared with the conventional exponential trending law (Mozayan et al., 2016), an improved exponential trending law is proposed as follows: d d t S 1 S 2 = − ε 1 | e p + | g s a t S 1 ε 2 | e q + | h s a t S 2 − k 1 S 1 k 2 S 2 (24) where, ɛ ₁, ɛ ₂, k ₁, k ₂ are positive control parameters; g and h are positive power parameters; sat (S _i ) denote the saturation function which can be expressed as s a t S i = 1 S i > β i S i β i | S i | ≤ β i − 1 S i < − β i (25) where, β _i represents the range of the boundary layer; i = 1, 2.

In Eq. 24, the saturation function is adopted to design a boundary layer on both sides of the sliding surface which can reduce the high-frequency chattering phenomena introduced by traditional sign-function sgn (S _i ). −ɛ _i |e| ^h sat (S _i ) and −k _i S _i are variable and constant velocity term, respectively, which work together to propel the state trajectory towards the sliding surface with adaptive speed. Thus, good dynamic performance and less chattering phenomena can be balanced effectively.

Substituting Eq. 24 into Eq. 21, the derivative of function V can be rewritten as d V d t = − 2 K S 1 S 2 ε 1 | e p + | h s a t S 1 ε 2 | e q + | j s a t S 2 + k 1 S 1 k 2 S 2 ≤ 0 (26)

Obviously, it can be seen from Eq. 20 to Eq. 26 that function V and its derivative are positive-definite and negative-definite, respectively, which means that the proposed ISMVC strategy is proved to be asymptotically stable according to the Lyapunov theory.

Combining Eqs 23, 24, the actual control variables of positive-sequence inner-loop controller U d i ff d r e f + and U d i ff q r e f + can be expressed as U d i ff d r e f + U d i ff q r e f + = L ε 1 | e p + | g s a t S 1 + k 1 S 1 + k p + e p + L ε 2 | e q + | h s a t S 2 + k 2 S 2 + k q + e q + + R − ω L ω L R i v d + i v q + + U s d + U s q + + L d d t i v d r e f + i v q r e f + (27)

Similarly, the actual control variables of the negative-sequence inner-loop controller U d i ff d r e f − and U d i ff q r e f − can be obtained from Eq. 3 to Eq. 27 as U d i ff d r e f − U d i ff q r e f − = L ε 3 | e p − | m s a t S 3 + k 3 S 3 + k p − e p − L ε 4 | e q − | n s a t S 4 + k 4 S 4 + k q − e q − + R ω L − ω L R i v d − i v q − + U s d − U s q − + L d d t i v d r e f − i v q r e f − (28) where e p − and e q − are the tracking errors of the negative-sequence d-axis and q-axis grid current, respectively; ɛ ₃, ɛ ₄, k ₃, k ₄ are positive control parameters; m and n are positive power parameters; k p − and k q − are the integral coefficients.

Figure 3 shows the control diagram of the proposed ISMVC for MMC. First, the ERP is adopted in the design of the outer-loop controller to obtain the reference value of the positive- and negative-sequence grid current in dq reference frame which are regarded as the input reference of the ISMVC-based positive- and negative-sequence inner-loop controllers. Then, the reference value of DMV can be deduced while the CMV reference can be easily obtained from loop current controller. Finally, the triggering signals for IGBT in SMs can be calculated by the nearest level modulation (NLM) method (Chen et al., 2020). In addition, it should be noted that the positive- and negative-sequence components of the grid voltage and current are extracted by delayed signal cancellation (DSC) strategy (Timbus et al., 2007).

This paper proposes an ISMVC strategy combined with ERP for MMC under balanced and unbalanced grid conditions. The proposed method guarantees the valve-side grid current sinusoidal when eliminating the 2ω oscillations in active power and ERP simultaneously under unbalanced grid conditions. Faster dynamic response, better robustness and less chattering phenomena are provided by the improved SMC adopted in the positive- and negative-sequence inner-loop controllers of MMC due to the novel design of the sliding surface and reaching law, which can fully exerts the advantages of the ERP. Contrastive simulations of a four-terminal MMC-based DC power system based on the ISMVC and PIVC combined with the TRP and ERP are carried out to confirm the effectiveness and superiority of the proposed method under both balanced and unbalanced grid conditions. Therefore, the proposed method can work effectively under any grid conditions with fast dynamic response and good robustness.