Online dynamic voltage stability assessment method of AC/DC power systems

The reactive power absorption of conventional line commutated converter–high-voltage direct current transmission systems challenges on the voltage stability of AC/DC power systems. Many online dynamic voltage stability assessment approaches have been put forward to AC power systems, but the impact of DC transmissions is seldom taken into account for AC/DC power systems. In this study, an online dynamic voltage stability assessment method of AC/DC power systems is proposed. The proposed method models the converter as an equivalent AC voltage source in series with the internal impedance by utilising the control mode model of DC transmission systems and the measured data of the converter bus. The equivalent AC voltage source can identify the dynamic of DC transmissions. Then the AC/DC power system can be modelled by the equivalent AC power system. Based on the nodal equivalent method, the proposed method establishes a voltage stability index of the equivalent AC power systems for the dynamic voltage stability assessment of AC/DC power systems. The simulation demonstrated the effectiveness of the proposed method.


Introduction
Due to the advantages of the asynchronous AC grid connection and the capacity of bulk power transmission over a long distance, the conventional line commutated converter-high-voltage direct current (LCC-HVDC) transmission systems are widely used in the power transmission of resources which are far from the load centers [1]. However, the reactive power absorption of conventional LCC-HVDC transmission systems, which can reach to 40-60% of the HVDC active power transmission, challenges on the voltage support capability of AC systems [2]. Therefore, the voltage stability problem of AC/DC power systems is getting more and more attention.
The voltage stability problem of power systems has so far been studied by many researchers. Previous studies show that the voltage stability is a dynamic problem and the dynamic aspects of loads, generators and other dynamic system models should be taken into account. Several different approaches have been put forward to analyse the dynamic voltage stability of AC power systems. In [3], the author presents a voltage stability index (VSI) which is related to the eigenvalues of the system matrices. The VSI can give a valuable advance warning before the voltage collapse. However, the nonlinear loads are linearised around an operating point in the calculation of the VSI. In [4], both the reduced and unreduced Jacobian matrix of the system are studied and compared. The dynamic voltage stability is analysed by a bifurcation analysis method. With the development of computer technology, the time domain simulation can provide accurate result of voltage collapse [5]. Nevertheless, these approaches mentioned above usually need appropriate and accurate dynamic models of the power systems and take a long time to calculate sometimes. They cannot be applied for online voltage stability monitoring. In [6], the time-series voltage data from phasor measurement units (PMU) is used to compute the Lyapunov exponent to predict voltage stability in real time. Based on the nodal equivalent method, Zhang et al. [7] proposed an online method to identify the voltage stability of AC power systems during transients by utilising the measured data. Many online voltage stability assessment approaches have been put forward to AC power systems, but the impact of DC transmissions is seldom taken into account for AC/DC power systems.
In this paper, an online dynamic voltage stability assessment method of AC/DC power systems is proposed. Considering the control mode of the conventional LCC-HVDC transmission systems, the proposed method models the converter as an equivalent AC voltage source in series with the internal impedance by utilising the measured data of the converter bus. Based on the nodal equivalent method, a VSI for the dynamic voltage stability assessment of AC/DC power systems is established.
The rest of the paper is organised as follows. The quasi-steadystate (QSS) model and the proposed equivalent voltage source model of the conventional LCC-HVDC are detailed in Section 2. Section 3 describes the proposed VSI of AC/DC power systems and procedures of the online voltage stability assessment. Simulation results for a modified WSCC 9 bus test system are presented in Section 4 to demonstrate the effectiveness of the proposed method. Section 5 eventually draws the conclusions.

Quasi-steady-state model of DC line
The diagram of a two terminals conventional LCC-HVDC system is shown in Fig. 1. The QSS model equations can be expressed as follows: where V dr and V di are DC voltages on the DC terminals of the rectifier and inverter, respectively. V r and V i are AC primary voltages at converter transformers of the rectifier and inverter sides, respectively. N r and N i are the bridge numbers of the rectifier and inverter stations, respectively. K r and K i are the converter transformer tap ratios of the rectifier and inverter sides, respectively. α is the firing angle of the rectifier. γ is the extinction angle of the inverter. X cr and X ci are converter transformer reactances. I d is the DC current of the HVDC line. I r and I i are the transformer primary AC currents of the rectifier and inverter sides, respectively. The parameter μ = 0.995 for the twelve-pulse AC/DC converter.

Equivalent voltage source model of HVDC
When the control mode of HVDC has been determined, the operation state of the LCC-HVDC system depends on the voltage magnitude of the commutation bus which is connected to the converter. The equivalent circuit with a voltage source connected with an impedance has been widely used to represent the external characteristics of the dynamic components or to simply an external power system. In this paper, the proposed method models the converter as an equivalent AC voltage source in series with the internal impedance by utilising the control mode model of DC transmission systems and the measured data of the converter bus.
The LCC-HVDC inverter model and the equivalent model are given in Fig. 2. The calculation method for identifying the parameters of the equivalent AC voltage source model is derived from the QSS model of the LCC-HVDC as the followings: By multiplying e jθ Vi to (1) for the inverter, the equation has where the AC voltage vector of the inverter V i = V i e jθ Vi . Then replacing I d with (4), we have With the AC current vector of the inverter I i = I i e jθ Ii , (7) can be replaced with Rearranging (8) one gets The formation of (9) can be replaced as where The same to the rectifier, the equivalent parameters for the rectifier side can be derived as Equations (10)-(15) give the parameters expressions of the equivalent AC voltage source in series with the internal impedance. The general form of the equivalent voltage source model of the HVDC converter can be expressed by where the subscript 'C' of V and I represent the commutation bus which are connected to the converter. φ represents α or γ for the rectifier and inverter, respectively. I DC represents the reversal direction of I r and I i for the rectifier and inverter, respectively. In this paper, the control mode is considered to be constant power control in rectifier and constant extinction angle control in inverter. Then the parameters N r , N i , K r , K i , X cr , X ci and μ for the converter are known. Based on the control mode function of HVDC, the operation state parameters of V dc , α and γ can be calculated by the measured AC voltage V r and V i . The equivalent voltage source model parameters E DCr and Z DCr can be estimated by measuring the converter bus AC voltage phase angle θ Vi and current phase angle θ Ii . The equivalent model for the LCC-HVDC system is derived from their mathematic equations. It can easily identify the dynamic of DC transmissions by utilising the timeseries voltage and current data from measurements.

VSI index for AC/DC power systems
A multi-port Thevenin equivalent AC/DC power system with multiple voltage sources and LCC-HVDC transmission lines supplying loads is shown in Fig. 3. The dynamic of the power system can be represented using the differential and algebraic equations [8] as follows: where x and y are state variables and algebraic variables, respectively. During the dynamic, state variables are related to the slow mechanical behaviours, while the algebraic variables are related to the electromagnetic transient behaviours. The function f and g are the differential and algebraic equations of the AC/DC power systems, respectively. In Fig. 3, all buses in the multi-port Thevenin equivalent AC system can be classified into four classes: load buses, generator buses, tie buses and commutation buses which are connected to converters. Since the injection currents of tie buses are zero, the injection currents from these three types can be expressed by where the matrix Y is known as the system admittance matrix. V and I are the voltage and current vectors of each bus, respectively. Subscripts L, G, T and C represent the load bus, generator bus, tie bus and commutation bus, respectively. −I L represents the reversal direction of the load current.
By the Gauss elimination method, the tie buses can be eliminated and the first row of loads of (20) can be rearranged as follow: where

Rearranging (21) yields
Using the general form of the equivalent voltage source model of the HVDC converter proposed in Section 2, replacing V C with (16), we have denotes the equivalent voltage source considering the influence of LCC-HVDC by E DC . The ith equivalent branch circuit of (23) for load can be expressed as represents the equivalent impedance of the nodal equivalent model considering the coupled impedance of other loads and LCC-HVDC. n load and n dc are the number of load buses and commutation buses, respectively. Based on the nodal equivalent method, (24) can be treated as a typical two-bus power system as shown in Fig. 4. By utilising the maximal power transfer condition, the equivalent impedance Z eq, i AC/DC will reach to the load impedance Z L, i when the power grid operation is approaching the collapse point. Then the VSI for AC/DC power systems is defined by where Z L, i = (V L, i /I L, i ) can be calculated by measured data.

Procedures of online voltage stability assessment
A PMU placed at a bus can measure the bus voltage and current phasors. It is assumed that proper PMU placements can ensure the complete observability of the AC power systems [9]. In this case, the voltage vectors and current vectors of load buses, generator buses, tie buses and commutation buses needed in (20) can be estimated from PMU real-time measurements. Based on the QSS model of the LCC-HVDC, the operation state parameters of V dc , α and γ can be calculated for the equivalent voltage source model of the HVDC converter by the measured AC voltage of commutation buses.
With the real-time measurements of PMU, the dynamic performance of AC/DC power systems can be tracked at high (iv) Calculate the VSI for each load buses according to (25) by the time-series equivalent impedance of (24) and dynamically update these values.
(v) Determine the voltage stability of AC/DC power systems. The load bus is voltage stable when VSI < 1 and in a critical state when VSI = 1.

Simulation
In this paper, power system analysis toolbox (PSAT) tool box and MATLAB software are used to simulate a long-term dynamic voltage instability scenario. The modified WSCC 9 bus AC/DC system used in the dynamic simulation is shown in Fig. 6. A LCC-HVDC transmission line is added in parallel with the AC transmission line of bus 7-8. The system base MVA is 100 MVA. The rated AC voltage is 230 kV. The rated DC voltage is 300 kV. The control mode of HVDC is considered to be constant power control in rectifier and constant extinction angle control in inverter. The active power order of the rectifier station is P drs = 0.8 pu. The extinction angle of inverter γ = 18°. Other parameters of DC line are showed in Fig. 7. The load model considers the exponential recovery (ER) load model. The parameters of ER loads are showed in Table 1. The line and generator parameters are the same as those in [10].
In this long-term dynamic voltage instability simulation, the transmission line of bus 6-9 is cut off at t = 10 s. The simulation results of load bus voltages and VSI corresponding to time scale is plotted in Figs. 8 and 9.
Figs. 8 and 9 show that after the fault of AC system, the voltage stability margin decreases gradually with the recovery of load power demands. When the power grid operation is approaching the collapse point, the VSI proposed in this paper approaches to 1. The VSI can identify the voltage instability in advance even the bus voltage of load might be in reasonable ranges.

Conclusion
This paper proposed an online dynamic voltage stability assessment approaches considering the impact of DC transmissions for AC/DC power systems. Based on the QSS model of the LCC-HVDC, the proposed method models the converter as an equivalent AC voltage source in series with the internal impedance. The equivalent model for the LCC-HVDC system is derived from its mathematic equations which can identify the dynamic of DC transmissions by utilising the time-series voltage and current data from measurements. The AC/DC power system can be modelled as an equivalent AC power system. Based on the nodal equivalent method, the proposed method establishes a VSI for the dynamic voltage stability assessment of AC/DC power systems. The procedures for the dynamic voltage stability assessment of AC/DC power systems are summarised. The proposed VSI can identify the voltage instability in advance when the power grid operation is approaching the collapse point.