As using the classical quasisteady state (QSS) model could not be able to accurately simulate the dynamic characteristics of DC transmission and its controlling systems in electromechanical transient stability simulation, when asymmetric fault occurs in AC system, a modified quasisteady state model (MQSS) is proposed. The model firstly analyzes the calculation error induced by classical QSS model under asymmetric commutation voltage, which is mainly caused by the commutation voltage zero offset thus making inaccurate calculation of the average DC voltage and the inverter extinction advance angle. The new MQSS model calculates the average DC voltage according to the actual halfcycle voltage waveform on the DC terminal after fault occurrence, and the extinction advance angle is also derived accordingly, so as to avoid the negative effect of the asymmetric commutation voltage. Simulation experiments show that the new MQSS model proposed in this paper has higher simulation precision than the classical QSS model when asymmetric fault occurs in the AC system, by comparing both of them with the results of detailed electromagnetic transient (EMT) model of the DC transmission and its controlling system.
By the end of 2013, China has built over 15 EHV and UHV DC transmission lines; both the total length and transmission capacity are the largest in the world [
In order to improve the precision and speed of electromechanical transient simulation in hybrid AC/DC system, researchers have developed a variety of models for the DC system, including equivalent circuit model, dynamic phasor model, small signal linearized model, and classic quasisteady state (QSS) model, for transient stability simulation.
The equivalent circuit DC system models mainly adopted the variable topology of converters, to establish the “centerprocess” method [
Dynamic phasor method was firstly presented in [
Small signal linearized model was established by applying the actual sampling data to model the DC transmission system in the
Classical QSS model for the DC transmission and its controlling system has the advantage of fast computation; thus it has been widely used in hybrid AC/DC system simulation [
The structure of the paper is as follows. The next section introduces the error causes for the classical QSS model. The modified QSS model is presented in Section
The basic function of DC transmission system is to complete the AC to DC (rectifier) and DC to AC (inverter) conversion and transmission of electrical energy [
The principal wiring diagram of single bridge rectifier.
This section describes the classical quasisteady state model succinctly [
The AC system is assumed to be threephase symmetric sinusoidal system, with a frequency of 50 Hz (60 Hz in other countries or regions), regardless of the harmonics and the influence of the neutral shift.
The inductance value of the series smoothing reactor on DC side is large enough, and the performance of DC filters is ideal, so that the influence of the ripples can be neglected in the direct current.
The converter transformer is thought of as ideal, regardless of the saturation effect, excitation impedance and copper loss, and so forth.
The characteristics of thyristor valves are ideal, namely, the voltage drop during conducting state and the leakage current during blocking state can be ignored, and the six valves are triggered to enter the conducting state in turn with an equal time interval of 1/6 cycle.
Define the firing delay angle as
DC voltage on rectifier side under symmetric AC voltage waveform.
The instantaneous threephase voltages can be expressed in (
In classical QSS model, the average DC voltage can be directly calculated according to the symmetric threephase commutation voltage waveform. Taking the commutation period of valve 3 to valve 4 (the shaded area in Figure
Substitute (
The commutation angle on rectifier side can be given as
The RMS value of AC current on rectifier side will be
The active power consumption by the converters and corresponding power factor on rectifier side can be written as
The formulas on inverter side of the DC transmission system are similar to the rectifier side in the classical QSS model; we only need to replace the variable of firing delay angle
DC line model can be generally classified as lumped parameter circuit model, segmented
The equivalent circuit of the DC transmission lines.
According to Figure
The DC control system model is adopted as the CIGRE HVDC control system, the block diagram is shown in Figure
The control system of CIGRE HVDC test system.
From the modeling of the three parts of DC systems from Sections
The asymmetric commutation voltage in the AC system may cause potential calculation errors for the classical QSS model in the following ways:
If the average voltage of DC side is still computed according to formulas under symmetric voltage assumption, all other parameters in the DC side will have inevitable deviation and thus affect the accuracy of calculation. Therefore, more precise formulas of the average DC voltage should be derived, according to the actual AC system operation status.
If the rectifier firing delay angle and the inverter extinction advance angle are calculated using the symmetric waveform, the triggering pulse cannot consider the influence of the commutation voltage zero offset, which might lead to commutation failure or pole blocking for the DC system.
It has been shown that the classical QSS model could not provide reliable simulation results under the condition of asymmetric commutation voltage. To address this main defect, a modified quasisteady state model of DC system is proposed in this paper. It would be better to use the actual voltage waveform on the DC terminals to calculate the average DC voltage, so as to avoid the error caused by symmetric assumption. Before this, it should be better to find the exact commutation voltage zero point for the triggering pulse of each valve so as to compute the DC voltage and extinction advance angle. Taking the inverter side of sixpulse converter as an example, it is easy to illustrate the situation.
Since we only care about the fundamental components during electromechanical simulation, the influence of harmonics and interharmonics is not considered in this study. For a given operation status of threephase voltage amplitude and phase angle, formulas can be derived for predicting the six zero points within one cycle; the detailed process is as follows.
Suppose the instantaneous threephase asymmetric commutation voltages are expressed as
using phase
At the line voltage zero point
Applying the trigonometric transformation, the zero point
The calculation formulas for the rest five line voltage zero points within one cycle can be acquired similarly, which are listed in
During the simulation process of asymmetric faults, due to the influence of the line voltage zero offset, phase locking device, and the DC control system, the actual firing delay angles for thyristor valves will not be equal to the initial angles given by the triggering pulse control system.
In order to address the influence of zero offset on the DC control system, the firing angle
Figure
The determination of triggering pulse for the valves on the inverter side.
It can be seen in Figure
During the commutation process, three valves are participating; thus the valves can be divided into three classes. In this study, the valve that is entering into the commutation status is defined as
The relationship between
Pulses 




























Similar to the classical QSS model in Section
According to Table
The commutation process and the corresponding DC voltage instantaneous value for the single bridge inverter within one cycle.
Trigger pulse  Time period  Upper bridge valve state  Lower bridge valve state  Commutation state  DC voltage waveform  Corresponding relationship  DC voltage calculation 



Valve 3 commutes to 5  Valve 4 conducting 





Valve 5 conducting  Valve 4 conducting  — 


 




Valve 5 conducting  Valve 4 commutes to 6 





Valve 5 conducting  Valve 6 conducting  — 


 




Valve 5 commutes to 1  Valve 6 conducting 





Valve 1 conducting  Valve 6 conducting  — 


 




Valve 1 conducting  Valve 6 commutes to 2 





Valve 1 conducting  Valve 2 conducting  — 


 




Valve 1 commutes to 3  Valve 2 conducting 





Valve 3 conducting  Valve 2 conducting  — 


 




Valve 3 conducting  Valve 2 commutes to 4 





Valve 3 conducting  Valve 4 conducting  — 



The expression of “
Using the last column of Table
Similar to classical QSS model, the voltage drop
Considering that the commutation voltage contains only the fundamental component, it is sufficient to integrate the average DC voltage by halfcycle voltage waveform. Taking the halfcycle voltage waveform from the triggering point of valve 6 to valve 3 of the inverter side as an example, the actual DC terminal voltage waveform on the inverter side can be shown as the shaded part in Figure
Actual DC voltage waveform analysis on the inverter side during asymmetric commutation line voltage.
Then the average DC voltage from
Solving the integral formula above, we can get the average DC voltage on the inverter side as shown in (
The calculation process of average DC voltage on the rectifier side is similar. After obtaining the DC voltages on both sides of each DC system, other variables of DC system can then be calculated.
In order to test the validity of the proposed MQSS model, simulations are performed on CIGRE HVDC benchmark test system; according to the new model, the classical QSS model, and the power system EMT model from PSCAD software, results from PSCAD are chosen as the comparison reference, because it contains the full electromagnetic transient models that can take into consideration all dynamic performance of the DC system.
At 1.0 s, a threephase symmetric shortcircuit grounding fault occurs on the AC bus of the inverter side, the grounding resistance is set as 0 Ω, the fault lasts for 0.1 s, and the total simulation time is 1.5 s. With the same initial steady state and faulted operation conditions, the calculation results of the three models for the DC system, namely, the new modified quasisteady state model (marked with “new”), the electromagnetic transient model (marked with “EMT”), and classical quasisteady state model (marked with “QSS”), are shown in Figures
DC current transient waveform on the inverter side during threephase grounding fault.
DC voltage transient waveform on the inverter side during threephase grounding fault.
DC voltage transient waveform on the rectifier side during threephase grounding fault.
It can be seen from the curves of Figures
Steady state DC values and errors during the 100 ms threephase shortcircuit fault for both QSS and MQSS models.
EMT model (pu)  QSS model (pu)  Modified model (pu)  Absolute error of QSS model  Absolute error of modified model  


0.5500  0.5500  0.5500 



0  0.0561  0.056 



0.0050  0.0670  0.0670 


It can be seen from Table
Under asymmetric fault occurring on the AC bus of the inverter side, the voltage asymmetry is mainly decided by fault resistance. If the fault resistances are different, the asymmetric degree of the AC bus voltage will also be different. In this study, we use different values of the AC fault resistance under asymmetric faults to investigate the validity of the proposed MQSS model.
At 1.0 s, a single phase grounding fault occurs on the AC bus of the inverter side, grounding resistance is 10 Ω, and the fault lasts for 0.1 s, and the simulation results for the new model, EMT model, and classical QSS model are shown in Figure
DC current transient waveform on the inverter side under single phase grounding fault.
It can be seen from Figures
DC voltage transient waveform on the inverter side under single phase grounding fault.
DC voltage transient waveform on the rectifier side under single phase grounding fault.
The comparison of steady state DC values during the single phase grounding fault (from time 1.0 s to 1.1 s) and corresponding errors of the two QSS models to EMT model are given in Table
Steady state DC values and errors during the 100 ms single phase grounding fault for QSS and MQSS models.
Single phase grounding 
Parameters  EMT model (pu)  QSS model (pu)  Modified model (pu)  Relative error 
Relative error of 

0 

0.5500  0.6000  0.5481 



0  0.4267  0.1954 




0.0507  0.4600  0.1843 





10 

0.5820  0.7338  0.5512 



0.2929  0.5925  0.3331 




0.3000  0.6139  0.3440 





20 

0.6823  0.8426  0.7164 



0.5400  0.6929  0.5846 




0.5600  0.7360  0.6001 


Since double phasegrounded fault usually induces more severe asymmetry than phasetophase shortcircuit, the phasetophase shortcircuit asymmetric fault type is not included in this study. At 1.0 s, a double phasegrounded fault (assuming to be phase
Typically, the smaller the shortcircuit grounding resistance, the lower the AC voltage; thus the commutation voltage asymmetry degree is larger. It is essential to simulate the more serious condition, such as commutation failure and pole blocking for the DC system.
In the accurate EMT simulation model, continuous commutation failure and HVDC pole blocking for the DC system can be encountered when the grounding resistance is from 0 to 40 Ω, with the initial firing delay angle of the inverter side
Simulation results for QSS and MQSS models under double phasegrounded faults with small grounding resistances.
Double phasegrounded 
EMT Model  QSS model 
Modified model 

0  Blocking  Blocking  Blocking 
20  Blocking  2.1817  Blocking 
40  Blocking  2.2689  Blocking 
For larger grounding resistance from 60 to 100 Ω under double phasegrounded fault, the simulation results and corresponding errors of the two QSS models to referenced EMT model are given in Table
Steady state DC values and errors during the 100 ms double phasegrounded fault for QSS and MQSS models.
Double phasegrounded 
Parameters  EMT model (pu)  QSS model (pu)  Modified model (pu)  Relative error of 
Relative error of 

60 

0.6500  0.8500  0.7470 



0.4800  0.7400  0.6217 




0.4750  0.7250  0.6380 





80 

0.7050  0.8840  0.7069 



0.5700  0.7790  0.5733 




0.5750  0.7800  0.6424 





100 

0.7900  0.9200  0.8104 



0.6500  0.8000  0.6486 




0.6520  0.8150  0.6623 


To conclude, the QSStype models are developed for electromechanical transient simulation, which has much larger time steps and simulation duration than electromagnetic transient simulation. Therefore, the requirements are to reduce the amount of calculation time of electromagnetic computation, while maintaining high accuracy. It is well accepted that the detailed electromagnetic transient simulation programs, such as PSCAD/EMTDC, can provide the reference values; thus we compare both the proposed MQSS and the conventional QSS with the results of PSCAD. The accuracy of our MQSS model has shown to be substantially improved comparing with the conventional QSS model during asymmetric faults. And the computation time does not increase much, as indicated by the additional multiplications and additions in (
In power system electromechanical transient stability studies, the classical quasisteady state (QSS) model is not able to accurately simulate the dynamic characteristics of DC transmission and its controlling system when asymmetric fault occurs in AC system; therefore, a new modified quasisteady state model (MQSS) is proposed in this paper. The new MQSS model utilizes the actual halfcycle voltage waveform on the DC terminals to predict the exact zero points of commutation voltages and then calculate the average DC voltages and the extinction advance angles, so as to avoid the negative effect of the asymmetric commutation voltage distortion. Simulation experiments show that the new MQSS model proposed in this paper can reduce the simulation error by 15% at least compared to the classical QSS model, under single phase grounding and double phasegrounded asymmetric faults in the AC system, by comparing both of the two models with the results of the detailed EMT model. Because the new MQSS model is capable of reflecting the dynamic characteristics of DC systems without considering the complicated electromagnetic transient processes in typical EMT models, it is very suitable for transient stability simulation in hybrid AC/DC power systems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by China Postdoctoral Science Foundation under Grant 2013M542349, in part by the Fundamental Research Funds for the Central Universities of China under Grant xjj2013026, and in part by the State Key Laboratory of Electrical Insulation and Power Equipment under Grant EIPE14314.