The Coordination Control Strategy for Improving Systems Frequency Stability Based on HVDC Quota Power Support

Abstract

This paper aims to improve the recovery characteristics of the receiving grid during DC blocking failures through DC emergency power support, revealing the mechanism of improving the power angle stability of the receiving grid through timely DC power increase and decrease. Based on the transmission pattern of “north-to-south power transmission” in a certain provincial power grid, the actual maximum long-term overload capacity of multiple DC lines is calculated, considering multiple constraints such as the strength of the receiving grid, the stability limit of AC evacuation channels, and DC overload capacity. A multi-DC emergency power support strategy after DC blocking is developed, avoiding the risk of AC transmission channel overload caused by power flow transfer, reducing the impact of large-capacity DC blocking and inappropriate DC power increases on the receiving grid, and achieving coordination between AC and DC systems. The effectiveness of the proposed strategy is verified through simulation.

1. Introduction

The flexible controllability of High Voltage Direct Current (HVDC) transmission has fundamentally altered the dynamic characteristics of AC-DC hybrid grids compared to traditional power systems [1,2]. Taking Jiangsu Power Grid as an example, during the “14th Five-Year Plan” period, Jiangsu Province accounted for approximately 35.5% of the total load in East China, ranking first among the four provinces and one municipality in East China. Jiangsu is a significant load center in the East China Power Grid, characterized by heavy provincial loads, substantial electricity demand, a compact grid structure, and a dense concentration of flexible AC-DC transmission equipment. As multiple DC lines feed into electrically close AC buses, the operational characteristics of the receiving-end AC-DC hybrid grid represented by Jiangsu Power Grid will become even more complex [3,4].

The AC-DC hybrid receiving-end grid has a high dependence on external power sources, with dense UHVDC (Ultra-High Voltage Direct Current) landing points and large DC transmission power. However, conventional LCC-HVDC (Line Commutated Converter—High Voltage Direct Current) systems utilize thyristors, which are semi-controlled power electronic devices with poor disturbance resistance. In the event of large disturbances, such as commutation failures or blockages in LCC-HVDC, it can significantly impact the active and reactive power balance and stability of the receiving-end grid [5]. This impact may not be mitigated solely by the hot standby capacity of the receiving-end grid and the primary and secondary frequency modulation of traditional units [6].

For instance, on 6 November 2013, a single-phase ground fault occurred on a 500 kV transmission line in the East China Power Grid. The line tripped due to unsuccessful reclosing, resulting in two commutation failures in the FuFeng DC system. The active power at the inverter side of the FuFeng DC dropped from 5.238 million kW to 800,000 kW, while the absorbed reactive power increased from 2640 Mvar to 5340 Mvar. On 19 September 2015, a bipolar blockage accident occurred in the JinSu DC, resulting in a power loss of approximately 4.9 million kW. Before the fault, the system frequency was 49.97 Hz. However, due to insufficient primary frequency modulation capacity to meet the actual demand, the system frequency dropped to a minimum of 49.56 Hz approximately 12 s after the fault, creating a significant power deficit [7].

During periods of low load in the receiving-end grid, factors such as fewer synchronized generators, low rotational inertia, poor frequency modulation performance of renewable energy units, and inadequate conventional primary frequency modulation standards are the main reasons for DC blockages that lead to significant frequency drops [8]. On 12 November 2021, a fault caused a trip on a 1000 kV transmission line in the Jiangsu Power Grid, resulting in simultaneous commutation failures on seven DC lines. The system power fluctuation exceeded 10 million kW, severely impacting both the sending and receiving-end AC grids [7]. Therefore, further research is needed on how to utilize healthy DC modulation techniques for power support and AC-DC coordination measures in cases of power deficits in the receiving-end grid caused by commutation failures or blockages in DC systems.

DC systems typically possess a long-term overload capacity of 1.05 to 1.1 times and a short-term overload capacity of 1.2 to 1.5 times for 3 s, while also being highly controllable [8]. Measures such as DC power modulation or DC emergency power support can be employed for transient emergency control during large system disturbances. The fundamental principle of DC power modulation involves integrating a DC power modulation controller into the DC control system. When the system encounters disturbances, by monitoring changes in certain operational parameters of the AC system, the active power transmitted by the DC system or the reactive power exchanged between the converter station and the AC system can be rapidly adjusted. This allows for the absorption or compensation of power in the AC system [9]. Compared to traditional safety and stability control measures such as generators or load shedding, DC power modulation offers faster adjustment speeds and greater adjustment capacity. Additionally, due to the more flexible control of flexible DC transmission technology, which eliminates commutation failures and enables decoupled control of active and reactive power, its participation in emergency power support becomes even more flexible and effective. It can provide reactive power support to weak AC grids without absorbing reactive power when delivering active power support [10,11].

In terms of related research and engineering practice, reference [12] introduces a computational method for determining the maximum support capability of a regional power grid, taking into account DC operating conditions, DC overload capacity, and the dispersal capacity of AC line power flow. Additionally, it proposes a coordinated optimization algorithm for the actual support volume of multiple DC lines, which has been applied in the East China power grid to enhance system frequency stability. Reference [13] employs a model prediction approach to present a frequency active response and support strategy tailored for back-to-back flexible DC interconnection systems, aiming to bolster frequency response support capabilities between asynchronous grids. Reference [14] derives the mechanism of DC power modulation to suppress low-frequency oscillations in AC systems, analyzes key factors influencing DC modulation effectiveness, and designs a DC additional controller based on the test signal method and extended residue ratio index. This approach has been validated in the actual system of the Northwest power grid. Reference [15] examine the limiting factors that affect DC system power support, noting that a stronger sending-end AC system and a weaker receiving-end AC system favor DC power enhancement. Moreover, maintaining the receiving-end grid voltage at a reasonable level enables the DC system to operate at 1.1 times its long-term overload capacity.

In reference [16], an active power modulation strategy based on a sliding-mode, robust controller is proposed. This strategy determines the direction and extent of converter modulation by assessing the effective inertia coefficient index of the converter, and it has been applied in a multi-terminal flexible DC transmission system containing wind turbines. Reference [17] utilizes a Thevenin equivalent parameter estimation method suitable for large perturbations to track the equivalent parameters of the AC-DC hybrid system in real-time. Considering commutation failure, converter capacity, and low-voltage current limiting constraints, it determines the adjustable capacity of DC and establishes a control strategy based on the Bang-bang control principle following system failures. Reference [18] designs additional power controllers for conventional DC and flexible DC, enabling reasonable power distribution in the DC sending-end grid. Focusing on the residual unit reserve and available transmission capacity of each provincial domain as optimization objectives, reference [19] formulates a strategy for allocating power shortages caused by DC locking across regional grids to address frequency offset issues, taking into account factors such as transmission section limits and regional reserve capacity constraints.

With a focus on an asynchronous networking system with multiple flexible DCs running parallel to each other, reference [20] proposes a coordinated control strategy between the dynamic area control error and flexible DC emergency power support. This strategy considers the mutual influence of the AC systems on both sides while addressing key issues such as power flow limit violations or power flow reversals at critical sections. Reference [21] introduces an emergency collaborative control strategy that considers the DC locking process. By determining the priority of DC participation in power support and the sequence of sending-end generator trips and receiving-end load shedding, it establishes emergency control measures to respond to DC locking. To address regional tie-line power fluctuations, reference [22] defines a power adjustment ratio index to quantitatively evaluate the modulation performance of different DCs and optimizes DC power modulator parameters using a particle swarm algorithm. Lastly, reference [23] applies a DC modulation controller to a multi-terminal DC grid and discusses the impact of controller installation location on suppressing regional oscillations.

Although previous studies have delved deeply into the supporting role of DC power for AC systems, they have not fully considered the various operational constraints encountered when increasing DC power in practice. On one hand, the electrical proximity between high-power DC access points and AC transmission channels can cause adverse effects on the AC system when supporting DC power. For instance, when a power shortage occurs in the Sunan power grid due to DC locking, the unbalanced power may shift to cross-river channels, potentially causing some of them to exceed their static stability limits. On the other hand, despite DC transmission theoretically possessing a transient overload capacity of 1.4 pu, its actual limit is often below this theoretical value due to factors such as operational stability and equipment defects. Therefore, this paper aims to address this situation by conducting a detailed analysis of the impact of multi-DC power limits and developing a coordinated multi-HVDC emergency power support strategy tailored to practical operational needs. This strategy is designed to achieve coordinated cooperation between AC and DC systems after system failures, thereby enhancing the frequency stability of the AC system.

2. Power Network Equation with Multi-HVDC System

The DC control system boasts characteristics such as rapid response, flexible and diverse control methods, high precision control, and large adjustment capacity, ensuring the safe and stable operation of the DC system under various working conditions. Additionally, in case of system failures, the DC control system can quickly adjust the active power it transmits and the reactive power absorbed by the converter station. This allows for mutual compensation of power surplus and deficiency in the AC grid, reducing the adverse effects caused by traditional safety and stability control measures such as generators and load shedding. Consequently, it mitigates the impact of unbalanced power generated by severe failures on the AC/DC system, enhancing the emergency support capability of the AC/DC system. Therefore, the control performance of the DC system plays a pivotal role in ensuring the safe and reliable operation of the AC/DC power grid.

The AC/DC hybrid power system network is contracted to include only the generator potential nodes and the DC converter station bus nodes of the studied object, as Figure 1 shows. The system network equation is as follows [24]:

$$\left[\begin{array}{c}{\dot{\mathit{I}}}_{\mathrm{M}}\\ {\dot{\mathit{I}}}_{\mathrm{S}}\\ {\dot{\mathit{I}}}_{\mathrm{A}}\\ {\dot{\mathit{I}}}_{\mathrm{N}}\end{array}\right]=\left[\begin{array}{cccc}{\mathit{Y}}_{\mathrm{MM}}& {\mathit{Y}}_{\mathrm{MA}}& \mathbf{0}& \mathbf{0}\\ {\mathit{Y}}_{\mathrm{SM}}& {\mathit{Y}}_{\mathrm{SS}}& {\mathit{Y}}_{\mathrm{SA}}& \mathbf{0}\\ \mathbf{0}& {\mathit{Y}}_{\mathrm{AS}}& {\mathit{Y}}_{\mathrm{AA}}& {\mathit{Y}}_{\mathrm{AN}}\\ \mathbf{0}& \mathbf{0}& {\mathit{Y}}_{\mathrm{NS}}& {\mathit{Y}}_{\mathrm{NN}}\end{array}\right]\left[\begin{array}{c}{\dot{\mathit{U}}}_{\mathrm{M}}\\ {\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\\ {\dot{\mathit{U}}}_{\mathrm{N}}\end{array}\right]$$

(1)

In the formula: ${\dot{\mathit{I}}}_{\mathrm{M}}$, ${\dot{\mathit{I}}}_{\mathrm{S}}$, ${\dot{\mathit{I}}}_{\mathrm{A}}$, ${\dot{\mathit{I}}}_{\mathrm{N}}$ represents the node injection current; ${\dot{\mathit{U}}}_{\mathrm{M}}$, ${\dot{\mathit{U}}}_{\mathrm{S}}$, ${\dot{\mathit{U}}}_{\mathrm{N}}$ represents the node voltage; subscripts S and A represent the generator potential node sets of group S and group A, respectively; subscripts M and N represent the DC converter station bus node sets at the receiving end and the sending end, respectively.

If the DC system’s transmission power is 0, i.e., ${\dot{\mathit{I}}}_{\mathrm{M}}={\dot{\mathit{I}}}_{\mathrm{N}}=0$, according to Formula (1), the network equation containing only nodes S and A is obtained as:

$$\left[\begin{array}{l}{\dot{\mathit{I}}}_{\mathrm{S}}\\ {\dot{\mathit{I}}}_{\mathrm{A}}\end{array}\right]=\left[\begin{array}{cc}{{\mathit{Y}}^{\prime }}_{\mathrm{SS}}& {\mathit{Y}}_{\mathrm{SA}}\\ {\mathit{Y}}_{\mathrm{AS}}& {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\end{array}\right]\left[\begin{array}{l}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]=\mathit{Y}\left[\begin{array}{l}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]$$

(2)

In the formula: ${{\mathit{Y}}^{\prime }}_{\mathrm{SS}}=-{\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}{\mathit{Y}}_{\mathrm{MA}}+{\mathit{Y}}_{\mathrm{SS}}$; ${{\mathit{Y}}^{\prime }}_{\mathrm{AA}}=-{\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}{\mathit{Y}}_{\mathrm{NS}}+{\mathit{Y}}_{\mathrm{AA}}$; Y is the system nodal admittance matrix when the system is contracted to only include generator potential nodes.

If the transmission power of the DC system is not 0, i.e., ${\dot{\mathit{I}}}_{\mathrm{M}}\ne 0$, and ${\dot{\mathit{I}}}_{\mathrm{N}}\ne 0$, Formula (2) becomes:

$$\left[\begin{array}{c}{\dot{\mathit{I}}}_{\mathrm{S}}+\Delta {\dot{\mathit{I}}}_{\mathrm{MS}}\\ {\dot{\mathit{I}}}_{\mathrm{A}}+\Delta {\dot{\mathit{I}}}_{\mathrm{NA}}\end{array}\right]=\left[\begin{array}{cc}{{\mathit{Y}}^{\prime }}_{\mathrm{SS}}& {\mathit{Y}}_{\mathrm{SA}}\\ {\mathit{Y}}_{\mathrm{AS}}& {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\end{array}\right]\left[\begin{array}{l}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]={\mathit{Y}}^{\prime }\left[\begin{array}{c}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]$$

(3)

In the formula: $\Delta {\dot{\mathit{I}}}_{\mathrm{MS}}=-{\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}{\dot{\mathit{I}}}_{\mathrm{M}}$; $\Delta {\dot{\mathit{I}}}_{\mathrm{NA}}=-{\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}{\dot{\mathit{I}}}_{\mathrm{N}}$.

Considering DC power as a load, and converting the increased injection current at the generator potential node into the grounding admittance of that node, we obtain Formulas (4) and (5):

$$\begin{array}{ll}{\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{SS}}\right]}_{ii}& =-{\displaystyle \frac{\Delta {\dot{I}}_{sm}}{{\dot{V}}_s}}\\ & =-{\displaystyle \frac{1}{{\dot{V}}_s}}{\displaystyle \sum _{m=1}^M\left[{\left(-{\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}{\dot{I}}_m\right]}\\ & =-{\displaystyle \sum _{m=1}^M\left[{\left(-{\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}\frac{P_m\left(1-j{k}_m\right)}{{\dot{V}}_m^{*}{\dot{V}}_s}\right]}\\ & =-{\displaystyle \sum _{m=1}^M\left[{\left(-{\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}\frac{P_m\left(1-j{k}_m\right)}{V_m{V}_s\angle \left({\theta }_s-{\theta }_m\right)}\right]}\\ & ={\displaystyle \sum _{m=1}^M\left[{\left({\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}\frac{P_m\left(1-j{k}_m\right)}{V_m{V}_s}\right]}\end{array}$$

(4)

$$\begin{array}{ll}{\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\right]}_{jj}& =-{\displaystyle \frac{\Delta {\dot{I}}_{an}}{{\dot{V}}_a}}\\ & =-{\displaystyle \frac{1}{{\dot{V}}_a}}{\displaystyle \sum _{n=1}^N\left[{\left(-{\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}{\dot{I}}_n\right]}\\ & =-{\displaystyle \sum _{n=1}^N\left[{\left(-{\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}\frac{-P_n\left(1+j{k}_n\right)}{{\dot{V}}_n^{*}{\dot{V}}_a}\right]}\\ & =-{\displaystyle \sum _{n=1}^N\left[{\left({\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}\frac{P_n\left(1+j{k}_n\right)}{V_m{V}_s\angle \left({\theta }_a-{\theta }_n\right)}\right]}\\ & =-{\displaystyle \sum _{n=1}^N\left[{\left({\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}\frac{P_n\left(1+j{k}_n\right)}{V_n{V}_a}\right]}\end{array}$$

(5)

In the formula: ${\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{SS}}\right]}_{ii}$ and ${\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\right]}_{jj}$ represent the increased grounding admittance at the internal nodes of the generators in groups S and A, respectively; ${\dot{V}}_s$ and ${\dot{V}}_a$ are the increased injection currents at the internal generator nodes of groups S and A, respectively; and are the voltages at the internal generator nodes of groups S and A, respectively; M and N represent the number of DC lines feeding into the receiving end and transmitting from the sending end, respectively; ${\dot{V}}_m$ and ${\dot{V}}_n$ are the AC bus node voltages at the DC converter stations at the receiving and sending ends, respectively; k_m = Q_m/P_m, k_n = Q_n/P_n; P_m and Q_m represent the active power fed into the m-th DC line at the receiving end and the reactive power absorbed by the converter station, respectively; Pn and Qn represent the active power transmitted by the n-th DC line at the sending end and the reactive power absorbed by the converter station, respectively. For DC nodes at the sending end, the load power is ${\tilde{S}}_n=P_n+j{Q}_n$; for DC nodes at the receiving end, the load power is ${\tilde{S}}_m=-P_m+j{Q}_m$. In a two-machine instability mode, θ_s ≈ θ_m, θ_a ≈ θ_n.

From Formulas (4) to (5), we obtain Formulas (6) to (7), where the real and imaginary parts of ${\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{SS}}\right]}_{ii}$ and ${\left[\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\right]}_{jj}$ are expressed as:

$$\{\begin{array}{l}{\left[\Delta {{\mathit{G}}^{\prime }}_{\mathrm{SS}}\right]}_{ii}=\mathrm{Re}\left\{{\displaystyle \sum _{m=1}^M\left[{\left({\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}\frac{P_m\left(1-j{k}_m\right)}{V_m{V}_s}\right]}\right\}\\ {\left[\Delta {{\mathit{B}}^{\prime }}_{\mathrm{SS}}\right]}_{ii}=\mathrm{In}\left\{{\displaystyle \sum _{m=1}^M\left[{\left({\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}\frac{P_m\left(1-j{k}_m\right)}{V_m{V}_s}\right]}\right\}\end{array}$$

(6)

$$\{\begin{array}{l}{\left[\Delta {{\mathit{G}}^{\prime }}_{\mathrm{AA}}\right]}_{jj}=\mathrm{Re}\left\{-{\displaystyle \sum _{n=1}^N\left[{\left({\mathit{Y}}_{AN}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}\frac{P_n\left(1+j{k}_n\right)}{V_n{V}_a}\right]}\right\}\\ {\left[\Delta {{\mathit{B}}^{\prime }}_{\mathrm{AA}}\right]}_{jj}=\mathrm{In}\left\{-{\displaystyle \sum _{n=1}^N\left[{\left({\mathit{Y}}_{AN}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}\frac{P_n\left(1+j{k}_n\right)}{V_n{V}_a}\right]}\right\}\end{array}$$

(7)

In other words, after considering the transmission power of DC, additional grounding access should be added to the generator potential nodes. Assuming that the matrices Y_SM and Y_MM in Formula (4) and the matrices Y_AN and Y_NN in Formula (5) do not consider grounding branches, and all system branches are purely inductive, then the elements in the matrices Y_SM and Y_AN are all positive imaginary numbers. Since the DC converter station buses are generally not directly connected, the admittance matrices Y_MM and Y_NN are both diagonal matrices, and their diagonal elements are all negative imaginary numbers. Thus, the diagonal elements of the matrices ${\mathit{Y}}_{\mathrm{MM}}^{-1}$ and ${\mathit{Y}}_{\mathrm{NN}}^{-1}$ are all positive imaginary numbers, i.e.,:

$$r_{sm}={\left({\mathit{Y}}_{\mathrm{SM}}{\mathit{Y}}_{\mathrm{MM}}^{-1}\right)}_{sm}<0$$

(8)

$$s_{an}={\left({\mathit{Y}}_{\mathrm{AN}}{\mathit{Y}}_{\mathrm{NN}}^{-1}\right)}_{an}<0$$

(9)

The network equation described by Formula (2) becomes:

$$\left[\begin{array}{l}{\dot{\mathit{I}}}_{\mathrm{S}}\\ {\dot{\mathit{I}}}_{\mathrm{A}}\end{array}\right]=\left[\begin{array}{cc}{{\mathit{Y}}^{\prime }}_{\mathrm{SS}}+\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{SS}}& {\mathit{Y}}_{\mathrm{SA}}\\ {\mathit{Y}}_{\mathrm{AS}}& {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}+\Delta {{\mathit{Y}}^{\prime }}_{\mathrm{AA}}\end{array}\right]\left[\begin{array}{l}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]={\mathit{Y}}^{″}\left[\begin{array}{l}{\dot{\mathit{U}}}_{\mathrm{S}}\\ {\dot{\mathit{U}}}_{\mathrm{A}}\end{array}\right]$$

(10)

3. Extended Equal-Area Criteria (EEAC) Theory Considering Multi-HVDC Emergency Power Support

Based on the previous definition, the rotor motion equation of the generator can be written as:

$$\{\begin{array}{l}\delta =\omega -{\omega }_0=\Delta \omega \\ T_{\mathrm{J}}p\Delta \omega =P_{\mathrm{m}}-P_{\mathrm{e}}\end{array}$$

(11)

The equivalent angles and equations of motion for groups S and A in the inertia center coordinate system are expressed by Formulas (12) and (13), respectively:

$$\{\begin{array}{l}{\delta }_{\mathrm{S}}={\displaystyle \frac{1}{T_{\mathrm{JS}}}}{\displaystyle \sum _{i\in \mathrm{S}}{T}_{\mathrm{J}i}{\delta }_i}\hfill \\ {\omega }_{\mathrm{S}}={\displaystyle \frac{1}{T_{\mathrm{JS}}}}{\displaystyle \sum _{i\in \mathrm{S}}{T}_{\mathrm{J}i}{\omega }_i}\hfill \\ \begin{array}{l}\Delta {\omega }_{\mathrm{S}}={\omega }_{\mathrm{S}}-{\omega }_0\\ T_{\mathrm{JS}}={\displaystyle \sum _{i\in \mathrm{S}}{T}_{\mathrm{J}i}}\end{array}\hfill \\ T_{\mathrm{JS}}p\Delta {\omega }_{\mathrm{S}}={\displaystyle \sum _{i\in \mathrm{S}}\left(P_{\mathrm{m}i}-P_{\mathrm{e}i}\right)}\hfill \end{array}$$

(12)

In the formula: δ_S represents the equivalent power angle of group S; ω_S represents the equivalent angular velocity of group S; T_Ji represents the inertia time constant of the ith generator in group S; P_mi and P_ei represent the mechanical power and electromagnetic power of the ith generator in group S, respectively.

$$\{\begin{array}{l}{\delta }_{\mathrm{A}}={\displaystyle \frac{1}{T_{\mathrm{JA}}}}{\displaystyle \sum _{j\in \mathrm{A}}{T}_{\mathrm{J}j}{\delta }_j}\hfill \\ {\omega }_{\mathrm{A}}={\displaystyle \frac{1}{T_{\mathrm{JA}}}}{\displaystyle \sum _{j\in \mathrm{A}}{T}_{\mathrm{J}j}{\omega }_j}\hfill \\ \begin{array}{l}\Delta {\omega }_{\mathrm{A}}={\omega }_{\mathrm{A}}-{\omega }_0\\ T_{\mathrm{JA}}={\displaystyle \sum _{j\in \mathrm{A}}{T}_{\mathrm{J}j}}\end{array}\hfill \\ T_{\mathrm{JA}}p\Delta {\omega }_{\mathrm{A}}={\displaystyle \sum _{j\in \mathrm{A}}\left(P_{\mathrm{m}j}-P_{\mathrm{e}j}\right)}\hfill \end{array}$$

(13)

In the formula: δ_A represents the equivalent power angle of group A; ω_A represents the equivalent angular velocity of group A; T_Jj represents the inertia time constant of the jth generator in group A; P_mj and P_ej represent the mechanical power and electromagnetic power of the jth generator in group A, respectively.

Based on this, the power angle stability of the entire system becomes a relative stability problem between the two complementary groups, S and A. In the literature [22], the DC sending-end generator group is considered group S, but in this paper, the DC receiving-end generator group is regarded as group S, and the sending-end generator group is regarded as group A. Considering DC feed-in power as equivalent mechanical power, the power angle changes of the DC receiving-end units are analyzed accordingly. For the system described by Formula (1), letting the single-machine rotor angle δ = δ_S − δ_A, the motion equation of its equivalent single-machine infinite bus system is:

$$T_{\mathrm{J}}{\displaystyle \frac{{\mathrm{d}}^2\delta }{\mathrm{d}{t}^2}}=P_{\mathrm{m}}-P_{\mathrm{e}}=P_{\mathrm{m}}+P_{\mathrm{c}}-P_{\mathrm{em}}\sin \left(\delta -v\right)$$

(14)

In the formula:

$$\{\begin{array}{l}{T}_{\mathrm{J}}={\displaystyle \frac{T_{\mathrm{JA}}{T}_{\mathrm{JS}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}\hfill \\ P_{\mathrm{m}}={\displaystyle \frac{1}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}\left(T_{\mathrm{JS}}{\displaystyle \sum _{j\in \mathrm{A}}{P}_{\mathrm{m}j}}-T_{\mathrm{JA}}{\displaystyle \sum _{i\in \mathrm{S}}{P}_{\mathrm{m}i}}\right)\hfill \\ P_{\mathrm{e}}=P_{\mathrm{c}}+P_{\mathrm{em}}\sin \left(\delta -v\right)\hfill \\ P_{\mathrm{c}}={\displaystyle \frac{T_{\mathrm{JS}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{j\in \mathrm{A}}{\displaystyle \sum _{l\in \mathrm{A}}{U}_j{U}_l{G}_{jl}}}-{\displaystyle \frac{T_{\mathrm{JA}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{i\in \mathrm{S}}{\displaystyle \sum _{k\in \mathrm{S}}{U}_i{U}_k{G}_{ik}}}\hfill \\ P_{\mathrm{em}}=\sqrt{C^2+D^2}\hfill \\ \begin{array}{l}v=-\arctan {\displaystyle \frac{C}{D}}\\ C={\displaystyle \frac{T_{\mathrm{JS}}-T_{\mathrm{JA}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{j\in A}{\displaystyle \sum _{i\in \mathrm{S}}{U}_i{U}_j{G}_{ij}}}\\ D={\displaystyle \sum _{i\in \mathrm{S}}{\displaystyle \sum _{j\in \mathrm{A}}{U}_i{U}_j{B}_{ij}}}\end{array}\hfill \end{array}$$

(15)

After considering the transmission power of the DC system, Formula (14) needs to be revised. Changes in DC power will not affect the values of P_em and v, but they will cause changes in P_c. The expression for P_c is rewritten as:

$$\begin{array}{ll}{P}_{\mathrm{c}}=& {\displaystyle \frac{T_{\mathrm{JS}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{j\in \mathrm{A}}{\displaystyle \sum _{l\in \mathrm{A}}{U}_j{U}_l{G}_{jl}}}-{\displaystyle \frac{T_{\mathrm{JA}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{i\in \mathrm{S}}{\displaystyle \sum _{k\in \mathrm{S}}{U}_i{U}_k{G}_{ik}}}-\\ & {\displaystyle \frac{T_{\mathrm{JS}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{j\in \mathrm{A}}\left[V_j{\displaystyle \sum _{m=1}^M\left(\frac{r_{sm}{P}_m}{V_m}\right)}\right]}-{\displaystyle \frac{T_{\mathrm{JA}}}{T_{\mathrm{JA}}+T_{\mathrm{JS}}}}{\displaystyle \sum _{i\in \mathrm{S}}\left[V_i{\displaystyle \sum _{n=1}^N\left(\frac{s_{an}{P}_n}{V_n}\right)}\right]}\end{array}$$

(16)

Since r_sm < 0 and s_an < 0, according to Formula (14), when the DC power increases, that is, when P_m and P_n increase, it will cause P_c to increase; conversely, P_c will decrease.

Based on the above conclusions, the principle of using emergency power support from healthy DC lines to improve power angle stability under DC blocking faults is analyzed using the changes in acceleration and deceleration areas of the equal-area criterion, as shown in Figure 1.

In Figure 2, DC power is equivalent to the mechanical power injected into the system, so adjusting DC power is equivalent to adjusting the mechanical power of the prime mover. In Figure 2a, the equivalent mechanical power output by the prime mover in the system is P_m + P_c0, and point a is the normal operating point where the generator operates at a power angle of δ₀. When a DC blocking fault occurs, the DC power decreases, causing P_c to decrease, and the equivalent mechanical power output by the prime mover becomes P_m + P_c1. Since the electromagnetic power is greater than the equivalent mechanical power, it causes the generator rotor to decelerate, moving the operating point from a to c. The deceleration area is represented by the curved triangle a-b-c. Upon reaching point c, the rotor begins to accelerate. If DC emergency power support measures are not taken, the acceleration area will be the curved triangle c-h-g, and the minimum power angle of the generator’s first swing will be δ₃. If emergency power support from healthy DC lines is implemented (supposedly at point c), the equivalent mechanical power output by the prime mover becomes P_m + P_c2. In this case, the minimum power angle of the generator’s first swing changes to δ₂, and the acceleration area is enclosed by c-d-e-f.

During the power angle swing back, as shown in Figure 2b, if the DC operating power is not changed, the acceleration area will be the curved triangle i-j-k, and the maximum power angle of the generator’s swing back will be δ₆. If DC power reduction measures are taken during the power angle swing back, the acceleration area will be enclosed by i-j-l-m, and the minimum power angle of the generator’s swing back will become δ₅.

Based on the above analysis, in a multi-DC hybrid system, from the perspective of improving the system’s power angle stability, after a DC lockout fault occurs, it is advisable to simultaneously increase the active power transmitted by multiple DC transmissions during the initial swing phase of the generator power angle. This maximizes the acceleration area of the forward swing and minimizes the amplitude of the initial power angle swing. Additionally, during the power angle return swing, it is appropriate to moderately reduce the DC operating power to decrease the acceleration area of the return swing, achieving a reduction in the amplitude of the return swing and ultimately enhancing the power angle stability of the system after a fault.

4. The HVDC Power Support Quota Impact Factor Analysis

When implementing DC power support measures, it is necessary to consider the impact of DC power increases on the AC grid, as improper DC power increases can cause secondary impacts on the AC grid. Since the long-term overload time of DC may reach 1 to 2 h after adopting DC power support, it is necessary to ensure that the basic operating indicators of the AC grid remain within the allowable range during the long-term overload operation of DC. This article mainly calculates the actual long-term overload capacity of each DC circuit based on constraints such as the strength of the receiving-end AC system and the stability limit of AC lines used to disperse DC power.

To align with practical engineering, this article establishes a model of a provincial power grid in China within the ADPSS Simulator, a hybrid electromagnetic-electromechanical simulation program for power systems. This model incorporates the entire province’s power system as well as portions of adjacent provincial power systems represented as equivalent networks. The model comprises 3591 stations, 866 generators, 11,537 buses, 3301 transformers, 10,006 AC lines, 10 DC lines, and 3886 loads. All DC lines shown in Figure 3 utilize electromagnetic transient models to guarantee accuracy.

4.1. The AC System Strength Impact and DC Support Quota Determination Method

Taking JS HVDC as an example, without considering the theoretical value of long-term overload capacity, we aim to determine the actual long-term overload allowance value of DC constrained by the strength of the receiving-end AC system. Under normal operating conditions, the firing angle at the rectifier side of JS HVDC is 17°. In a time-domain simulation, the DC power is increased by continuously and equally increasing the reference value of the DC current.

Table 1. Simulation results of the maximum long-term overload capacity of JinSu DC constrained by AC system strength.

Increase Times	Current Orders/kA	Active Powers/MW	AC Voltages of Receiving End AC System/p.u.	Firing Angles/°
1	4.50	7200.0	0.9721	17.00
2	4.55	7286.5	0.9687	16.05
3	4.60	7359.7	0.9653	15.35
4	4.65	7444.5	0.9641	14.56
5	4.70	7511.2	0.9620	13.89
6	4.75	7608.1	0.9598	12.56
7	4.80	7670.3	0.9580	11.72
8	4.85	7759.1	0.9561	10.41
9	4.90	7836.9	0.9535	9.46
10	4.95	7930.8	0.9513	8.27
11	5.00	8003.0	0.9493	7.33

shows the changes in the firing angle at the rectifier side and the AC bus voltage at the receiving-end converter station during this process. When the AC bus voltage at the receiving end falls below the minimum allowable value of 0.95 p.u. or the firing angle reaches the minimum value of 5°, the current command value is no longer increased.

Based on the simulation results listed in Table 1, since increasing DC power requires absorbing a large amount of reactive power from the AC system, considering only the constraint of AC voltage level, the maximum long-term overload capacity of JS HVDC is approximately 800 MW.

Using the same simulation calculation method, the maximum long-term overload capacity ΔP_d,max1 of each DC circuit constrained by the strength of the AC system is obtained, as shown in

Table 2. Maximum long-term overload capacity of each HVDC constrained by AC system strength.

HVDC	Maximum Long-Term Overload Capacity/MW
LZ	580
JS	800
JG (Monopole)	1030
XT	1070
YH	900
YZ	770

.

4.2. AC Line Stability Limits Impact Analysis

When increasing DC power in emergency situations, it is necessary to consider the constraints on the power flow evacuation capacity of the receiving-end AC evacuation corridors to avoid long-term over-limit operation of the receiving-end AC lines caused by DC power increases. The AC evacuation corridors in this paper denote the nearby AC lines that transfer the active power brought by the HVDC system. The stability limit of AC lines is determined by both the thermal stability performance of the AC lines and the transient stability performance of the system. Therefore, during the process of emergency DC power increases, it should be ensured that the power flow in the AC evacuation corridors of the receiving-end grid does not exceed its stability limit after the DC power increase, to avoid unavoidable losses to the grid due to long-term over-limit operation of AC lines.

To analyze the impact of different DC transmission powers on the power flow decentralization of AC evacuation corridors at the DC receiving end, the power flow decentralization sensitivity index is introduced. As shown in Figure 4, let P_ij denote the power flow in a certain AC evacuation corridor ij before DC power adjustment, and let P_ij’ represent the power flow in the same corridor after the DC power change. Given a DC power variation of ΔP_d, the power flow decentralization sensitivity δ_ij of this DC with respect to the AC evacuation corridor ij can be defined as:

$${\delta }_{ij}={\displaystyle \frac{{P^{\prime }}_{ij}-P_{ij}}{\Delta P_{\mathrm{d}}}}$$

(17)

Generally, δ_ij is related to the grid structure of the power network. Assuming the grid structure remains unchanged and variations in load, generator operation, and network losses are not considered, δij is typically a constant value.

If the stability limit of the AC evacuation corridor is known as P_ij,max, then the maximum power increase ΔP_d,ij,max for this DC circuit, constrained by the stability limit of the AC evacuation corridor, can be calculated as:

$$\Delta P_{\mathrm{d},ij,\max }={\displaystyle \frac{P_{ij,\max }-P_{ij}}{{\delta }_{ij}}}$$

(18)

Since multiple AC evacuation corridors play a role in decentralizing the power flow from DC feed-in, when calculating the maximum long-term overload capacity ΔP_d,max2 of a single DC circuit constrained by the stability limits of multiple AC evacuation corridors, the minimum value among the maximum long-term overload capacities calculated based on the stability limits of each AC evacuation corridor should be taken. That is:

$$\Delta P_{\mathrm{d},\max 2}=\min \left\{\Delta P_{\mathrm{d},ij,\max },\Delta P_{\mathrm{d},ik,\max },\Delta P_{\mathrm{d},il,\max },\cdots \right\}$$

(19)

Calculating the maximum long-term overload capacity using the YH HVDC as an example: The power constraint imposed by the HA-SCW four-line is (650–600)/0.77 = 650,000 kW, and the power constraint imposed by the HA-AL four-line is (350–180.6)/0.23 = 7,365,000 kW. Therefore, limited by the stability limit of the HA-SCW four-line, the maximum power increase for YH DC is 650,000 kW. Using the same method, the maximum long-term overload capacity values for various DC circuits in this province, constrained by the stability limits of AC evacuation corridors, are calculated and presented in

Table 3. Maximum long-term overload capacity of each HVDC constrained by the stability limits of AC evacuation corridors.

HVDC	Maximum Long-Term Overload Capacity/MW
LZ	820
JS	660
JG (Monopole)	1560
XT	1200
YH	650
YZ	530

.

4.3. Actual Values of the Maximum Long-Term Overload Capacity for Each HVDC

Taking into account the above two constraints and combining them with the theoretical value of DC long-term overload capacity of 1.1 p.u., the actual value of DC long-term maximum overload capacity ΔP_d,max is calculated as:

$$\Delta P_{\mathrm{d},\max }=\min \left\{\Delta P_{\mathrm{d},\max 0},\Delta P_{\mathrm{d},\max 1},\Delta P_{\mathrm{d},\max 2}\right\}$$

(20)

where ΔP_d,max0 represents the theoretical value of DC long-term overload capacity. Based on Table 1, Table 2 and Table 3, the actual short-term and long-term overload capacities for each DC circuit are obtained and presented in

Table 4. Actual short-term and long-term overload capacities for each HVDC.

HVDC	Maximum Short-Term Overload Capacity/MW	Maximum Long-Term Overload Capacity/MW
LZ	900	300
JS	2160	660
JG (Monopole)	1200	400
XT	3000	1000
YH	2400	650
YZ	360	120

.

5. Simulation Verifications

To verify the above analysis, the simulations are implemented in this section. The AC grid used in the simulation adopts a power system network covering the entire province and neighboring provinces, which is modeled in the electromechanical-electromagnetic hybrid simulation ADPSS of the power system. Among them, generators, DC projects, transformers, and lines of 220 kV and above are consistent with actual projects, and the network below 220 kV is equivalent to a load model. The model contains 3591 stations, 11,537 nodes, 10,006 AC lines, 1565 two-winding transformers, 1736 three-winding transformers, 1266 generators, 3886 loads, and 9 DC lines.

The systems used for the case study are shown in Figure 5. As this system is so large and it is hard to present the total topology, we supply the part of the topology as shown in Figure 4, where the inverters of YH HVDC, XT HVDC, and YZ HVDC and nearby 500 kV grids are included.

Assuming that the voltage level of the AC system at the DC sending end can ensure the required power increase for each DC circuit, when a DC locking fault occurs, from the perspective of improving the power angle stability of the receiving end grid, according to the data listed in Table 4, the short-term overload capacity of the non-faulted DC should be fully utilized first to increase the transmission power of each DC circuit as much as possible, in order to increase the maximum possible acceleration area of the first swing and reduce the swing amplitude of the power angle. When the power angle of the unit swings back, power reduction measures are taken for each DC circuit, utilizing the long-term overload capacity of each DC circuit to reduce the maximum possible acceleration area of subsequent swings, thereby improving the power angle stability of the system.

The control diagram of each HVDC is presented in Figure 6. Here, the low-pass filter is used to filter the noise and the washout component is used to ensure the control will act only if the frequency is fluctuating. The subscript 1 to n of each ω_LP and ω_D denote different HVDC’s parameters. Especially, the hard limiter ΔP_maxn and ΔP_minn of the controller has considered the short-term and long-term overload capacities for each HVDC, which is according to Table 4’s results.

The input signals of the coordinated control are unified as frequency fluctuations Δf, and the outputs of the controls are the active power ΔP_d of each HVDC.

Due to the overflow of the river crossing channel caused by power flow transfer, namely the overload of the AC lines crossing Changjiang River, and SCW-LWS and FC-ML serving as secondary power flow decentralization channels for YH DC and XT DC, respectively, these two DC circuits may further exacerbate the overload risk of the river crossing channel during the power increase process. Therefore, the power control strategy for the provincial YZ DC is the same as that for the UHV DC. The DC power is first increased to the short-term overload capacity value. When the UHV DC power is reduced, the YZ DC power is simultaneously reduced to the long-term overload value. Here, YZ DC adopts the controllable-line-commutated-converter (CLCC) topology to eliminate commutation failures.

According to the above strategy, a comparison of the load angle difference response curves between the HS power plant and the YZ power plant without and with the DC power support strategy is shown in Figure 7. As can be seen from the figure, taking appropriate emergency power increase and reduction measures after a DC locking fault can significantly suppress the swing amplitude of the power angle difference between the units on the south and north of the river, improving the transient stability margin of the system’s load angle.

The comparison charts of active power response for the three river-crossing channels, namely QT-QH, SCW-LWS, and FC-ML, without and with the DC power support strategy are shown in Figure 8, Figure 9, and Figure 10, respectively. According to the charts, compared to not adopting the power support strategy, implementing appropriate emergency power supports and reductions after a DC locking fault not only improves the swing amplitude of the load angle difference between units on both sides of the river but also suppresses power fluctuations in the river-crossing channels. The active power of the three river-crossing channels is 2.94 million kW, 2.78 million kW, and 2.82 million kW, respectively, all below the stability limit of 3 million kW, indicating that the risk of exceeding the limit in the river-crossing channels has been mitigated. This improvement is due, on one hand, to the power support from the South Jiangsu DC, which compensates for the power shortage in the South Jiangsu power grid, and, on the other hand, to the power increase of the YZ DC, which shares the active power load with the parallel AC channels.

The comparison of voltage curves for a 220 kV DS busbar and a 500 kV DS busbar, without and with a DC power support strategy, is shown in Figure 11 and Figure 12, respectively. According to the analysis of the charts, due to the emergency power supports of multiple DC lines, the surplus reactive power in the nearby grid at the receiving end of the locked DC is absorbed, bringing the voltage level of the receiving grid closer to the rated value.

Therefore, adopting a coordinated DC power support strategy not only compensates for the power shortage in the receiving grid but also achieves coordination between AC and DC systems. This improves the load angle stability of the receiving grid, mitigates the risk of power flow exceeding limits in AC river-crossing channels caused by unbalanced power, and maintains the voltage level of the receiving grid within an acceptable range.

To further verify the effectiveness of the proposed strategy, under a certain operating mode in the summer, a unipolar blocking fault at JS DC was set. The comparison curves of the transmission power of the SCW-LW line and the FC-ML line before and after DC power support are shown in Figure 13 and Figure 14. As can be seen from the figure, the coordination strategy of multi-DC power support and Yangzhou-Zhenjiang DC can significantly reduce the transmission burden of the river-crossing channel, avoid the risk of AC channel overload, and at the same time make up for the power shortage of the fault DC receiver near-area power grid and improve the power angle stability of the system.

6. Conclusions

This article proposes a coordinated control strategy for multi-HVDC emergency power support considering DC power support quotas, aiming to address the stability issues in provincial power grids caused by DC locking failures. This provides a reference for emergency response plans in power grids under severe faults. The main conclusions are as follows:

(1)

For the actual long-term overload capacity of HVDC under emergency control conditions, this study fully considers multiple constraints, such as the impact of DC power supports on the strength of the receiving AC system, the stability limits of AC evacuation channels, and the theoretical value of the DC line’s overload capacity. Through this comprehensive analysis, the actual maximum long-term overload capacity of DC is determined, avoiding adverse effects on the receiving grid caused by inappropriate DC power support.

(2)

After a DC locking failure occurs, the first step is to maximize the active power of multiple healthy DC lines by utilizing their short-term overload capacity. This aims to compensate for the power shortage at the receiving end, increase the maximum possible acceleration area during the first swing, and suppress the swing amplitude of the load angle. During the load angle swingback, appropriate power reduction measures are taken for each DC line, utilizing their long-term overload capacity. This aims to reduce the maximum possible acceleration area during subsequent swings, thereby improving the overall load angle stability of the system and reducing the probability of grid desynchronization.

(3)

Based on the proposed method, the simulation results prove that the stability of the system can be enhanced. The difference between the HS power plant and the YZ power plant can be reduced by 3.6 degrees after a DC blocking fault. Additionally, the over voltages at 500 kV line DS can also be decreased by 2.06% after a DC blocking fault.

(4)

For practical power system operation, the power shortage of the multi-DC-fed receiving grid should be orderly allocated to the remaining non-fault DC, and the AC cross-river channel power flow over-limit risk caused by the emergency power increase of UHV DC can be suppressed by adjusting the YZ DC operating power, which can effectively solve the system stability problem in the power shortage state.

As the main topic of this paper concentrates on the coordination strategy between HVDCs, the virtual inertia control strategy of renewable power is not considered because of the limited number of pages. In future work, the coordination strategy between HVDC and renewable power can be improved to further enhance the frequency stability of the power system.