Dynamic Phasor Modelling of LCC-HVDC System Based on a Practical Project

. This paper presents a detailed dynamic phasor modeling process of a line-commutated convert-er-based HVDC (LCC-HVDC) system. Firstly, the dynamic phasor models of the single-ended LCC rectifier station, inverter station and the DC line are established, respectively. Secondly, LCC-HVDC is an AC-DC-AC system. The interfaces are explained to connect the converter stations with the DC line. Through block modeling, it is helpful to simplify the process and verify the accuracy of each block. Finally, based on a practical project, the model is compared with the electromagnetic-transient (EMT) simulation results in PSCAD/EMTDC to verify the accuracy of the dynamic phasor model.


Introduction
Because the line-commutated converter-based HVDC (LCC-HVDC) project has the advantage of long-distance and large-capacity transmission, it is widely used in the world. However, the transmission capacity of the LCC-HVDC project increases continuously, so that the strength of the AC system at the receiving end weakens relatively [1]. Besides, with a number of LCC-HVDC putting into operation, the coupling between the receiving and the sending end is getting closer [2]. Therefore, it is urgent to study the stability of the AC and DC hybrid power system.
As a key device for connecting AC and DC systems, LCC converters have very complex nonlinear characteristics. It is not only difficult to study them directly, but also not conducive to explore the stability mechanism [3]- [4]. Therefore, developing the linearization model in time-domain or frequency-domain is a basic and important method to analyze the stability.
Reference [5]- [7] develop a time-domain linearization model based on the quasi-steady-state characteristics of the LCC converter. Reference [8] presents an averagevalue model for a LCC-HVDC system using dynamic phasors. Compared with electromagnetic-transient (EMT) model, the average-value model reduces unnecessary computational complexity when the high-frequency dynamics of the system are not concerned. Reference [9]- [10] develop a small-signal model of the LCC inverter to investigate the impact of the control system on stability. By using the switching function, the relationship of the voltage and current of AC and DC side are described. However, it only aims at the single side modeling including the LCC inverter station, AC system and control system. Although many scholars have researched on small-signal modeling of LCC-HVDC, there is almost no literature on explaining the idea of simplifying the modeling process of a complex LCC-HVDC system.
In this paper, the idea of block modeling is adopted. First, the LCC-HVDC system is divided into three subsystems including the LCC rectifier side, the LCC inverter side and the DC line. Second, the dynamic phasor models of three sub-systems are established, independently. Third, the three sub-systems are connected through the interfaces. Finally, a dynamic phasor model of LCC-HVDC is established and validated by the dynamic model in MATLAB and the EMT simulation from PSCAD/EMTDC. The results show the accuracy of the dynamic phasor model of LCC-HVDC based on a practical project.

Study system 2.1Description of the study system
The study system is a unipolar 12-pulse LCC-HVDC system, shown in figure 1. The rectifier station and the inverter station are connected by the DC line. On the rectifier side, the AC system is connected to the rectifier by transformers. AC filter banks and reactive power compensation devices are also connected at the point of common coupling (PCC) to filter out harmonics and compensate reactive power. The DC side of the rectifier is connected to the DC line through the smoothing reactor. The system structure is symmetric, therefore, the same devices and connections are also available on the inverter side.
As for the control system, the phase-locked loop (PLL) locks the phase of the PCC voltage to provide a phase basis for the synchronization of the LCC-HVDC. The fixed current controller and the fixed voltage controller are adopted by the rectifier and inverter, respec-tively. Idref and Udref are the reference current and voltage. Idm_r and Udm_i can be obtained by the Id_r and Ud_i through the measurement. The command value output by the controllers are the delay trigger angle (αr) of the rectifier and the advance trigger angle (βi) of the inverter. AC Figure 1. The configuration of LCC-HVDC system.

2.2The interfaces of subsystems
In order to simplify modelling, the LCC-HVDC system can be divided into three subsystems, which are bounded by the smoothing reactors. The equivalent circuit of the rectifier and inverter subsystems are shown in figure 2.  In figure 2(a), the subscript r represents all the variables on the rectifier side. Correspondingly, the variables on the inverter side are represented by the subscript i, shown in figure 2(b). us_r and is_r are the voltage and current of the AC system, respectively. The impedance of AC system consists of Rs_r and Ls_r, which are in series. uPCC_r is the voltage of PCC. ZF_r is the impedance of the AC filters. RT_r and LT_r represent the impedance of the transformer. uc_r and ic_r are the voltage and current on AC side of the rectifier, respectively. Lsm_r is the smooth-ing reactor. Udc_r and Idc_r are the DC voltage and current, respectively.
Due to the fixed voltage controller on the inverter side, Udc_r can be substituted by a DC voltage source. Similarly, Idc_r can be substituted by a DC current source for the fixed current controller on the inverter side. The variables in brackets are obtained by d-q transformation of the corresponding electrical quantities.
The DC line adopts the T-type equivalent circuit, shown in figure 3.  In which, Rdc_1, Rdc_2, Ldc_1 and Ldc_2 are the resistances and reactors of the DC line. Cdc is the ground capacitance. ZFdc_r represents the impedance of the DC filter banks on the rectifier side, which is the same as ZFdc_i on the inverter side. Idc_r and Udc_i are the interfaces to connect the rectifier and inverter. The commutation inductor of the transformer (Leq_r) should be considered on the DC side because the fixed current control on the rectifier side cannot keep the current unchanged 11. Leq_i on the inverter side is similar. Leq_r can be expressed as eq_r T_r In which, μ is commutation overlap angle. LT_r is the leakage inductor of the transformer.
The voltage of the AC system and its d-q variables can be represented by s_r 0 0_r In which, Vm_r, ω0 and α0_r are the peak to peak voltage, angular frequency and phasor of the AC system. The state variable of the AC system on both side are

The AC filter banks on the rectifier/inverter side.
According to the practical project, the configuration of the AC filter banks on the rectifier and inverter side are shown in figure 4. According to the configuration of the filter banks, they can be divided into three groups as A, B and C. The state-space equation of the AC filters on the rectifier side can be obtained by Kirchhoff's laws as Cb3q_r Lb3d_r Similarly, the state-space equation of the AC filters on the inverter side are the equation (7) and (8) (11) is the angle of power factor, in front of which the plus is for the inverter and the minus is for the rectifier. Taking the commutation of the LCC into account, Au/i is a coefficient to correct the voltage/current. In which, the plus is for the inverter and the minus is for the rectifier.

6)
In which, ωr and θr are the angular frequency and phase of the output of PLL. KpPLL_r and KiPLL_r are the proportion and the integral gain, respectively.
The state space equation of PLL on the inverter side is the same as equation (14), except for changing the subscript r to i.
The fixed current control and the fixed voltage control block diagram on each side are shown in figure 6.

8)
Kp_r and Ki_r are the proportion and the integral gain, respectively. x1_r is the difference between the reference current (Idcref) and the measured current (Idcm_r), which is a state variable. Adopting the first order inertia element to measure the DC current on the rectifier side (Idc_r), the measured current (Idcm_r) can be obtained by dcm_r m_r dc_r dcm_r d d I T I I t (9) Tm_r is the time constant of the first order inertia element.
Similarly, the state-space equation based on figure  6(b) can be expressed as Kp_i and Ki_i are the proportion and the integral gain, respectively. x1_i is the difference between the reference voltage (Udcref) and the measured voltage (Udcm_r), which is a state variable. Tm_i is the time constant.

Modelling of the DC line
According to the T-type equivalent circuit in figure 3

21)
There are two banks of the DC filter, whose configuration is same. The DC filter bank on the rectifier side is shown in figure 7. It is important to note that the state variables including the capacitor voltage and inductor current must be independent. Thus, for the convenience of modelling, choose IFdc_r and Idc_r as the state variables on the rectifier side and choose IFdc_i and Iline_r on the inverter side.

Small-signal dynamic modelling of LCC-HVDC
Based on the state-space model above, the small-signal dynamic model can be obtained by linearization as d dt Based on the practical project, the order of LCC-HVDC system is 67. The state variable X consists of the AC system, the AC filter banks, the PLL, the control system, the DC line and the DC filter banks. The input variable ΔU=[Idcref, Udcref] T .

Simulation Results and Model Validation
In this section, a dynamic phasor model for a practical LCC-HVDC system is developed in MATLAB/SIMULINK. In order to validate the accuracy of the dynamic phasor model, the time-domain response obtained from the model in MATLAB is compared with the detailed time-domain simulation in PSCAD/EMTDC.

4.1The parameters of the LCC-HVDC system
The parameters of the practical LCC-HVDC system are shown in table Ⅰ and table Ⅱ. It should be noted that the project parameters of the AC filter given in Table 2 are for the bipolar system, so they need to be converted.

Results and analysis
In order to investigate the step response of the system, two cases are set as follows: Case 1: Udcref step changes from 0.93 p.u. to 0.90 p.u. at time t=5s and from 0.90 p.u. to 0.93 p.u. at time t=8s.
In each case, the small-signal step responses of DC current, voltage of PCC, active power on the rectifier side and DC voltage, voltage of PCC, active power on the inverter side are compared, completely. The comparison results are shown in figure 8 and 9.
It can be seen from the figures that the results of the dynamic phasor model in MATLAB have great consistency with the results of the detailed model in PSCAD. Besides, the simulation results in PSCAD has a larger range of fluctuation than the results in MATLAB, because all frequencies are included in the detailed model, but only fundamental frequency characteristics are considered in the dynamic phasor model in MATLAB. However, the average value of both is the same.

Conclusion
This paper presents a detailed dynamic phasor modeling process of a complex double-ended LCC-HVDC system. Through the idea of block modeling, it is helpful to simplify the process and verify the accuracy of each block. The blocks can be connected by the reserved interfaces. Besides, a practical project is provided. Based on the practical project, the accuracy of the dynamic phasor model is verified.