Research on the STATCOM Mathematical Model of Battery Storage in HVDC Transmission System

When traditional STATCOM (Static Synchronous Compensator) performs large-capacity reactive power compensation, the control accuracy of the DC side voltage will be affected and the DC side voltage will fluctuate greatly. Therefore, this study proposes to use battery energy storage STATCOM (STATCOM/BESS) and gives the main circuit topology of STATCOM/BESS. By analyzing the working principle of STATCOM/BESS, the mathematical model and control method are derived and modeled, in which the Shepherd model is used as the energy storage battery in STATCOM/BESS. Aiming at the possible imbalance of the battery state of charge (SOC) in STATCOM/BESS, the phase-to-phase SOC balance control and phase control are proposed. At last, in PSCAD/EMTDC simulation software, STATCOM/BESS is tested for dynamic response and steady-state response performance of active power control, reactive power control, and transformation failure suppression test, which verifies the effectiveness and superiority of STATCOM/BESS commutation failure.


INTRODUCTION
High Voltage Direct Current (HVDC) plays an important role in China's strategy of "West-East electricity transmission project and national network" by virtue of its advantages in long-distance and large-capacity power transmission and grid interconnection (Zhang et al., 2010). In HVDC transmission projects put into operation in China, the characteristics of "strong HVDC and weak AC, multi-infeed DC" are increasingly prominent, which leads to the safety, stability, and reliability of the power grid . When a fault occurs at the receiving end of the HVDC system, it is easy to cause the voltage sag of the AC busbar at the inverter side, leading to commutation failure of the HVDC system. STATCOM can realize dynamic reactive power compensation, suppress the bus voltage flicker, and improve the voltage transient stability of the system. When the traditional STATCOM performs high-capacity reactive power compensation, the control accuracy of the outer DC side voltage will be affected, leading to the DC side voltage oscillation and large amounts of harmonic components in the output current. At this time, the traditional STATCOM only compensates for the reactive power, and its supporting effect on the grid voltage will be not ideal (Li et al., 2014).
STATCOM by itself has no ability to control and compensate active power, and the energystorage STATCOM combined with reactive power compensation technology can make up for the deficiency of traditional STATCOM in HVDC system applications (Authors Anonymous, 2016). At present, the energy storage in the field of power systems mainly includes pumped storage, flywheel energy storage, supercapacitor, and battery energy storage (Cheng et al., 2007). In the study of Castaneda et al. (2010), energy-storage STATCOM was applied to the grid-connected wind power planning, which effectively avoids voltage collapse in case of sudden failure and shorts the voltage recovery time. Yang et al. (2001) proposed a STATCOM/BESS device with parallel batteries on the DC side of STATCOM to improve the dynamic and transient stability and power transmission capacity of the system. Xie et al. (2009) studied the effect of energy-storage STATCOM on suppressing voltage phase change in the weak power grid. Through experiments and simulations, it is found that when the system fails, compared with traditional STATCOM, energy-storage STATCOM can suppress voltage phase transformation and reduce voltage sag by 7%. For the study of the topology and control method of energy storage type STATCOM, a static reactive generator/battery energy storage (STATCOM/BESS) integrated system consisting of highpressure-resistant, high-power, and low-switching switching devices such as GTO, IGBT, and so on is designed (Fei et al., 2005), which can compensate for the power loss of the system itself while improving the system's compensation capacity, thus damping oscillation and improving the system's temporary stability. Compared with traditional STATCOM, STATCOM/ BESS integrated system can effectively reduce fault voltage landing and significantly suppress oscillations during voltage recovery.
In this study, according to the existing research results and the actual system requirements, a STATCOM/BESS with an angle cascaded H-bridge structure is carried out, and the performance of the designed STATCOM/BESS is verified by simulation.

DC/DC Section
Energy storage batteries are connected to the DC side in direct parallel connection, non-isolated two-way DC/DC connection, isolated two-way DC/DC connection, and so on. Isolated twoway DC/DC connection requires a low-or high-frequency transformer for boost, which is large in volume and high in cost (Fei et al., 2005;Xie et al., 2009;He, 2016). The non-isolated bi-directional DC/DC connection has a simple structure, is of low cost, and has good energy conversion efficiency. Its disadvantages are that the adjustment range of input and output is small and there is no electrical isolation between the battery pack and the power grid .
The actual STATCOM/BESS phase contains 36/42 cascade H bridge modules, which is a large number. Therefore, considering the cost and technical difficulty of the device, the parallel capacitor of the energy storage battery is directly connected to the DC side of STATCOM, in which the parallel capacitor plays the role of energy buffer and flattens the power difference between the H bridge module and energy storage battery module.

DC/AC Section
In the conventional low-voltage distribution network, the DC/AC part of STATCOM usually adopts the low-voltage two-level topology. However, due to its low-voltage grade, high switching device frequency, and large loss, the multi-level topology is usually adopted in the HVDC transmission system (Shen and Raksincharoensak, 2021a). STATCOM/BESS uses the multilevel topology to make the output waveform have a better harmonic spectrum, and the voltage stress borne by each switching device is smaller, effectively reducing the capacity and voltage level of each battery pack (Li et al., 2014). Figure 1 shows the angle-cascaded H-bridge STATCOM/ BESS topology . There is a zero-sequence current in the angle structure, which can exchange active  power between three-phase bridge chains. Several H-bridge submodules are cascaded to form a bridge chain. Each H-bridge submodule can output three levels of voltage, and the bridge chain series reactor can be directly integrated into the grid and withstand the gridline voltage. When the number of H-bridge sub-modules is n, 2n+1 levels can be output through modulation, and the AC side output of the converter is a multilevel waveform, so changing the number of H-bridge submodules can change the output voltage of the converter (He, 2016). Compared with the diode clamp three-level topology and capacitor clamp three-level topology, the cascaded H-bridge topology has a simple structure and convenient maintenance, requiring only the minimum number of components in the case of the same number of output levels (Cui et al., 2011), and avoiding the problem of unbalanced charge and discharge when multiple battery groups are connected in series.
Based on the above considerations, this study adopts the angle-cascaded H-bridge topology.

STATCOM/BESS Working Principle
The H-bridge-cascaded STATCOM/BESS each phase bridge chain can be directly integrated into the power grid after series reactor L. The network between each phase bridge chain and the connection point is equivalent to the two-terminal network as shown in Figure 2A; Yang et al., 2022. Each phase bridge chain can be regarded as a controllable voltage source, with output voltage U A , connection point grid U S, and internetwork current I A . By controlling the amplitude and phase angle of the output voltage U A , mutual compensation of the active and reactive power between STATCOM/BESS and the power grid is realized.
In Figure 2B, the grid voltage at the U S grid-connected point controls U A and U S in the same direction and makes U A amplitude greater than U S amplitude. At this point, STATCOM/BESS sends perceptual reactive power to the grid. In Figure 2C, the U A is controlled in the same direction as the U S , and the U A amplitude is smaller than the U S amplitude. At this time, the STATCOM/BESS sends capacitive reactive power to the grid. In Figure 2D, the U A phase is controlled to lead the U S phase, and the STATCOM/ BESS sends active power to the grid at this time. In Figure 2E, the U A phase is controlled to lag BEHIND U S phase, at which time STATCOM/BESS absorbs active power from the grid. By controlling the amplitude and phase angle of the output voltage U A , STATCOM/BESS, and the power grid can complement each other with active and reactive power.

The STATCOM/BESS Mathematical Model
Assuming that the parameters of each phase of the angle cascade STATCOM/BESS are completely consistent, one three-phase STATCOM can be decomposed into three single-phase STATCOM to facilitate the analysis and derivation of the mathematical model. Taking the AB phase as an example, its single-phase equivalent circuit is shown in Figure 3;  According to Kirchhoff's voltage theorem, The singlephase bridge chain voltage drop under the angle-cascaded structure is the grid-connected point line voltage u s , u ab is the bridge chain output voltage, and i ab is the grid-connected phase current.
The H-bridge sub-module is composed of four switching devices in parallel with a DC capacitor . Suppose that the switch state of G j1 is 1 when it is on and 0 when it is off. The same is true for G j3 . At this time, the difference between the switch states of G j1 and G j3 is used to define the switch state H j of the entire H-bridge module; then H j is Eq. 3 can be used to obtain the current i cj of the DC container in parallel with the jth H bridge sub-module: According to Eq. 3, the state equation of the capacitor voltage can be obtained, where u dabj is the capacitor voltage of the jth H bridge module on the AB bridge chain: Define the sum of the switching states of the H bridge at all levels as the switching state N of the phase converter, then Using Eq. 6, one can obtain the output voltage of the jth H bridge sub-module u abj : Frontiers in Energy Research | www.frontiersin.org June 2022 | Volume 10 | Article 827914 Assume that the capacitor voltages of all modules in the bridge chain are equal; then, u dab is the sum of capacitor voltages of all modules in the bridge chain, and the output voltage u ab of the bridge chain can be obtained by combining Eq. 5: Combining Eq. 4, we can get After substituting Eq. 7 into Eq. 1, we can get Assume that the grid-connected current i ab and the total capacitor voltage u dab are state variables, and establish their state equation by combining Eqs 8 and 9; Shen and Raksincharoensak, 2021b. After discretization, the discrete state equation can be obtained as where i ab (k+1) and u dab (k+1) indicate the AB-phase gridconnected current and bridge-chain capacitor voltage value at time (k+1), and N(k) and u sab (k) indicate the AB bridge-chain switch at time k and state and grid-connected point line voltage, respectively and, T k is the sampling period (Shen et al., 2021).

ENERGY STORAGE ELEMENT AND SOC EQUALIZATION CONTROL Parameter Extraction of the Shepherd Model
The Shepherd model directly describes the electrochemical behavior of batteries including the port voltage, open-circuit voltage, internal resistance, discharge current, and state of charge (SOC) through an equation. The model can be used to describe the charging and discharging process of the battery. The circuit diagram of the Shepherd model is shown in Figure 4, where E b is the no-load voltage of the battery, U b is the output voltage of the battery, R b is the internal resistance of the battery and i b is the output current of the battery. The nonlinear function shown in Eq. 11 describes the relationship between battery no-load voltage and output current (Zhu et al., 2020): In Eq. 11, E 0 is the voltage constant (V), K is the polarization voltage (V), Q is the battery capacity (Ah), A is the voltage constant in the exponential discharge zone (V), B is the time constant in the exponential discharge zone (Ah −1 ) (Zhou et al., 2016), and it = i b dt is the actual cumulative output charge of the battery. Figure 5 shows the charge and discharge waveform of the battery ( Shen et al., 2017). It can be seen that the change in the battery terminal voltage can be roughly divided into three areas: exponential change area, relatively stable area, and rapid change area. The terminal voltage of the battery in the area of exponential change rises or falls with the change of charge in the form of exponential, and the corresponding voltage value of the battery in the state of full charge is E full . When the battery runs in the relatively stable region, the terminal voltage changes slowly with the charge quantity, and the critical voltage corresponding to the transition from the exponential changing region to the relatively stable region is E exp . In the rapidly changing region, the battery terminal voltage changes rapidly with the amount of charge, and  the critical voltage corresponding to the transition from the relatively stable region to the rapidly changing region is E nom . Estimation of battery internal resistance R: The internal resistance R affects the output voltage of the battery and then affects the working efficiency of the battery. Therefore, the internal resistance of the battery can be estimated by measuring the working efficiency of the battery. The working efficiency of the battery η can be easily calculated as follows (Liu et al., 2020): where I nom is the nominal discharge current of the battery and V nom is the nominal voltage. Typical values of the working efficiency of batteries can be obtained by statistical methods in the practical project, and the calculation equation of internal resistance R can be obtained from Eq. 13: To determine the remaining parameters of the model, three sets of data can be obtained from the measured discharge curve given by the manufacturer: full charge voltage E full , exponential area boundary voltage E exp , discharge capacity Q exp , stable discharge area boundary voltage E nom ,and boundary capacity Q nom .
The exponential zone discharge voltage time constant A can be determined by Eq. 14: The exponential zone discharge voltage time constant B can be determined by Eq. 15: After extracting the parameters A and B, the polarization voltage K can substitute the first group of data and the third group of data in Eq. 11 to obtain Eq. 16: Finally, the voltage time constant E 0 can be derived from the full charge voltage E full to obtain Eq. 17:

SOC Equalization Control
There are many chained STATCOM/BESS modules, which may lead to different charge and discharge times of each phase energy storage battery and sub-module batteries in the phase. Especially when the grid-connected voltage is not symmetrical, the disequilibrium of charge and discharge between energy storage batteries will be more serious, reducing the efficiency of energy storage batteries. Therefore, a state of charge (SOC) equalization control strategy for STATCOM/BESS is proposed, as shown in Figure 6; Yang et al., 2021a.
In Figure 6A, SOC ab -SOC ca is the SOC of each phase, SOC 0 is the average SOC of the phase, and are calculated using Eqs 18 and 19, respectively: SOC m SOC mi (m ab, bc, ca) The basic idea of phase-to-phase SOC equalization control is as follows: the difference between each phase SOC m and the phase average SOC 0 passes through the PI controller G soc(s) to obtain the phase SOC equalization active current command I socm (Yang et al., 2021b), which is then sent to the current inner loop control. If the SOC m is less than SOC 0 , a positive active power command is obtained, and the energy storage battery of this phase is in the state of charge as a whole so that the SOC m increases and the SOC balance of the phase is achieved, and vice versa.
In Figure 6B, SOC mi is the SOC of the sub-module battery in the phase and SOC m0 is the average SOC of the sub-module battery in the phase. The basic idea of SOC control of sub-module in phase is as follows (Yang et al., 2018): If SOC mi is greater than SOC m0 , then the difference value is proportionally controlled to obtain the submodule SOC to adjust the additional modulation signal v socmi . Since v socmi is ahead of the line voltage corresponding to the link, the submodule will output active power to the grid and SOC mi decreases so as to achieve the SOC balance of the submodule, and vice versa.

SYSTEM CONTROL STRATEGY
The goal of the STATCOM/BESS control system is to maintain a stable phase transformation voltage and reduce the probability of its occurrence when it detects that the commutation failure of the   DC transmission system may occur . In order to suppress commutation failure of the DC transmission system, after STATCOM/BESS detects the AC system failure, it switches the corresponding control strategy according to the severity of the failure and proposes the STATCOM/BESS power coordinated control method. When the AC system fails, STATCOM/BESS performs fast with reasonable compensation of reactive and active power, provides effective voltage support to the receiving end power grid, and effectively suppresses the occurrence of commutation failure. The power coordination control system is shown in Figure 7, Feng et al., 2021. On the basis of receiving the upper active power command, the active power control module adds active power adjustment control based on frequency deviation, which can enhance the damping of the interconnected system and suppress power oscillation (Qi et al., 2006); the active power command is set to 0 under fault conditions and reactive power control The module starts to work, including steady-state constant reactive power control and constant AC voltage control, while the transient voltage control mode is based on the dynamic  reactive power support based on voltage sag, with additional DC transmission system inverter station shutting-off the angle control. It can provide more reactive power compensation for the grid under fault conditions to ensure the normal progress of the phase transformation process. The dynamic reactive power QT injected into the grid by STATCOM/BESS should track the voltage changes at the grid-connected point in real time, and satisfy the following equation (Oghorada et al., 2021): In the equation, v represents the STATCOM/BESS gridconnected point voltage per unit value, I N represents the STATCOM/BESS rated current, v max and v min , respectively, represent the upper and lower limits of the bus voltage at the grid-connected point during the dynamic reactive power support process, and k T1 represents the reactive power proportional coefficient in the process of dynamic reactive power compensation.

SIMULATION ANALYSIS
In this study, PSCAD/EMTDC software is used to build the electromagnetic transient simulation model. The overall framework of the electromagnetic transient model is shown in Figure 8. In the model, the main circuit of the LCC-HVDC system adopts the single-pole 12 pulse structure, and the main parameters are selected by referring to Yongfu DC Project, as shown in Table 1. The DC control system is the same as the controller in the CIGRE standard model.
Limited by the simulation speed, the STATCOM/BESS electromagnetic transient simulation model in this article is equipped with three H-bridge modules per phase, with a total capacity of 150 Mvar, which is connected to the LCC-HVDC inverter side AC bus through a 35/525 kV transformer.

Active and Reactive Power Control
The response performance of STATCOM/BESS is simulated. The STATCOM output active power instruction is set as 60 MW and reactive power instruction as 60 Mvar, respectively, and the output power changes are observed. Figure 9A shows the constant active power control power response graph. During the dynamic response process, the active power output by STATCOM quickly follows the active command; the rise time is about 0.18 s, the maximum output is 66 MW, and the overshoot is less than 11%. In the steady-state process, the active power output by STATCOM fluctuates between 59 and 61.5 MW, and the steady-state response error is less than 3%. Figure 9B shows the power response diagram of constant reactive control. Its dynamic response process is also good, with a rising time of 0.178 s, a maximum output value of 67.6 Mvar, and an overshoot of less than 13%. It enters the steady state at 0.24 s. The STATCOM output reactive power fluctuates between 58 Mvar and 62.5, and the steady-state response error is less than 5%. The simulation results show that the STATCOM/BESS used in this study has good output performance and can respond quickly to a given power command.

Single-Phase Ground Fault
When the setting is 2 s, a single-phase ground fault occurs in the AC system on the inverter side and the fault lasts for 0.2 s. Figure 10 shows the effective value of the AC voltage on the inverter side before and after the fault, the extinction angle of commutation failure, the transmission power of the system, and the DC. In Figure 10A, when STATCOM is not connected, the AC voltage on the inverter side drops to 0.7 pu after the fault occurs and basically recovers to 0.96 pu after 2.5 s. After STATCOM is connected, the voltage sag is reduced to 0.83 pu, and the recovery is increased to 0.96 pu, reducing the impact of faults on the AC system. In Figure 10B, when STATCOM is not connected, the extinction-angle γ decreases to 0°after the fault occurs, indicating that the first commutation failure occurs. After STATCOM access, the extinction-angle γ is about 10°, and commutation failure is inhibited. In Figures 10C,D, when STATCOM is not connected, the power transmitted by the system drops dramatically and the DC increases sharply, seriously affecting the system operation and equipment safety. After STATCOM is connected, the transmission power and DC voltage of the system are less affected by the fault and will be stabilized in a short time. According to this group of simulation, when a single-phase grounding short-circuit occurs on the inverter side, voltage sag will occur, which will lead to commutation failure. After STATCOM/BESS is connected, the terminal voltage can be effectively supported, constantextinction-angle can be improved, and the commutation failure and subsequent chain reaction caused by the fault can be inhibited.

Three-Phase Ground Fault
When the setting is 2 s, a three-phase ground fault occurs in the AC system on the inverter side, and the fault lasts for 0.2 s. Figure 11 shows the effective value of the AC voltage on the inverter side before and after the fault, the extinction angle of commutation failure, the transmission power of the system, and the DC current. As shown in Figure 11A, when STATCOM is not connected, AC voltage sag on the inverter  side is obvious after the fault occurs, and the lowest drops to 0.5 pu, which basically recovers to 0.96 pu after 2.5 s. After STATCOM is connected, the voltage sag is reduced to 0.7 pu, and the power supply is restored more quickly. As shown in Figure 11B, when STATCOM is not connected, the extinctionangle γ decreases to 0°after the fault occurs and the first commutation failure occurs, and again to 0°at 2.15 s, indicating that continuous commutation failure occurs during the fault process.

With/Without BESS Simulation Verification
In order to compare the commutation failure suppression effect between STATCOM/BESS and conventional STATCOM, a serious single-phase grounding fault is set at 1 s in this experiment. Figures 12A,B show the AC voltage and inverter current at the inverter side of the HVDC system when an A-phase at the inverter side suddenly has a grounding fault after a period of normal operation. At this time, the system is equipped with conventional STATCOM. After the fault occurred, the A-phase voltage begin to drop, and the AC voltage on the inverter side is greatly distorted. The inverter current dropped to 0 at the same time in the vicinity of 1 s, indicating that commutation failure occurred.
When STATCOM/BESS is installed in the system, the grounding fault occurs in A-phase in 1 s. It can be seen from Figures 13A,B that the AC voltage distortion on the inverter side is relatively small, and the phenomenon that the three-phase current is zero at the same time during the fault period does not occur, which indicates that STATCOM/BESS has restrained the occurrence of commutation failure under the single-phase fault condition.

CONCLUSION
In this study, the mathematical model of STATCOM/BESS is established, a suitable battery model is selected as its energy storage element, and a SOC equilibrium control strategy is put forward. The model is built in PSCAD/EMTDC simulation software and its function is tested and the following conclusions are obtained: 1) STATCOM/BESS adopts coordinated power control, which is divided into two modules: active power control and reactive power control. It can compensate active power and reactive power quickly and reasonably in case of fault. 2) Shepherd model is selected as the energy storage element, and the SOC equilibrium control strategy is proposed to improve the working efficiency of the energy storage battery. 3) Compared with the traditional STATCOM that can only compensate for reactive power, STATCOM/BESS can track active power commands and respond quickly when the active power output is required. At the same time, it can also compensate reactive power quickly when a serious fault occurs, thereby supporting the grid voltage by suppressing the commutation failure, which is of great significance in improving the operation stability of the AC/DC system with a weak receiving end. 4) The STATCOM/BESS device is installed in the single-pole 12pulse HVDC transmission system, a single-phase ground fault is set, and a comparison experiment is carried out with the traditional STATCOM system; by comparing the two typical electrical quantities of AC voltage and inverter current, the effectiveness of the STATCOM/BESS designed in this study is verified in suppressing commutation failure.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
CX was responsible for providing ideas and methods and providing an experimental platform. SC was responsible for deriving formulas, reviewing, and verifying. JC was responsible for model building, simulation, data analysis, and manuscript writing. XX, ZX, XH, and SL wrote sections of the manuscript. All authors participated in the reading and approved the submitted version.

FUNDING
This work was supported by the National Natural Science Foundation of China (52067009).