An Evaluation Method of Renewable Energy Resources’ Penetration Capacity of an AC-DC Hybrid Grid

Abstract

With increasingly more renewable energy being integrated into the AC-DC hybrid grid, the grid shows more complex dynamic characteristics due to the mutual coupling of HVDC and renewable energy. To evaluate the renewable energy resources’ penetration capacity of the AC-DC hybrid grid, this paper proposes an evaluation method of the renewable energy resources’ penetration capacity of an AC-DC hybrid grid, which considers both economy and safety. Firstly, indicators are proposed for an evaluation of the economy and safety of the AC-DC hybrid grid integrated with renewable energy, where both static and transient stability indicators are considered. Secondly, to maximize the renewable energy penetration capacity and minimize the network loss, an optimization model of the renewable energy penetration capacity of the AC-DC hybrid grid is established considering the static and transient stability constraints. Then, a heuristic solution method for solving the renewable energy penetration capacity optimization model is proposed. Finally, based on the improved IEEE 39 node system, a case analysis is carried out. The simulation results verify the correctness and effectiveness of the proposed method.

1. Introduction

To cope with global climate change, many countries are actively promoting the transformation to low-carbon, clean, and sustainable energy systems. For example, China has put forward the grand goal of “achieving carbon peak by 2030 and carbon neutrality by 2060” and has set the construction of a new power system with renewable energy as the main development direction in the next step. In the future power system, a large number of renewable energy generation technologies, such as wind turbines and photovoltaics, will be integrated into the power grid and become the main energy form on the power supply side. As the core hub for absorbing large-scale renewable energy, the power grid will make full use of the advantages of HVDC transmission in terms of the transmission distance, transmission capacity, transmission flexibility, transmission cost, etc., and form a future scene where massive renewable energy is integrated into the AC-DC hybrid grid [1].

With the continuous expansion of the scale of renewable energy integrated into the AC-DC hybrid grid, the grid shows significant features, such as reduced system rotational inertia and weakened frequency regulation [2]. When the power grid is disturbed or faulted, the transient frequency of the power system can easily exceed the limit. In addition, the mutual coupling characteristics of DC and renewable energy results in the power system having a more complex transient process, and it is very easy to cause the cascading fault of multi circuit DC commutation failure and renewable energy off grid. Therefore, to ensure the safe and stable operation of the power grid, research on the maximum accessible renewable energy generation capacity in the AC-DC hybrid grid, i.e., the renewable energy penetration capacity, is of great significance for the development of power systems in the future [3].

In the existing literature, the renewable energy penetration capacity of the power grid is defined as the maximum renewable energy generation capacity that can be integrated into the power grid under various constraints and operation modes, which is also known as the penetration limit power, penetration capacity, maximum permeability, grid connection limit [4,5,6,7,8,9], etc. Based on the Local Thevenin Index (LTI) technology of Wide Area Monitoring System (WAMS), the authors of [4] analyzed the impact of the renewable energy penetration capacity on the voltage stability of a large-scale power grid, which is helpful for determining the best location for integrating a wind farm into an existing power system. The authors of [5] used the active power-voltage analysis method to determine the static voltage stability constraints, and then determined the maximum penetration capacity of renewable energy generation. On the basis of static security constraints, the authors of [6] established a detailed mathematical model of the transmission power constraints based on DC power flow and put forward a method to calculate the capacity of large-scale wind farms integrated into a power system. The authors of [7] established the optimal access point and maximum penetration capacity optimization model of renewable energy based on the constraints of the line ampacity, transformer capacity, segmented active power, and node voltage, and proposed an intelligent optimization algorithm to effectively solve the model. The authors of [8,9] proposed an evaluation model of the maximum wind power penetration capacity considering the system’s static voltage stability. When evaluating the renewable energy penetration capacity of the power grid, the above literature has only considered the static security and stability of the power grid and did not consider the impact of integrated renewable energy on the transient stability of the power grid.

For this problem, some literature has considered the transient stability of the system when evaluating the penetration capacity of renewable energy. Based on the input state stability (ISS) theory, the authors of [10] constructed the power grid transient security and stability indicator system, and proposed a method for quantitative evaluation of the renewable energy penetration capacity. The authors of [11,12,13,14] considered the transient frequency stability of the system after renewable energy integration, and proposed their own characteristics in the indicator design, model establishment, and solution methods. Based on the actual power grid model, the authors of [15] analyzed the impact of the access location on the transient voltage stability and renewable energy consumption capacity of the power grid. The authors of [16] proposed a high-precision real-time transient instability detection (TID) method to analyze the impact of the wind power penetration capacity on the power grid transient stability. The above literature proposed an evaluation method for the renewable energy penetration capacity considering the transient stability of the power grid but failed to consider the static and transient stability of the system at the same time nor the interaction between DC and renewable energy.

In summary, the shortcomings of the existing research on the evaluation of the renewable energy penetration capacity of the power grid are mainly reflected in two aspects: Firstly, the evaluation of the renewable energy penetration capacity is not comprehensive, where the static and transient stability of the power grid have not been considered simultaneously. With the increasing proportion of renewable energy in the power system, renewable energy will have a great impact on the static and transient stability of the power grid. Therefore, both static and transient stability constraints need to be considered when evaluating the renewable energy penetration capacity of the power grid. Secondly, the existing research has not studied the system characteristics of the AC-DC hybrid grid when it is integrated with a large number of renewable energy technologies, and the mutual coupling between DC and renewable energy has not been considered when evaluating the renewable energy penetration capacity of the power grid. Therefore, the renewable energy penetration capacity evaluation results are not fully applicable to the AC-DC hybrid grid.

To overcome the shortcomings of existing research, this paper proposes an evaluation method for the renewable energy penetration capacity of the AC-DC hybrid grid considering the static and transient stability constraints.

The contributions of this paper are summarized as follows:

• Considering the interaction between DC and new energy sources, the static and transient stability of the system are evaluated.
• Indicators for evaluating the economy and safety of the AC-DC hybrid grid integrated with renewable energy are proposed, where both static and transient stability indicators are considered.
• An optimization model of the renewable energy penetration capacity of the AC-DC hybrid grid is constructed to maximize the renewable energy penetration capacity and minimize the network loss.
• A heuristic solution method for solving the renewable energy penetration capacity optimization model is proposed.

This paper is organized as follows: Section 2 describes the economic and security evaluation indicators of the AC-DC hybrid grid integrated with renewable energy. Then, the renewable energy penetration capacity evaluation model of the AC-DC hybrid grid is constructed in Section 3. Section 4 proposes a solution method for the renewable energy penetration capacity evaluation model. A case is analyzed using the IEEE 39 node system to verify the correctness and effectiveness of the algorithm in Section 5. Finally, the conclusion of this work is given in Section 6.

2. Evaluation Indicators of the AC-DC Hybrid Grid Integrated with Renewable Energy

To evaluate the impact of renewable energy integration on the AC-DC hybrid grid, this paper constructs an evaluation indicator system considering both economy and security. In the indicator system, the most important indicators are the static voltage stability indicator (network loss sensitivity), transient voltage stability indicator, and transient frequency stability indicator. The indicators help to comprehensively evaluate the stability of the power system with renewable energy integration.

2.1. Economic Indicator

The economic indicator is the system network loss, which is defined as Equation (1):

$$P_{\mathrm{LOSS}}={\displaystyle \sum _{i\in {\mathsf{\Omega }}_N}{\displaystyle \sum _{j\in {\mathsf{\Omega }}_N}{U}_i{U}_j{G}_{i,j}\cos \left({\delta }_i-{\delta }_j\right)}}$$

(1)

where $P_{\mathrm{LOSS}}$ is the system network loss; $U_i$, $U_j$ are the voltage amplitude of nodes i and j, respectively; ${\delta }_i$, ${\delta }_j$ are the phase angle of nodes i and j, respectively; $G_{i,j}$ is the conductance part of the node admittance matrix; and ${\mathsf{\Omega }}_N$ is the set of nodes in the power grid.

2.2. Safety Indicator

2.2.1. Line Ampacity

Line ampacity refers to the carrying capacity of each line in the power grid, expressed as $I_b,b\in {\mathsf{\Omega }}_B$, where $I_b$ is the current of branch b and ${\mathsf{\Omega }}_B$ is the set of branches in the power grid.

2.2.2. Line Current Ampacity after N − 1 Fault

The line ampacity after the N − 1 fault is the carrying capacity of the lines after a single line in the power grid is a fault, expressed as $I_{b,d}^{N-1}, b,d\in {\mathsf{\Omega }}_B$, where $I_{b,d}^{N-1}$ is the current of branch b after branch d is a fault.

2.2.3. Node Voltage Amplitude

The node voltage amplitude is the voltage amplitude of each node (bus) in the power grid $U_i, i\in {\mathsf{\Omega }}_N$.

2.2.4. Network Loss Sensitivity

With the increase in the load, when the power grid operation point approaches the critical point of static voltage stability, the system network loss will increase suddenly. Therefore, the sensitivity of network loss to the active and reactive power of each node load can be used as an indicator to characterize the static voltage stability of the system, and its calculation formula is:

$$\left[\begin{array}{c}\frac{d{P}_{\mathrm{LOSS}}}{dP}\\ \frac{d{P}_{\mathrm{LOSS}}}{dQ}\end{array}\right]={\left[{\mathit{J}}^T\right]}^{-1}\left[\begin{array}{c}\frac{d{P}_{\mathrm{LOSS}}}{d\delta }\\ \frac{d{P}_{\mathrm{LOSS}}}{dU}\end{array}\right]$$

(2)

where $\frac{d{P}_{\mathrm{LOSS}}}{dP}$ and $\frac{d{P}_{\mathrm{LOSS}}}{dQ}$ are the column vectors composed of the sensitivity of the network loss to the active and reactive power of each node load, respectively; J is the Jacobian matrix of the system. $\frac{d{P}_{\mathrm{LOSS}}}{d\delta }$ and $\frac{d{P}_{\mathrm{LOSS}}}{dU}$ are the column vectors composed of the sensitivity of the network loss to the voltage phase angle and amplitude of each node, where the calculation formulas of the i-th element $\frac{d{P}_{\mathrm{LOSS}}}{d{\delta }_i}$ and $\frac{d{P}_{\mathrm{LOSS}}}{d{U}_i}$ are shown in Equations (3) and (4), respectively:

$$\frac{d{P}_{\mathrm{LOSS} }}{d{\delta }_i}=2\sum _{j\in {\mathsf{\Omega }}_N} U_i{U}_j{G}_{i,j}\sin \left({\delta }_j-{\delta }_i\right),i\in {\mathsf{\Omega }}_N$$

(3)

$$\frac{d{P}_{\mathrm{LOSS} }}{d{U}_i}=2\sum _{j\in {\mathsf{\Omega }}_N} U_j{G}_{i,j}\cos \left({\delta }_j-{\delta }_i\right),i\in {\mathsf{\Omega }}_N$$

(4)

When renewable energy is integrated into the AC/DC hybrid grid, the Jacobian matrix needs to be modified under the different control modes of DC and renewable energy to measure the interaction between DC and renewable energy. For example, when the DC adopts constant current control at the rectifier side and constant arc extinguishing angle control at the inverter side (C/E), which is the most used, the power expressions at the rectifier side and inverter side are:

$$\{\begin{array}{l}{P}_{\mathrm{IN}}=U_{\mathrm{dIN}}{I}_{\mathrm{DC}}\hfill \\ Q_{\mathrm{IN}}=-I_{\mathrm{DC}}\sqrt{U_{\mathrm{d}0\mathrm{IN}}^2-U_{\mathrm{dIN}}^2}\hfill \\ P_{\mathrm{RC}}=U_{\mathrm{dRC}}{I}_{\mathrm{DC}}\hfill \\ Q_{\mathrm{RC}}=-I_{\mathrm{DC}}\sqrt{U_{\mathrm{d}0\mathrm{RC}}^2-U_{\mathrm{dRC}}^2}\hfill \end{array}$$

(5)

$$\{\begin{array}{l}{U}_{\mathrm{d}0\mathrm{IN}}=K_{\mathrm{m}}{T}_{\mathrm{IN}}{U}_{\mathrm{IN}}\hfill \\ U_{\mathrm{d}0\mathrm{RC}}=K_{\mathrm{m}}{T}_{\mathrm{RC}}{U}_{\mathrm{RC}}\hfill \\ U_{\mathrm{dIN}}=U_{\mathrm{d}0\mathrm{IN}}\cos \left({\gamma }_{\mathrm{IN}}\right)-(\sqrt{2}/2)K_{\mathrm{m}}{X}_{\mathrm{IN}}{I}_{\mathrm{IN}}\hfill \\ U_{\mathrm{dRC}}=U_{\mathrm{dIN}}+R_{\mathrm{RC}}{I}_{\mathrm{DC}}\hfill \end{array}$$

(6)

where $P_{\mathrm{IN}}$, $P_{\mathrm{RC}}$ are the active power at the rectifier side and inverter side, respectively; $Q_{\mathrm{IN}}$, $Q_{\mathrm{RC}}$ are the reactive power at the rectifier side and inverter side, respectively; $U_{\mathrm{d}0\mathrm{IN}}$ is the no-load voltage at the inverter side; $U_{\mathrm{d}0\mathrm{RC}}$ is the no-load voltage at the rectifier side; $U_{\mathrm{dIN}}$ is the DC voltage at the inverter side; $U_{\mathrm{dRC}}$ is the DC voltage at the rectifier side; $I_{\mathrm{DC}}$ is the DC current; $U_{\mathrm{IN}}$ and $U_{\mathrm{RC}}$ are the AC voltage at the inverter side and rectifier side, respectively; $I_{\mathrm{IN}}$ and $I_{\mathrm{RC}}$ are the inverter side and rectifier side, respectively; $T_{\mathrm{IN}}$ and T_RC are the transformation ratio of the transformer at the inverter side and rectifier side, respectively; $K_{\mathrm{m}}$ is the proportion coefficient of the inverter; ${\gamma }_{\mathrm{IN}}$ is the inverse angle; $X_{\mathrm{IN}}$ is the single bridge commutation reactance of the inverter; and $R_{\mathrm{RC}}$ is the equivalent resistance of the DC line.

Thus, the sensitivity of the DC node power to the converter bus voltage can be obtained, as shown in Equations (7) and (8):

$$\{\begin{array}{l}\frac{\partial P_{\mathrm{IN}}}{\partial U_{\mathrm{RC}}}=0\hfill \\ \frac{\partial P_{\mathrm{IN}}}{\partial U_{\mathrm{IN}}}=K_{\mathrm{m}}{T}_{\mathrm{IN}}{I}_{\mathrm{DC}}\cos {\gamma }_{\mathrm{IN}}\hfill \\ \frac{\partial Q_{\mathrm{IN}}}{\partial U_{\mathrm{RC}}}=0\hfill \\ \frac{\partial Q_{\mathrm{IN}}}{\partial U_{\mathrm{IN}}}=-I_{\mathrm{DC}}\frac{K_{\mathrm{m}}{T}_{\mathrm{IN}}\left(U_{\mathrm{d}0\mathrm{IN}}-U_{\mathrm{dIN}}\cos \left({\gamma }_{\mathrm{IN}}\right)\right)}{\sqrt{U_{\mathrm{d}0\mathrm{IN}}^2-U_{\mathrm{dIN}}^2}}\hfill \end{array}$$

(7)

$$\{\begin{array}{l}\frac{\partial P_{\mathrm{RC}}}{\partial U_{\mathrm{RC}}}=0\hfill \\ \frac{\partial P_{\mathrm{RC}}}{\partial U_{\mathrm{IN}}}=K_{\mathrm{m}}{T}_{\mathrm{IN}}{I}_{\mathrm{DC}}\cos \left({\gamma }_{\mathrm{IN}}\right)\hfill \\ \frac{\partial Q_{\mathrm{RC}}}{\partial U_{\mathrm{RC}}}=I_{\mathrm{DC}}\frac{K_{\mathrm{m}}{T}_{\mathrm{RC}}{U}_{\mathrm{d}0\mathrm{RC}}}{\sqrt{U_{\mathrm{d}0\mathrm{RC}}^2-U_{\mathrm{dRC}}^2}}\hfill \\ \frac{\partial Q_{\mathrm{RC}}}{\partial U_{\mathrm{IN}}}=-I_{\mathrm{DC}}\frac{K_{\mathrm{m}}{T}_{\mathrm{IN}}{U}_{\mathrm{dRC}}\cos \left({\gamma }_{\mathrm{IN}}\right)}{\sqrt{U_{\mathrm{d}0\mathrm{RC}}^2-U_{\mathrm{dRC}}^2}}\hfill \end{array}$$

(8)

PQ control is often adopted for renewable energy generation to meet the maximum power point tracking (MPPT) of active power and zero reactive power at the same time. Under this control mode, the sensitivity of the output power of renewable energy generation to the node voltage is zero, that is:

$$\{\begin{array}{l}\frac{\partial P_{\mathrm{DG}}}{\partial U_{\mathrm{DG}}}=0\hfill \\ \frac{\partial Q_{\mathrm{DG}}}{\partial U_{\mathrm{DG}}}=0\hfill \end{array}$$

(9)

When there is DC and renewable energy generation in the power grid, it is necessary to calculate the partial derivatives of the active and reactive power of the DC and renewable energy to the node voltage amplitude and phase angle according to Equations (7)–(9), and correct the corresponding elements of the Jacobian matrix J. The modified Jacobian matrix can be brought into Equation (2) to calculate the network loss sensitivity and used as the static voltage stability indicator of the power grid when DC and renewable energy are integrated.

2.2.5. Transient Voltage Stability Indicator

To consider the mutual coupling between DC and renewable energy when faults occur in the AC-DC hybrid grid, this paper further expands the meaning and application scope of the multi-infeed interaction factor (MIIF), and uses it as the transient voltage stability indicator of the AC-DC hybrid grid. MIIF is an indicator used for measuring the interaction relationship between multiple DC lines, which represents the transient voltage drop degree of the DC converter bus after a power grid failure, expressed as Equation (9) [17,18]:

$$MII{F}_{i,j}=\frac{\mathsf{\Delta }{U}_i}{\mathsf{\Delta }{U}_j}$$

(10)

In [19], a new stability classification, i.e., converter-driven stability, is proposed to describe the effects of the penetration of converter-interfaced generation, such as HVDC and renewable energy. Converter-driven stability is used to measure the stability of the power system after the converter-interfaced generation with different control algorithms is integrated. The transient voltage stability indicator proposed in this paper is used to consider the transient voltage stability of the power system after the renewable energy and HVDC with different control algorithms is integrated. The connotation of the indicator is consistent with the slow-interaction converter-driven stability of the converter-driven stability classification. To evaluate the converter-driven stability of the power system, this paper further expands the meaning and application scope of MIIF, and uses it to measure the mutual coupling between DC and renewable energy during the fault of the AC-DC hybrid grid and its impact on the transient voltage drop of the system to evaluate whether the cascading fault of DC commutation failure and renewable energy off-grid will occur in the grid. The calculation method of MIIF in the AC-DC hybrid grid is as follows:

$$MII{F}_{i,j}=\frac{{\left({\mathit{J}}_R^{-1}\right)}_{i,j}}{{\left({\mathit{J}}_R^{-1}\right)}_{j,j}}$$

(11)

$${\mathit{J}}_R={\mathit{J}}_{QU}-{\mathit{J}}_{Q\delta }{\mathit{J}}_{P\delta }^{-1}{\mathit{J}}_{PU}$$

(12)

where ${\mathit{J}}_{QU}$, ${\mathit{J}}_{Q\delta }$, ${\mathit{J}}_{P\delta }$, and ${\mathit{J}}_{PU}$ are the sub matrices of the Jacobian matrix system corresponding to $\frac{\partial Q}{\partial U}$, $\frac{\partial Q}{\partial \delta }$, $\frac{\partial P}{\partial \delta }$, and $\frac{\partial P}{\partial U}$, respectively.

Based on Equations (11) and (12), considering the specific characteristics of renewable energy and DC, it is necessary to modify the calculation method of MIIF. On the one hand, the corresponding elements of the Jacobian matrix need to be modified according to Equations (7)–(9) to consider the influence of the different control modes of DC and renewable energy on the transient voltage sag of the system at the same time. On the other hand, in Equation (11), i should include two sets ${\mathsf{\Omega }}_{\mathrm{DG}}$ and ${\mathsf{\Omega }}_{\mathrm{DC}}$ at the same time, i.e., $i\in \left\{{\mathsf{\Omega }}_{\mathrm{DG}},{\mathsf{\Omega }}_{\mathrm{DC}}\right\}$, where ${\mathsf{\Omega }}_{\mathrm{DG}}$ is the set of nodes with renewable energy integrated and ${\mathsf{\Omega }}_{\mathrm{DC}}$ is the set of the converter bus of HVDC. Comparing the MIIF of the nodes with renewable energy integrated and the DC converter bus with the critical voltage of the renewable energy low-voltage crossing and the critical voltage of the DC commutation failure, respectively, we can judge whether the system voltage drop during the fault will lead to the cascading fault of DC commutation failure and a renewable energy off-grid accident, and then measure the transient voltage security of the system.

2.2.6. Transient Frequency Stability Indicator

With the gradual replacement of synchronous generators in the power grid, the moment of inertia and the frequency modulation capacity of the system are continuously reduced, resulting in the deterioration of the transient frequency stability. To describe the impact of renewable energy and DC on the system transient frequency stability, the maximum frequency deviation and steady-state frequency deviation are selected as transient frequency stability indicators.

After the renewable energy and DC are integrated, the equivalent inertia time constant and equivalent unit regulated power of the system are shown in Equations (13) and (14):

$$T_{\mathrm{jeq}}^{*}=\frac{{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{G}}}}{T}_{\mathrm{G}}^i{C}_{\mathrm{G}}^i}{{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{G}}}}{C}_{\mathrm{G}}^i+{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{DG}}}}{C}_{\mathrm{DG}}^i}$$

(13)

$$K_{\mathrm{G}}^{*}=\frac{{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{G}}}}{K}_{\mathrm{G}}^i{C}_{\mathrm{G}}^i}{{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{G}}}}{C}_{\mathrm{G}}^i+{\displaystyle \sum _{i\in {\mathsf{\Omega }}_{\mathrm{DG}}}}{C}_{\mathrm{DG}}^i}$$

(14)

where $T_{\mathrm{jeq}}^{\ast }$ and $K_{\mathrm{G}}^{\ast }$ are the equivalent inertia time constant and unit regulated power, respectively; $C_{\mathrm{G}}^i$ is the capacity of the synchronous generator integrated into node i; $T_{\mathrm{G}}^i$ and $K_{\mathrm{G}}^i$ are the inertia time constant and unit regulated power of the synchronous generator, respectively; and ${\mathsf{\Omega }}_{\mathrm{G}}$ is the set of all synchronous generator integrated nodes.

By listing the differential equation of the transient frequency variation characteristics of the system, the maximum frequency deviation and steady-state frequency deviation can be deduced, as shown in Equations (15) and (16) [20]:

$$\mathsf{\Delta }{f}_{\max }^{\prime }=M^{\prime }+N^{\prime }{e}^{a^{\prime }{t}_{\min }}\cos \left(b^{\prime }{t}_{\min }-\arctan \frac{h^{\prime }}{g^{\prime }}\right)$$

(15)

$$\mathsf{\Delta }{f}_{\infty }^{\prime }=\frac{\mathsf{\Delta }P}{K_{\mathrm{G}}^{\ast }+K_{\mathrm{D}}^{\ast }}$$

(16)

where $\mathsf{\Delta }{f}_{\max }^{\prime }$ is the maximum frequency deviation of the power grid; $\mathsf{\Delta }{f}_{\infty }^{\prime }$ is the steady-state frequency deviation of the power grid; $\mathsf{\Delta }P$ is the power shortage of the power grid; $K_{\mathrm{D}}^{\ast }$ is the frequency regulation coefficient of the load; and $t_{\min }$ is the time when the frequency drop is greatest, and its expression is shown in Equation (17):

$$t_{\min }=\frac{\arctan \frac{a^{\prime }}{b^{\prime }}+\arctan \frac{h^{\prime }}{g^{\prime }}}{b^{\prime }}$$

(17)

All parameters $a^{\prime }$, $b^{\prime }$, $g^{\prime }$, $h^{\prime }$, $M^{\prime }$, and $N^{\prime }$ in Equations (15) and (17) are functions of $T_{\mathrm{jeq}}^{\ast }$ and $K_{\mathrm{G}}^{\ast }$ [20].

3. Evaluation and Optimization Model of the Renewable Energy Penetration Capacity of the AC-DC Hybrid Grid

Based on the evaluation indicators established in Section 2, the evaluation and optimization model of the renewable energy penetration capacity of AC-DC hybrid grid is established. The optimization variables of the model are the access location and capacity of renewable energy in the power grid.

3.1. Objective Function

The optimization objective of the renewable energy penetration capacity evaluation model is to maximize the renewable energy penetration capacity of the power grid. When the maximum renewable energy penetration capacity has multiple feasible access locations in the power grid, the access scheme is selected to minimize the network loss, that is:

$$\{\begin{array}{l}{F}_1=\max C_{\mathrm{DG}}\hfill \\ F_2=\min P_{\mathrm{LOSS}}\hfill \end{array}$$

(18)

In Equation (18), F₁ is the primary optimization objective function, that is, it maximizes the penetration capacity of renewable energy; F₂ is the secondary optimization objective function, that is it minimizes the system network loss; and $C_{\mathrm{DG}}$ refers to the renewable energy penetration capacity in the power grid, which is defined as:

$$C_{\mathrm{DG}}=\sum _{i\in {\mathsf{\Omega }}_{\mathrm{DG}}} C_{\mathrm{DG}}^i$$

(19)

where $C_{\mathrm{DG}}^i$ is the renewable energy capacity integrated into node i.

3.2. Constraint Condition

3.2.1. System Power Flow Constraints

The power grid shall meet the power flow equation, which is:

$$U_i\sum _{j\in {\mathsf{\Omega }}_N} U_j\left(G_{i,j}\cos \left({\delta }_i-{\delta }_j\right)+B_{i,j}\sin \left({\delta }_i-{\delta }_j\right)\right)=P_i, i\in {\mathsf{\Omega }}_N$$

(20)

$$U_i\sum _{j\in {\mathsf{\Omega }}_N} U_j\left(G_{i,j}\sin \left({\delta }_i-{\delta }_j\right)-B_{i,j}\cos \left({\delta }_i-{\delta }_j\right)\right)=Q_i, i\in {\mathsf{\Omega }}_N$$

(21)

where $P_i$ and $Q_i$ are the active and reactive power injected by node i, respectively.

3.2.2. Line Ampacity Constraints

The ampacity of each line in the power grid shall be less than the maximum ampacity of the line, which is:

$$\left|I_b\right|\le I_b^M, b\in {\mathsf{\Omega }}_B$$

(22)

where $I_b^M$ is the maximum ampacity of branch b, $b\in {\mathsf{\Omega }}_B$.

3.2.3. Line Ampacity Constraints after N − 1 Fault

After a single line in the power grid trips, the ampacity of the remaining lines shall be less than the maximum ampacity of the line, that is:

$$\left|I_{b,d}^{N-1}\right|\le I_d^M, b,d\in {\mathsf{\Omega }}_B$$

(23)

To solve Equation (23), the power flow transfer matrix $\mathit{\tau }$ of the system can be solved first, and then Equation (23) can be transformed into the constraints shown in Equation (24):

$$\left|{\tau }_{b,d}{I}_b\right|\le I_d^M, b,d\in {\mathsf{\Omega }}_B$$

(24)

where ${\tau }_{b,d}$ is the corresponding element of the matrix $\mathit{\tau }$.

3.2.4. Node Voltage Amplitude Constraints

The voltage amplitude of each node in the power grid shall be within the allowable range, i.e.:

$$U^D\le U_i\le U^M, i\in {\mathsf{\Omega }}_N$$

(25)

where $U^D$ and $U^M$ are the lower limit and upper limit of the allowable voltage, respectively.

3.2.5. Static Voltage Stability Constraints

When the operating point of the system approaches the critical point of the static voltage stability, the network loss sensitivity will tend to infinity. Therefore, a large upper limit value is set for the network loss sensitivity. When the network loss sensitivity exceeds this upper limit, it is considered that the static voltage stability margin of the system is insufficient, that is:

$$\frac{d{P}_{\mathrm{LOSS}}}{d{P}_i}\le {\left[\frac{d{P}_{\mathrm{LOSS}}}{dP}\right]}^M, i\in {\mathsf{\Omega }}_N$$

(26)

$$\frac{d{P}_{\mathrm{LOSS}}}{d{Q}_i}\le {\left[\frac{d{P}_{\mathrm{LOSS}}}{dQ}\right]}^M, i\in {\mathsf{\Omega }}_N$$

(27)

where ${\left[\frac{d{P}_{\mathrm{LOSS}}}{dP}\right]}^M$ and ${\left[\frac{d{P}_{\mathrm{LOSS}}}{dQ}\right]}^M$ are the upper limit values of the active and reactive network loss sensitivity, respectively.

3.2.6. Transient Voltage Stability Constraints

In case of short-circuit fault in the power grid, in order to avoid cascading fault caused by the transient voltage drop, that is, DC commutation failure and disconnection of renewable energy generation occurring at the same time, the transient voltage stability constraint of the system is set so that after the three-phase short-circuit fault of any bus, DC commutation failure and renewable energy generation off-grid will not occur at the same time, which is:

$$\underset{i\in {\mathsf{\Omega }}_{\mathrm{DG}}}{\max }H\left(\frac{MII{F}_{i,j}}{\mathsf{\Delta }{V}_{\mathrm{DG}}^i}\right)+\underset{i\in {\mathsf{\Omega }}_{\mathrm{DG}}}{\max }H\left(\frac{MII{F}_{i,j}}{\mathsf{\Delta }{V}_{\mathrm{DC}}^i}\right)\le 1,j\in {\mathsf{\Omega }}_N$$

(28)

where $\mathsf{\Delta }{V}_{\mathrm{DG}}^i$ is the low-voltage crossing critical voltage of node i with the renewable energy integrated; $\mathsf{\Delta }{V}_{\mathrm{DC}}^i$ is the critical voltage of commutation failure of node i with DC integrated, which is calculated from the critical commutation angle of DC commutation failure, and it is the general engineering judgment criterion for judging whether DC commutation failure will occur [18]; and $H\left(x\right)$ is the step function: when $x<1$, $H\left(x\right)=0$; when $x\ge 1$, $H\left(x\right)=1$.

3.2.7. Transient Frequency Stability Constraints

In case of power shortage in the power grid, the maximum transient frequency deviation and steady-state frequency deviation of the system shall be within the constraint range, that is:

$$\mathsf{\Delta }{f}_{\max }^{\prime }\le {\left[\mathsf{\Delta }{f}_{\max }^{\prime }\right]}^M$$

(29)

$$\mathsf{\Delta }{f^{\prime }}_{\infty }\le {\left[\mathsf{\Delta }{f^{\prime }}_{\infty }\right]}^M$$

(30)

where ${\left[\mathsf{\Delta }{f}_{\max }^{\prime }\right]}^M$ and ${\left[\mathsf{\Delta }{f}_{\infty }^{\prime }\right]}^M$ are the maximum allowable transient frequency deviation and steady-state frequency deviation of the system, respectively.

Under the transient voltage stability constraints proposed in Section 3.2.6, a single fault in the power grid will not lead to the cascading fault of DC commutation failure and renewable energy generation off-grid. Therefore, in the transient frequency stability constraints, the power shortage of the power grid is taken as the maximum value of the power shortage caused by the tripping of single power sources, such as DC unipolar locking and renewable energy generation off-grid.

3.2.8. Renewable Energy Penetration Capacity Constraints

Due to the renewable energy resource volume and geographical location constraints, the capacity of renewable energy that can be integrated into each access location has an upper limit, that is:

$$C_{\mathrm{DG}}^i\le {\left[C_{\mathrm{DG}}^i\right]}^M i\in {\mathsf{\Omega }}_{\mathrm{DG}}$$

(31)

where ${\left[C_{\mathrm{DG}}^i\right]}^M$ is the maximum allowable renewable energy capacity integrated into node i.

4. Solution Method of the Renewable Energy Penetration Capacity Evaluation Model of the AC-DC Hybrid Grid

The renewable energy penetration capacity evaluation model of the AC-DC hybrid grid established in Section 3 is a nonlinear mixed integer programming model, which is difficult to solve directly using the analytical method. Therefore, according to the specific characteristics of different constraints, this section proposes a heuristic solution method for the renewable energy penetration capacity evaluation model. In the proposed solution method, only the typical operation mode of the power grid is considered for simplification, and factors, such as the load fluctuation and network structure changes, are ignored. The process of the proposed method is shown in Figure 1.

In the renewable energy penetration capacity evaluation model, since the transient frequency stability constraints (29) and (30) are only related to the renewable energy penetration capacity and have nothing to do with the access location, the method shown in Figure 1 can be divided into two steps: first, the maximum renewable energy penetration capacity meeting the transient frequency stability constraints is solved. Secondly, the optimal access location of renewable energy is determined, that is, for different access locations, whether they meet other optimization constraints is checked, and the access scheme with the minimum system network loss is selected as the optimal solution of the model. If all access locations are not feasible, the penetration capacity of renewable energy is reduced and a search is performed again until a feasible solution is found.

The renewable energy penetration capacity and location obtained according to the above solution process not only meet all the constraints of the optimization model but also maximize the renewable energy penetration capacity and minimize the system network loss. Therefore, this optimal scheme is the optimal solution of the renewable energy penetration capacity optimization model proposed in Section 3. At this time, the capacity of renewable energy is the renewable energy penetration capacity of the power grid.

5. Example Test

In this paper, based on the IEEE-39 node system, 2 DC lines are integrated into form an AC-DC hybrid network, as shown in Figure 2. The parameters of each element in the system are as follows: 2 ± 270 kV DC are bipolar operation, the rated power is 540 MW, and the critical voltages of the commutation failure ($\Delta V_{\mathrm{DC}}^i$) are 0.126 and 0.183 p.u., respectively; the inertia time constant of all synchronous motors is 6.7 s, the difference adjustment coefficient is 12%, and the time constant of the servo mechanism is 0.15s; the maximum current ampacity of all lines is 10 MW; the lower limit and upper limit of bus voltage are 0.9 and 1.1 p.u., respectively; the upper limit values ${\left[\frac{d{P}_{\mathrm{LOSS}}}{dP}\right]}^M$ and ${\left[\frac{d{P}_{\mathrm{LOSS}}}{dQ}\right]}^M$ of the network loss sensitivity are 100; the maximum allowable renewable energy capacity ${\left[C_{\mathrm{DG}}^i\right]}^M$ of the nodes is 500 MW; the maximum allowable transient frequency deviation of the system is ${\left[\mathsf{\Delta }{f}_{\max }^{\prime }\right]}^M=1 \mathrm{Hz}$ and the steady-state frequency deviation is ${\left[\mathsf{\Delta }{f^{\prime }}_{\infty }\right]}^M=0.2 \mathrm{Hz}$; and the low-voltage ride through critical voltage ($\Delta V_{\mathrm{DC}}^i$) of renewable energy generation is 0.3 p.u. The simulation platform used in this paper is PSASP 7.35.

5.1. Evaluation of the Renewable Energy Penetration Capacity

For the example network, an optimization model for the evaluation of the renewable energy penetration capacity is established, and the method shown in Figure 1 is used to solve the model. Firstly, the maximum renewable energy penetration capacity satisfying the transient frequency constraint is 300 MW according to Equations (29)–(31). Secondly, for different access points with renewable energy integrated in the network, whether they meet other constraints when the current capacity of renewable energy is integrated is calculated, and finally, the feasible access location when the renewable energy penetration capacity is 300 MW is obtained, as shown in Figure 3. The yellow nodes in the figure indicate that the nodes can satisfy all optimization constraints when the renewable energy is integrated, and the red nodes indicate that the constraints will be exceeded when the renewable energy is integrated into the nodes.

It can be seen from Figure 3 that under the current renewable energy penetration capacity, the infeasible access locations are mainly near the DC inverter converter bus. On the one hand, when the electrical distance between the node with renewable energy integrated and the DC inverter converter bus is short, the possibility of DC commutation failure and renewable energy off-grid is higher after the short-circuit fault of the system, which does not meet the transient voltage security constraints of the system. On the other hand, the concentrated transmission of DC power and renewable energy power may lead to overloading of the transmission line or out of limit voltage amplitude of the terminal bus. The network loss when the renewable energy is integrated into all feasible locations us calculated, and the results are shown in

Table 1. Comparison of the network loss when the renewable energy resource is integrated into feasible buses.

Bus	${\mathit{P}}_{\mathbf{LOSS}}/\mathbf{MW}$	Bus	${\mathit{P}}_{\mathbf{LOSS}}/\mathbf{MW}$
1	0.988	23	0.857
2	0.865	24	0.802
3	0.803	25	0.822
12	0.845	26	0.735
15	0.791	30	0.866
16	0.799	32	0.861
17	0.781	33	0.926
18	0.786	35	0.859
19	0.857	36	0.879
20	0.856	37	0.94
21	0.823	36	0.879
22	0.857	39	1.025

.

It can be seen from Table 1 that when renewable energy is integrated into bus 26, the system network loss is the smallest, which is 0.735 MW. Therefore, the renewable energy penetration capacity of the AC-DC hybrid grid shown in Figure 2 is 300 MW, and the optimal access position of the renewable energy is bus 26.

5.2. Transient Frequency Stability Verification

To verify the transient frequency stability of the system under the optimal access scheme of renewable energy, the transient process of the system after power shortage is simulated to verify whether the transient frequency deviation is within the allowable range. Since the renewable energy capacity is greater than the DC unipolar capacity, the power shortage is set as the power shortage caused by the renewable energy trip. Further, 300 and 350 MW renewable energy are integrated into bus 26, respectively, and the system transient frequency obtained by the simulation is shown in Figure 4.

It can be seen from Figure 4 that when the renewable energy penetration capacity is 300 MW, the maximum frequency deviation of the system is less than 1 Hz and the steady-state frequency deviation is less than 0.2 Hz, satisfying the transient frequency stability constraints. When the renewable energy penetration capacity is 350 MW, the maximum frequency deviation is greater than 1 Hz, and the constraint shown in Equation (29) exceeds the limit. This example shows the correctness of the evaluation results of the renewable energy penetration capacity and the effectiveness of the transient frequency stability constraints.

5.3. Transient Voltage Security Verification

To verify the transient voltage stability of the system under the optimal access scheme of renewable energy, 300 MW of renewable energy generation is integrated into bus 26 and 29, respectively. It is assumed that the 3-phase grounding short-circuit fault of bus 28 occurs at 0.5 s, and the fault is removed after 0.25 s. The transient process of the system is simulated, and the transient voltages of 2 DC inverter converter buses (bus 9 and bus 28) and buses with renewable energy integrated are obtained, as shown in Figure 5 and Figure 6, respectively.

As can be seen from Figure 5, when the renewable energy is integrated into bus 26, during the fault period, because the transient voltage of bus 28 is lower than the critical voltage of DC commutation failure, only a single DC commutation failure occurs, which meets the transient voltage security constraint. As can be seen from Figure 6, when the renewable energy is integrated into bus 29, the transient voltage of bus 29 during the fault will also be lower than the low-voltage crossing critical voltage of the renewable energy, resulting in the cascading fault of DC commutation failure and renewable energy off-grid in the system, which does not meet the transient voltage stability constraint. This example shows the correctness of the evaluation results of the system penetration capacity and the effectiveness of the transient voltage stability constraint.

5.4. Discussion

5.4.1. Impact of Renewable Energy on the Power System Stability

(1)

Voltage stability. According to [19], when renewable energy does not adopt voltage control, the voltage stability of the system is reduced due to the unfavorable reactive power “load” characteristics of the converters. When the renewable energy adopts voltage control, it can provide reactive power support after a system failure to prevent the bus voltage from dropping sharply. The voltage stability is enhanced. In this paper, the influence of renewable energy on the voltage stability is evaluated by adding correcting values to the relevance elements of the Jacobian matrix. For different control algorithms of renewable energy, the correcting values of the $\frac{\partial P}{\partial U}$ and $\frac{\partial Q}{\partial U}$ elements of the Jacobian matrix corresponding to the integration node of the renewable energy are also different, which affects the system voltage stability. Since the renewable energy commonly adopts the maximum power point tracking control and does not participate in voltage control, the correcting values of the Jacobian matrix are 0, as shown in Equation (9).

(2)

Frequency stability. According to [19], when renewable energy does not adopt frequency control, the system inertial is reduced since renewable energy replaces synchronous generators, thereby reducing the frequency stability. When the renewable energy adopts frequency control, the system inertia and frequency response speed is increased, thereby enhancing the frequency stability. In this paper, the frequency stability is evaluated by establishing a frequency response model of the power system. Since the renewable energy commonly adopts the maximum power point tracking control and does not participate in frequency regulation, it is not considered in the frequency response model. Consideration of the frequency regulation control of renewable energy in frequency stability evaluation will be covered in our future research.

5.4.2. Reliability and Applicability to Real Power Systems

In current commonly used engineering methods, the economics and safety of the power system with a small capacity for renewable energy integration are evaluated. Since the capacity of the renewable energy is relatively small, it has little impact on the stability of the power system. There is no need to consider the stability constraints of the power system in the engineering methods. The method proposed in this paper is to evaluate the maximum penetration capacity of renewable energy in the power system. When the capacity of the renewable energy accounts for a large proportion, it may have a significant impact on the stability of the power system. Therefore, the stability constraints of the system need to be considered. The method proposed in this paper considers system stability constraints, including static voltage stability constraints, transient voltage stability constraints, and transient frequency stability constraints. The proposed method is an extension of existing engineering methods.

On the other hand, when the proposed method is applied to real power systems, further work is needed according to the actual situation of the power grid, such as considering different operation modes of the power system (load fluctuation, network structure changes, etc.) and the influence of different control algorithms of renewable energy on the system stability. These topics will be covered in our future research.

5.4.3. Comparison with Other Methods

In the method proposed in [5,6,7,8,9], only the static stability constraints and safety constraints, such as the line ampacity capacity and node voltage amplitude constraints, are considered. The maximum penetration capacity of the renewable energy obtained is relatively large. If the renewable energy of the obtained penetration capacity is integrated into the power system, the transient stability constraints of the system will be violated.

In the method proposed in [10,11,12,13,14,15,16], only the transient stability constraints and safety constraints, such as the line ampacity capacity and node voltage amplitude constraints, are considered. The maximum penetration capacity of the renewable energy obtained is relatively small. If the renewable energy of the obtained penetration capacity is integrated into the power system, the static stability constraints of the system will be violated.

The method proposed in this paper considers the static voltage stability constraints, transient voltage stability constraints, transient frequency stability constraints, and safety constraints, such as the line ampacity capacity and node voltage amplitude constraints. The obtained maximum penetration capacity of the renewable energy is the smallest. If the renewable energy of the obtained penetration capacity is integrated into the power system, all the static and transient stability constraints of the system will be met.

6. Conclusions

To evaluate the renewable energy penetration capacity of the AC-DC hybrid grid, this paper established an evaluation indicator system of the renewable energy penetration capacity of the AC-DC hybrid grid considering both the static and transient stability, proposed an optimization model of the renewable energy penetration capacity of the AC-DC hybrid grid and its solution method, and carried out simulation verification. The main conclusions are as follows:

• According to the specific characteristics of the renewable energy integrated into the AC-DC hybrid grid and the mutual coupling between DC and renewable energy, the static and transient stability of the system were comprehensively evaluated, and a more comprehensive evaluation indicator system was established.
• The evaluation and optimization model of the renewable energy penetration capacity of the AC-DC hybrid grid was established, and a heuristic solution method for the optimization model was proposed according to the characteristics of the model, which realizes the evaluation of the renewable energy penetration capacity of the power grid and obtains the optimal access scheme.
• The example test showed that the proposed method can correctly evaluate the renewable energy penetration capacity of the AC-DC hybrid grid and produces the optimal access scheme.