Abstract
With various components and complex topologies, the applications of high-voltage direct current (HVDC) links bring new challenges to the interconnected power systems in the aspect of frequency security, which further influence their reliability performances. Consequently, this paper presents an approach to evaluate the impacts of the HVDC link outage on the reliability of interconnected power system considering the frequency regulation process during system contingencies. Firstly, a multi-state model of an HVDC link with different available loading rates (ALRs) is established based on its reliability network. Then, dynamic frequency response models of the interconnected power system are presented and integrated with a novel frequency regulation scheme enabled by the HVDC link. The proposed scheme exploits the temporary overload capability of normal converters to compensate for the imbalanced power during system contingencies. Moreover, it offers frequency support that enables the frequency regulation reserves of the sending-end and receiving-end power systems to be mutually available. Several indices are established to measure the system reliability based on the given models in terms of abnormal frequency duration, frequency deviation, and energy losses of the frequency regulation process during system contingencies. Finally, a modified two-area reliability test system (RTS) with an HVDC link is adopted to verify the proposed approach.
HIGH-VOLTAGE direct current (HVDC) transmission technology has become an ideal solution for the bulk power transfer over long distances between the economic generation areas and the large load centers in recent decades [
The deployment of HVDC links brings challenges in terms of frequency security to the interconnected power systems. To begin with, the frequent failures of the HVDC link may raise great power imbalances that cause severe frequency deviation. With massive components and complex topologies, it is more likely that failures occur at HVDC links compared with AC links. Besides, since the HVDC link usually contains millions of megawatts of power flow, its failures may cause severe power imbalances for both the sending-end (SE) and receiving-end (RE) AC power systems. Such severe power imbalances caused by the outages of HVDC link are much likely to cause terrible frequency deviation in either side of the interconnected power system. For instance, the bi-polar block accident in the Jinping-Sunan ultra-HVDC caused a power loss of over 4000 MW and then led to a frequency drop of 0.44 Hz in the Eastern China grid on September 19, 2015 [
Considering the significant impacts of the HVDC link faults, several studies have been conducted on reliability modeling. Some of them focus on component-level analysis. In [
Meanwhile, due to the low-inertia feature, it has become a real concern how the abnormal frequency duration of interconnected power systems caused by the HVDC link failures can be effectively reduced. Some works have focused on regulating frequency with spinning reserves. In [
Furthermore, the conventional approaches for reliability assessment utilize reliability indices such as the expected energy not supplied (EENS) and the loss of load probability (LOLP), which do not depict the frequency-related reliability such as the frequency deviation and the abnormal frequency duration. Although [
Given this background, this paper proposes an approach for system operators to evaluate the reliability performance of interconnected power systems with HVDC links considering the frequency regulation process during system contingencies. Firstly, a multi-state model of an HVDC link is developed with different available loading rates (ALRs) and the corresponding probabilities. Then, the models for the primary frequency response (PFR) and secondary frequency response (SFR) of the interconnected power systems are integrated with a support scheme provided by the HVDC link for system frequency regulation. In this scheme, the converters under normal operation will temporarily carry out overload transmission to mitigate the power imbalance that causes frequency disturbance for both SE and RE AC power systems under system contingencies. Meanwhile, the proposed scheme can transmit the frequency deviation at the fault end to the opposite end through the DC voltage variation. The opposite end then feeds such deviation signal into its frequency regulation framework to dispatch reserves against the power vacancy. In this way, the mutual energy aid between reserves of both sides can be characterized, and the energy losses during the frequency regulation process can be detected for reliability analysis. Finally, several reliability indices are proposed to represent the frequency dynamics and energy losses during the frequency regulation process, which is evaluated by the Monte Carlo simulation (MCS) technique.
In this section, the multi-state model of an HVDC link is proposed. The multi-state model can effectively reflect different performance levels and the corresponding probabilities of composite systems in a simplified expression, and has been used in many reliability analyses [

Fig. 1 Reliability network of a bipolar HVDC link.
Based on the impact of different component failures on the ALRs, the reliability network of an HVDC link can be divided into three kinds of subsystems, namely filter system (FS), converter system (CS), and transmission system (TS). These three kinds of subsystems combine a monopole system (MS) of an HVDC link. Except for an FS that is composed of an ACF, each CS or TS can be regarded as an equivalent series element with several components, as shown in
(1) |
(2) |
In this paper, the lifespan of the main component in the HVDC link is assumed to be exponentially distributed, and these components work during the “constant failure rate period” [
After series equivalence, an MS can be modeled as an equivalent series-parallel element as shown in the red dash rectangle of
In the following subsections, the state space diagram for an MS of a bipolar HVDC link is firstly developed. Then, two separate state space diagrams of MS are aggregated to build the multi-state model of a bipolar HVDC link.
As shown in

Fig. 2 State space diagram of an MS.
Meanwhile, based on the usage of the state space diagram defined in [
(3) |
The row position of ZOC is the OC, from which the transition occurs and the column position of ZOC is the OC at which the transition occurs. Since there are 11 OCs for an MS in total, ZOC is a matrix whose elements are:
(4) |
equals to failure or repair rate of the corresponding subsystem according to the changes between the two OCs (e.g., , , , , , and ).
Besides, different OCs may lead to the same ALR, and the OCs with identical ALR can be merged into an equivalent state. Hence, the OCs of an MS can be aggregated into three equivalent ALR states, including the normal (N) state, the partial failure (PF) state, and the total failure (TF) state, as given in the lower left number of each block in

Fig. 3 Aggregated state space diagram of an MS.
Through the state merging technique given in [
(5) |
(6) |
Taking the transition from state 1 to state 2 in
(7) |
As observed from
Since each MS has three ALR states as given in the last subsection, by combining each state of MS in pairs, a state space diagram of a bipolar HVDC link with nine system states can be obtained, as illustrated in

Fig. 4 State space diagram for a bipolar VSC-HVDC link.
The probability of system state of the bipolar HVDC link can be calculated by solving the differential equations describing the Markov process in the same form as (3) based on the stochastic transitional probability matrix of the bipolar HVDC link ZHVDC built by the state transition rates, e.g., , in
State | Explanation |
---|---|
100% ALR | The system is in normal state with no MS failures |
75% ALR | Partial failure occurs in one MS while the other MS is normal, causing a 25% derating of loading |
50% ALR | One MS totally fails while the other MS is normal, or both MSs partially fail |
25% ALR | Partial failure occurs in one MS, while the other MS is in total failure state |
0% ALR | It is the worst operation state as both MSs totally fail |
In addition to the failure of the HVDC link, the outages of generators may also result in the loss of power supply and cause power imbalance. In this section, the multi-state model to characterize the reduction of power supply and the corresponding probabilities of generation system in either side of the interconnected power system is proposed.
Typically, the reliability performance of generators is evaluated by a two-state model with constant failure and repair rate [
(8) |
(9) |
Meanwhile, the generation systems in the SE or RE AC power systems are usually made up of multiple generators. The outage of each generator will cause a reduction in the output of the whole generation system. For a generation system with NG generators, assuming that a reduction of in the output is caused by NG,F failed generators, the corresponding probability pR for the reduced output is:
(10) |
In this section, the frequency regulation scheme provided by the HVDC link is given. Firstly, the conventional frequency regulation scheme with PFR and SFR models is presented. Then, based on the multi-state models of the HVDC link and generation systems, the power imbalance causing frequency disturbance during system contingencies is obtained. Finally, according to the obtained power imbalance as well as the PFR and SFR models, a frequency support scheme provided by the HVDC link is formulated and applied to different contingencies.
The conventional frequency regulation scheme includes the PFR and SFR models and has been widely adopted to maintain the frequency stability. A block diagram of the conventional frequency regulation scheme is shown in

Fig. 5 Block diagram of conventional frequency regulation scheme.
The model of PFR can be represented in the upper part of
(11) |
The equivalent governor speed regulation and time constants, R and TG, can be evaluated by:
(12) |
(13) |
where Gc is the set of generators. It can be observed that 1/R and TG are the averages of and TG,g weighted by Sg. The equivalent parameters can also be obtained by calculating the corresponding weighted averages of generators. The saturation block of PFR is represented by SatP(x), where the input of the block x is restricted between the downward reserve capacity and upward reserve capacity of PFR.
Meanwhile, SFR is integrated with the PFR, as shown in the lower part of
(14) |
During system contingencies, although the failures of HVDC link may lead to severe power imbalance, the temporary overload capability of normal converters in the HVDC link can still be exploited to compensate for the imbalanced power, and thus, regulate the following frequency deviation [
1) When the ALR of the HVDC link is 75% or higher, the transmission power of the normal converters can be increased by overload rate rk1.
2) When the ALR of the HVDC link decreases to 50% or below, the transmission power of the normal converters can be lifted by another overload rate rk2, which is higher than rk1. Hence, the transmission power increment can be as close as possible to the one with higher ALR.
For instance, assuming that an exemplary system contingency contains a partial failure at the HVDC link and failure of NG,F generators in the RE AC power system, the power imbalance that causes the frequency disturbance can be formulated as:
(15) |
and can be obtained by:
(16) |
Based on the multi-state model of the HVDC link, C contains five equivalent ALR states given in
Since the reliability parameters such as failure rates of AC transmission networks are typically much lower than those of generators, the transmission networks in the SE and RE AC power systems are assumed to be fully reliable in this paper. Besides, the transmission constraints and power flows of SE and RE AC power systems are also not considered to ensure the overall efficacy of the proposed solution, which should be improved in further studies. To mitigate the transmission congestions during system contingencies, the dynamic transmission rating (DTR) system is a potential solution [
Based on the frequency response models of PFR and SFR, a frequency support scheme provided by the HVDC link is formulated to eliminate the frequency deviation caused by the power imbalance during system contingencies. Still using the same system contingency as an example, a diagram of the frequency support scheme with the exemplary system contingency is given in

Fig. 6 Diagram of frequency support scheme with exemplary system contingency.
Specifically, when the system contingency occurs, the power imbalance can be obtained through (15). Then, based on the combined PFR and SFR models of the RE AC power system, the frequency deviation of the RE AC power system is measured. Then, a two-step frequency-voltage control of the HVDC link is activated. In detail, the DC voltage of the connected converter between the HVDC link and RE AC power system is adjusted based on . Furthermore, the DC voltage variation is used to produce an equivalent frequency deviation signal to the SE AC power system. The relationship between frequency deviation and voltage variation during the whole control process is given by [
(17) |
(18) |
The frequency deviation signal is transmitted to the SE AC power system through communication system. Based on , the generators in the SE AC power system will put in their reserves to support the RE AC power system, which is achieved by the PFR and SFR mechanisms of the SE system. The increased output will be transmitted to the RE AC power system through the HVDC link and converted into the transmission power increment by a saturation block , which denotes the transmission limit of the HVDC link. The saturation threshold of is adjusted subject to the ALR of the HVDC link and the temporary overload rate of normal converters, as given in (16).
As observed from the above scheme, by utilizing the temporary overload capability of normal converters, the power imbalance can be compensated by , and the frequency regulation resources of the RE AC power system will be greatly enhanced. For better demonstration, the schematic frequency dynamics of SE and RE AC power systems subject to the exemplary system contingency with and without the proposed frequency support scheme is shown in

Fig. 7 Schematic frequency dynamics of SE and RE AC power systems. (a) SE AC power system. (b) RE AC power system.
In the SE AC power system, there is an excess of power supply due to the power transmission block caused by the HVDC link outage, which leads to a rising tendency of frequency. On the other hand, the falling tendency of frequency in the RE AC power system is quite obvious due to the shortage of power supply. With the proposed frequency support scheme, the frequency dynamics at both ends are modified. Specifically, the characteristics of frequency profile change in many aspects, including the variation sharp rate, the greatest deviation to the nominal, the duration and the abnormal frequency area. Meanwhile, for the system contingencies where the generator outage occurs in the SE AC power system, the support scheme is similar. In such a case, the generators in the RE AC power system will increase their power outputs, and therefore, the HVDC link can reduce its transmission power to relieve the power imbalance in the SE AC power system.
With the proposed frequency support scheme, the system operators can quickly initiate the frequency regulation process when either the SE, RE AC power systems, or the HVDC link suffers from a contingency that fluctuates the system frequency. To describe the whole contingency management process, a schematic diagram is illustrated in

Fig. 8 Schematic diagram of contingency management process.
To evaluate the impact of the HVDC link and the proposed frequency regulation scheme on the reliability performance of the interconnected power system in terms of frequency and energy, four reliability indices are given as follows.
EAFD is used to evaluate the expected duration of abnormal system frequency, which can be represented by the shaded areas in
(19) |
The rate-of-change-of-frequency (RoCoF) and the frequency nadir are the two typical metrics adopted to evaluate the system frequency security [
(20) |
Typically, when the RoCoF surpasses its limit shown in
(21) |
The security indicator (IS) including ISS and ISR can be obtained by:
(22) |
During system contingencies, the PFR and SFR models are adopted to recover the frequency stability. However, the system frequency cannot be restored to the nominal value given in
(23) |
These durations are illustrated in
The regulation ability of the frequency response model is limited by the reserve capacity. In some system contingencies that cause under-frequency events in the RE AC power system, the frequency may not be restored to nominal value due to the shortage of reserve capacity. Under such a circumstance, load shedding is inevitable. EEC is used to evaluate the total energy losses caused by load shedding, which is defined as:
(24) |
(25) |
The proposed system frequency indices can be evaluated by the MCS technique [
Step 1: for the current sample n, generate the state sequences of the HVDC link utilizing its multi-state model described in Section II and the state sequences of the generating units based on their two-state model described in Section III. Then, the system contingency is formulated.
Step 2: perform the frequency regulation scheme developed in Section IV to obtain the regulated frequency dynamics of both SE and RE AC power systems, which is similar to that shown in
Step 3: update the value of reliability indices EAFD, ENFI, AFDE and EEC based on (19)-(25).
Step 4: determine the MCS stopping criteria . Update and go to Steps 1-3 to calculate the reliability indices again until is not satisfied. Otherwise, output the final EAFD, ENFI, AFDE, and EEC indices.
In this section, a modified two-area reliability test system (RTS) is used to illustrate the proposed reliability model [
ALR (%) | Probability | Frequency (occ/year) |
---|---|---|
100 | 0.9870 | 10.7324 |
75 | 0.0096 | 10.3249 |
50 | 0.0034 | 0.5373 |
25 |
3.0215×1 | 0.0353 |
0 |
1.2587×1 |
3.4739×1 |
The control parameters of the frequency response models in SE and RE AC power systems are derived from [
Parameter | Value | Parameter | Value |
---|---|---|---|
R | 20 | FHP | 0.3 |
H | 2864.35 | D | 2.5 |
TG | 0.2 | KAGC | 0.5 |
TCH | 0.3 | TRH | 7 |
rk1 | 15% | rk2 | 20% |
The nominal frequency is 50 Hz, the RoCoF limit is 0.6 Hz/s, and the over- and under-frequency limits are 50.5 and 49.5 Hz, respectively. The primary and secondary reserve capacities are both set to be 10%. rv and rf are 0.8 and 0.7 Hz/V, respectively. The sampling number N for MCS technique is set to be 300000. The units are committed based on power rating in descending order. Besides, all the units are assumed to share the reserve proportionally based on power rating, and the time period is one hour. The following case study is simulated though MATLAB 2016b combined with Simulink toolbox.
In this case, based on the evaluation procedure depicted in Section V-B, the reliability assessment results of the interconnected power system with an HVDC link with different frequency support schemes are given in
Scheme | Area | EAFD (hour/year) | ENFI (occ/year) | AFDE (Hz/year) | EEC (MWh/year) |
---|---|---|---|---|---|
No scheme | RE | 0.15 | 262.5 | 1603.7 | 918.9 |
SE | 0.24 | 262.5 | 2121.3 | 510.0 | |
Total | 0.39 | 525.0 | 3725.0 | 1428.9 | |
Conventional scheme | RE | 0.15 | 262.5 | 1367.9 | 732.6 |
SE | 0.06 | 262.5 | 1459.1 | 362.8 | |
Total | 0.21 | 525.0 | 2827.0 | 1095.4 | |
Proposed scheme | RE | 0.11 | 179.3 | 943.2 | 598.5 |
SE | 0.06 | 179.3 | 1128.7 | 275.5 | |
Total | 0.17 | 358.6 | 2071.9 | 874.0 |
Through comparing the reliability indices with different frequency regulation schemes, it can be observed that by utilizing the local frequency reserve sources, the conventional frequency regulation scheme can enhance the reliability performance for both SE and RE AC power systems. Meanwhile, since the proposed frequency regulation scheme allows the frequency reserve sources to be mutually available through the HVDC link, it reduces the value of the reliability indices further and improves the system reliability performance more significantly.
This case analyzes the effect of frequency response reserve capacity on system reliability. Both the primary and secondary reserve capacities increase from 5% to 15%. The results are shown in

Fig. 9 Effect of reserve capacities on reliability indices. (a) EAFD. (b) ENFI. (c) AFDE. (d) EEC.
In this case, the effect of the different HVDC penetration levels on system reliability is analyzed. Here, the HVDC penetration level refers to the ratio between the installed transmission capacity of the HVDC link and the installed generation capacity of the RE AC power system. Through adding the installed HVDC link transmission capacity, the HVDC penetration level increases from 20% to 40%. The effects of HVDC penetration levels on reliability indices are given in

Fig. 10 Effects of HVDC penetration levels on reliability indices. (a) EAFD. (b) ENFI. (c) AFDE. (d) EEC.
Meanwhile, all the four indices present a sharp rise when the HVDC penetration level is higher than a specific value; while in other intervals, they change quite slowly. For instance, ENFI changes smoothly when the HVDC penetration level is lower than 35% while lifting sharply after the HVDC penetration level exceeds 35%. Such a profile may stem from the fact that when the HVDC penetration level exceeds a critical high value, the existing reserve of the RE AC power system cannot cover the power imbalance caused by the HVDC link outages. Therefore, both the energy losses and frequency deviation of the RE AC power system have a significant increase. Besides, the reduced system inertia also makes the abnormal frequency duration become longer. It can be concluded that the HVDC penetration level should match the existing reserve level to keep a good reliability performance of the whole power system.
This paper develops a multi-state model of an HVDC link by investigating its different ALRs and the corresponding probabilities subject to component outages. A framework to evaluate the reliability of the interconnected power system is established with a novel frequency regulation scheme provided by the HVDC link.
The effects of different HVDC penetration levels and reserve capacities on the system reliability performance in terms of frequency dynamics during its regulation process are analyzed. Simulation results obtained from a modified two-area RTS indicate that the frequency regulation scheme provided by the normal converters of the HVDC link can improve the reliability performance of the interconnected power system. Specifically, it can effectively decrease the abnormal frequency duration, reduce the frequency deviation, and cut down the energy loss compared with the conventional frequency regulation scheme, which should therefore be adopted for frequency regulation during the system contingencies. Besides, it is also found from the simulation results that different kinds of the reserve can lift the frequency reliability of the interconnected system in different aspects. In detail, the primary reserve can better help reduce the duration and deviation of abnormal frequency, while the secondary reserve can better relieve the energy loss during system contingencies. Furthermore, the increase of HVDC penetration level may significantly deteriorate the system reliability if the existing reserve is unable to cover the power imbalance caused by the HVDC link outage. Therefore, more reserve is needed when the transmission capacity of the HVDC link increases.
The above results provide a reference to evaluate the risks of interconnected power systems with the ever-spreading HVDC projects in terms of frequency security at the system planning stage. Besides, the proposed technique can guide the system operators to comprehensively dispatch the reserves in the interconnected power system according to their frequency reliability preference by fully utilizing the overload transmission capability of the HVDC link.
Nomenclature
Symbol | —— | Definition |
---|---|---|
, | —— | Failure and repair rates of series subsystem |
λi, μi | —— | Failure and repair rates of component i |
λCS, μCS | —— | Failure and repair rates of a converter system (CS) |
λTS, μTS | —— | Failure and repair rates of a transmission system (TS) |
λFS, μFS | —— | Failure and repair rates of a filter system (FS) |
λg, μg | —— | Failure and repair rates of a generator |
Ag, Ug | —— | Availability and unavailability of a generator |
Ag,i, Ug,i | —— | Availability and unavailability of normal and failed generators in generation system |
ck | —— | Available loading rate (ALR) state of high-voltage direct current (HVDC) link |
C | —— | Set of potential ALRs |
D | —— | Load damping rate |
FHP | —— | Equivalent high-pressure power fraction of reheat turbine |
—— | Frequency limit | |
ΔfS,t | —— | Frequency deviation of sending-end (SE) AC power system |
ΔfR,t | —— | Frequency deviation of receiving-end (RE) AC power system |
Hg, Sg | —— | Inertia and rated apparent power of generator g |
ISS, ISR | —— | Security indicators of SE and RE AC power systems |
KAGC | —— | Auto generation control (AGC) system gain |
KAGC,i | —— | AGC system gain of generation i |
LSRs | —— | Energy loss in RE AC power systems with system contingency s |
LSRt | —— | Measured energy curtailment in time t |
M, N | —— | Sets of outage combinations (OCs) included in ALR states X and Y |
NG | —— | Sets of all generators |
NG,F | —— | Sets of all failed generators |
NG,C | —— | Sets of committed generators |
P | —— | Vector of state probability |
pi | —— | Probability of OC i obtained from P |
—— | Power reserve value during period t in RE AC power system | |
ΔPRt | —— | Power imbalance value during period t in RE AC power system |
—— | Reduced transmission power | |
—— | Increased transmission power | |
—— | Rated transmission power of HVDC link | |
—— | Probability of system contingency s during period t | |
pX | —— | Probability of equivalent ALR state |
qi→j | —— | Transition rate from OC i to j |
R, TG | —— | Equivalent governor speed regulation and time constants |
Ri, TG,g | —— | Governor speed regulation and time constants of generator g |
—— | Limit of rate-of-change-of-frequency (RoCoF) | |
, | —— | Downward and upward reserve capacities of primary frequency response (PFR) |
rk | —— | Overload rate of HVDC link |
rv, rf | —— | Preset control gain parameters |
S | —— | Set of potential contingencies |
SatP(x) | —— | Saturation block of PFR |
—— | Saturation block denoting ramping capacities of SFR | |
—— | Saturation block denoting reserve capacities of SFR | |
T | —— | Whole period of system contingency |
TCH, TRH | —— | Equivalent steam chest and reheat turbine time constants |
TP | —— | Required time period to stabilize the system frequency by PFR |
, | —— | Durations of frequency abnormal of SE and RE AC power systems |
, | —— | Durations of PFR and SFR models of SE AC power system |
, | —— | Durations of PFR and SFR models of RE AC power system |
ΔVDC,t | —— | DC voltage variation |
ZHVDC | —— | Stochastic transitional probability matrix of bipolar HVDC link |
ZOC | —— | Stochastic transitional probability matrix |
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