Stability modeling of MMC-HVDC transmission system with differential flat control

AC asymmetry grid conditions put forward higher requirements for the stability control and performance of MMC-HVDC transmission systems. This thesis proposes a small-signal modeling method of an MMC-HVDC transmission system with differential flat control for AC asymmetry grid conditions, incorporating dynamic models of MMC internal characteristics, new control systems, circulation suppressors, and DC lines. To prove the accuracy of the pre-sended small-signal model, a two-terminal MMC-HVDC test system with differential flat control is constructed on the PSCAD/EMTDC platform.


Introduction
The utilization of modular multilevel converters (MMC) in high voltage DC (HVDC) transmission systems is widespread due to its benefits, such as a high switching frequency, low harmonic content, excellent redundancy performance [1], and powerful transmission capacity.However, when voltage imbalance, load imbalance, or parameter asymmetry causes asymmetry in the AC system, the valve-side voltage of MMC generates a large negative-sequence (NS) component, which brings a challenge to the safety of converter equipment.The controller, designed to operate in symmetrical conditions, will be adversely affected if it is operated in asymmetric ones due to the second-order fluctuations of the powers between the converter station and AC system occurring concurrently.So, it is necessary to study the modeling of stability in MMC-HVDC transmission systems under asymmetric operating conditions.
Scholars [2][3] currently concentrate on the stability modeling of a flexible DC transmission system under typical AC system conditions.At the same time, some have studied the control strategies of an asymmetric MMC-HVDC transmission system [4].In [5] and [6], researchers have carried out dynamic modeling of single-terminal MMC systems under asymmetric conditions based on the impedance method or small signal method.However, most of them have only adopted the traditional proportion integration (PI) control strategy and have not considered the influence of the control system at the converter station.
The differential flat controller has the advantages of small output fluctuation and simple structure.Combined with the flat output of the MMC-HVDC system at both terminals, the differential flat controller is applied in the MMC-HVDC transmission system.It can effectively suppress the changes in DC side voltage, negative sequence current, and reactive power in the MMC-HVDC system [7].The small-signal model (SSM) of the two-terminal MMC-HVDC transmission system under AC asymmetrical conditions is deduced in this paper, which also introduces a differential flat control strategy.To prove the validity of the strategy and the correctness of the small-signal model, PSCAD/EMTDC software is employed to construct the test system.

Ssm of Mmc Main Circuit
MMC is the core element of the small signal model of the MMC-HVDC transmission system at both ends.When the AC system operates asymmetrically, MMC introduces more state variables.Fig. 1 shows the basic structure of the MMC station.
Basic structure of MMC.Taking the upper arm as an example, a switching function is introduced to establish the state equation of the MMC main circuit by coupling the arm current and the capacitor voltage of the sub-module (SM).The average switching functions of the upper arm can be written as follows.
( 1) where Spa is the switch status of the upper arm, ipa is the upper arm current, ucp is the SM capacitor voltage of the upper arm, and upa is the upper arm voltage.The corresponding variables of the lower arm are represented with subscript "n" instead of "p".Here, the switch state Spa can be described as:  2) and (3) into Equation (1a), the state-space model of SM capacitor voltages in the DQ rotating reference frame by Park transformation can be derived.Similarly, by substituting Equations ( 2) and ( 4) into Equation (1b), and combining the voltage equation of the AC circuit, the state-space model of the arm current can also be obtained.
At long last, linearization at the operational point enables the acquisition of SSM of MMC internal, AC-and DC-side dynamics, following analogous deduction techniques as outlined in [8].
where, T c dc qc dc qc i r dc i r qs ds qs ds q [,,,,

Ssm of Mmc Control System With Differential Flat Theory
The differential flat theory-based MMC-HVDC system has the rectifier station set to a constant DC voltage-reactive power control and the inverter station to a constant active power-reactive power control.Two-terminal control systems consist of several parts, including PS and NS differential flat controllers (DFC), phase-locked loop (PLL), position-sequence measurement links (PSML), negative-sequence measurement links (NSML), and circulating current suppressing controller (CCSC), as shown in Fig. 2. Taking the rectifier station as an example, the SSM process of the MMC control system with DFC is introduced in this section.The difference between the inverter stations is only that the outer loop controller is different.The corresponding state-space model of PLL, PSML(NSML), and CCSC can be found in [8] [9].

2.2.1
Positive sequence differential flat controller for constant DC voltage rectifier station.For a specific nonlinear system, a set of output variables y is set.If all state variables x and input variables u of this system can be represented by this set of output variables y and its NTH derivative, then the system has differential flatness.It is proven that the inner and outer loop controls of the MMC-HVDC system are flat, and its output is flat [7].
Fig. 3 is the control diagram of the PS differential flat controller of the rectifier station.The outer ring is equipped with a constant DC volt-reactive power control.In contrast, the inner ring is composed of two components, both designed in accordance with differential flat theory, namely, feedforward reference trajectory generation and error feedback compensation.When the actual AC system has asymmetric operation, the control parameters of the conventional control strategy are difficult to determine, and the controller has poor adaptability.So, the differential flat control theory is introduced into the MMC-HVDC control system.It can restrain the harmonic component of DC voltage, active power, and negative sequence current.

Small signal model of outer ring controller.
Suppose the state variables of the DC voltage and reactive power outer loop PI controller are x1 and x2, respectively.Then, the state equation of the outer loop controller is as follows: The output equation of the outer ring controller is: where kpu1 and kpu2 are the proportional gain of the outer ring, respectively.kiu1 and kiu2 are the outer ring integral gain, respectively.

Small signal model of inner loop
where sdm i * and sqm i * are the measured values of PS d-q components of current, respectively.
The expression for the PS current inner loop error feedback compensation quantity based on differential flatness theory in Fig. 2 9) and linearizing the joint Equations ( 7), (9), and (11), the small-signal model of the positive-sequence differential-flat controller for a fixed DC voltage converter station is obtained as: where the state variable matrix, input quantity matrix, and output quantity matrix are, respectively: Linearizing the state equations of the positive sequence differential flat controller, negative sequence differential flat controller, phase-locked loop, phase sequence measurement link, and circulating current suppressor of the joint-defined DC voltage rectifier station yields a small-signal model of the rectifier station's control system.respectively represent d-q component increment of state variable of NS measurement link.

Dc Line Small Signal Modeling
Fig. 4 shows the equivalent circuit of the DC line of the MMC-HVDC transmission system.In this section, the "π" type equivalent circuit is used with centralized parameter representation.From the laws of KVL and KCL, the equations of state for the currents and voltages on the DC line are obtained as follows: where dc1

I and dc2
I are composed of the MMC DC side current and the zero-sequence two-fold loop current.
After linearizing Equation ( 14), the DC transmission line small signal model is obtained as follows:

Global Ssm of Mmc-Hvdc Transmission System
In summary, the combination of Equations ( 5), (13), and (15) is organized to obtain the 96 th order global SSM expression for the two-terminal MMC-HVDC transmission system as:

Effectiveness of Differential Flat Control Strategies
In this paper, the differential flat control introduced is compared with the PI-based positive and negative sequence (PNS) current vector control and the proportion integration resonant (PIR)-based PNS current vector control.When the imbalance phenomenon occurs at 3 s, 10 kV negative sequence voltage is added, and the grid returns to normal operation at 3.5 s.Fig. 5 shows that the NS current d-axis component outputs under three control strategies. .NS currents with three strategies when the AC system is unbalanced.In Fig. 5, when the system operates in the asymmetrical condition after 3 s, the d-axis component ripple of NS current is smaller under the differential flat control than the other two methods, and the amplitude is close to zero, which proves that the differential flat control effectively limits the NS current.

Accuracy of Small-Signal Models
It is assumed that an imbalance occurs in AC grids of two-terminal stations, and a 10 kV negative sequence voltage is added.At the MMC1 station, the DC voltage of the converter station is 315 kV, and the active power is 300 MW.Voltage steps from 315 kV to 320 kV at 4 s, and the simulation time is 5 s.The dynamic response results of the SSM output from MATLAB computation and the results from simulation are shown in Fig.In Fig. 6, it can be found that the dynamic responses of DC active and reactive power outputs from the proposed SSM are basically consistent with the results of the simulation when the AC system is operated under asymmetric conditions.This proves the correctness of the SSM of the MMC system proposed in this paper.

Conclusion
A comparison of differential flat control to traditional PNS current vector control reveals that it is more effective in restraining the augmentation of NS current.Consequently, a miniature signal model of a two-terminal MMC-HVDC transmission system with a differential flat controller under asymmetric conditions is proposed in this paper.Verification of the correctness of an SSM and the validity of a differential flat control strategy provides a theoretical basis for examining the effect of differential flat controller parameters on the steadiness of MMC-HVDC transmission systems.

Figure 3 .
Figure 3. Positive sequence differential flat controller for rectifier station.
controller.The state variables of the two current inner-loop PI controllers are x3 and x4, respectively, then the state equations of the inner-loop PI controllers are: variable increments of the inner and outer loop PI controllers of the differential flat controller, respectively.The last five elements of c Χu are actually the state variables of the phase sequence measurement link and the DC line, thus reflecting the coupling relationship between the controller, the positive sequence measurement link, and the DC line.

Figure 5
Figure5.NS currents with three strategies when the AC system is unbalanced.In Fig.5, when the system operates in the asymmetrical condition after 3 s, the d-axis component ripple of NS current is smaller under the differential flat control than the other two methods, and the amplitude is close to zero, which proves that the differential flat control effectively limits the NS current.

Figure 6 .
Figure 6.Comparison of two results under the DC voltage step.(a) DC voltage; (b) Active power; (c) Reactive power.
) are the amplitudes and PAs of fundamental PS, NS, and ZS components of SM capacitor voltage.The amplitudes and PAs of the PS, NS, and ZS components of the second-order SM capacitor voltage are denoted by ( c_2 U * , Uis the DC component of phase, a SM capacitor voltage.By substituting Equations ( is: * are error feedback compensation amounts, respectively; kpu3 and kpu4 are proportional gains of feedback compensation links, respectively.kiu3 and kiu4 are the integral gain of the feedback compensation link, respectively.

Table 1 .
EMTDC software-based two-terminal MMC-HVDC test system is constructed to assess the efficacy of the differential flat control approach and the precision of the proposed SSM.Table1lists the main circuit parameters and controller parameters of the two-terminal MMC-HVDC system.Parameters of the converter stations in the MMC-HVDC system.