Study of a Numerical Integral Interpolation Method for Electromagnetic Transient Simulations

Abstract

In the fixed time-step electromagnetic transient (EMT)-type program, an interpolation process is applied to deal with switching events. The interpolation method frequently reduces the algorithm’s accuracy when dealing with power electronics. In this study, we use the Butcher tableau to analyze the defects of linear interpolation. Then, based on the theories of Runge–Kutta integration, we propose two three-stage diagonally implicit Runge–Kutta (3S-DIRK) algorithms combined with the trapezoidal rule (TR) and backward Euler (BE), respectively, with TR-3S-DIRK and BE2-3S-DIRK for the interpolation and synchronization processes. The proposed numerical integral interpolation scheme has second-order accuracy and does not produce spurious oscillations due to the size change in the time step. The proposed method is compared with the critical damping adjustment method (CDA) and the trapezoidal method, showing that it does not produce spurious numerical oscillations or first-order errors.

1. Introduction

In the development and construction of ultra-high-voltage DC lines, the proportion of power electronic components in the power system has been increasing year by year. Therefore, electromagnetic transient (EMT) simulations have become an important tool for analyzing the stability of AC/DC hybrid power systems [1,2,3,4]. EMT simulations can essentially be solved as differential–algebraic equations (DAEs), which are more complex, with stricter algorithmic requirements than ordinary differential equations (ODEs) [5]. The error in the values of algebraic equations causes numerical oscillation, and this should be suppressed by applying an L-stable algorithm to solve the DAE problem [6], especially when dealing with a circuit’s sudden changes [7].

The most commonly used EMT-type program [8,9] employs the nodal analysis method, which uses a fixed step-size integration method to discretize the differential equation of the circuit elements into a current source with a shunt resistor. The admittance matrix of the network is unchanged during the computation process, which can improve the speed of the computation [10]. The EMT-type program is specially designed for power system simulation, provides a variety of power system models, and has been used in engineering computations for many years [11].

Trapezoidal rule (TR) integration is an implicit method with the feature of symmetrical A-stability, which can show the unstable modes in the system. Further, it is most widely used in power system simulation. This algorithm enables the original instability patterns in the system to be manifested and eliminates the problem of superstability, but causes numerical oscillations [12]. The numerical oscillation of EMT can be suppressed by adding a buffer circuit [13] or using the damping trapezoid method, critical damping adjustment method (CDA) [12], two-stage diagonally implicit Runge–Kutta method (2S-DIRK) [14,15,16], root-matching method [17], trapezoidal method with the second-order backward difference formula (TR-BDF2) [17], or Rosenbrock numerical integration method [18]. The CDA method uses a two-step backward Euler method [19], which does not need to modify the admittance matrix and has good adaptability. As a result, it is widely used in EMT-type software [20].

On the other hand, switching operation events do not occur exactly at the constant-step intervals, and delays in the switching operation cause the emergence of unreal “spikes” [21] in the waveform and, thus, harmonics, which requires an accurate simulation of the switching operation through the variable time-step algorithm or interpolation method [22,23]. This requires multiple interpolations [22,24] or flexible integration for readjustment in the simulation of transients [25] to achieve resynchronization during real-time simulations. The interpolation method is a good choice because it does not change the admittance matrix of the network. However, linear interpolation reduces the simulation accuracy depending on when exactly the discontinuity occurs [26]. And interpolation requires a smaller time step [24]. In addition, exponential matrix methods can solve this problem [27,28], but their implementation requires significant modification of the CDA architecture, rendering it a less practical choice. Therefore, to develop an enhanced algorithm, the following conditions should be met:

(1)

L-stable, to effectively eliminate oscillations;

(2)

Variable step size, to handle switching events at any time;

(3)

High computational accuracy, at least secondary-order, during interpolation;

(4)

Consistent equivalent impedance, and the conductivity matrix does not need to re-triangulated;

(5)

It is better to construct it on the CDA scheme to minimize modifications of EMT-type tools.

Recent developments in DIRK methods have advanced the field of computational mathematics. Kennedy et al. [29,30] evaluated DIRK methods for stability and accuracy in solving differential equations, despite increasing computational demands. Da Silva et al. [31] introduced iso-spectral Runge–Kutta methods using the reduced Hamiltonian principle. Kamiński et al. [32] applied a novel Runge–Kutta–Fehlberg algorithm to the stochastic Duffing equation, enhancing structural response calculations. Despite these advancements, Runge–Kutta methods should be redesigned to meet all above conditions for EMT computations.

In this study, we use the Butcher tableau [33] to analyze the computation error caused by interpolation in the CDA method and discuss the defects of linear interpolation algorithms in Section 2. It is found that the linear interpolation algorithm cannot maintain the high-order simulation accuracy. Based on the Runge–Kutta theories, two three-stage diagonally implicit Runge–Kutta (3S-DIRK) methods combined with TR and BE are proposed: TR-3S-DIRK and BE2-3S-DIRK. Then, a new integration-based interpolation scheme is proposed, and the integral formulas for inductors, capacitors, and mutual inductors are derived. The proposed algorithms and interpolation scheme have similar advantages to CDA, such as an invariant admittance matrix and numerical oscillation suppression. Moreover, the proposed algorithms have second-order accuracy throughout the whole simulation.

To verify the feasibility of the proposed method, a special case was constructed combined with the line-commutated converter-based high-voltage direct-current (LCC-HVDC) system and the IEEE9 system, in which the calculation algorithm needs to be switched to obtain accurate LCC-HVDC results, while the errors in algorithm switching are reflected in the curves of IEEE9. Then, the proposed method is compared with the CDA method in a well-constructed case.

2. Interpolation in CDA and Its Defects

2.1. Switching Interpolation Architecture

The CDA method is the most commonly used method in EMT simulation tools, which adopts the TR method in the normal computation process. As is partly shown in Figure 1, when a switching action is detected, stage 1 involves interpolating to the moment of the switching action, point K, using linear interpolation; stage 2 and stage 3 suppresses the numerical oscillation using the BE method twice; and stage 4 is resynchronization using the linear interpolation algorithm, so the computation point is synchronized back to the original time step. At the same time, if a switching action is detected in stage 2, it will return to stage 1 to perform interpolation using the linear interpolation algorithm.

Resynchronization is needed, as all controls and system models need to recognize the effective time-step shift in their local integration scheme. Therefore, two types of interpolation were used, i.e., the interpolation after the TR and the interpolation after the BE method. The former interpolation process was used to obtain the exact time of the switching action in stage 1, and the latter synchronization process was used for the resynchronization computation in stage 4.

2.2. Linear Interpolation after TR Method

EMT-type tools simulate the transient phenomenon of dynamic elements, which can be written in standard differential equation form as,

$${\displaystyle \frac{dy}{dt}}=f(t,y)$$

(1)

Assuming that interpolation is performed at point K and $t_{n+k_T}=t_n+k_{T}h$, then, using the TR and linear interpolation algorithms, we have the following formulas:

$$\{\begin{array}{l}{y}_{n+1}=y_n+{\displaystyle \frac{h}{2}}f(t_n,y_n)+{\displaystyle \frac{h}{2}}f(t_{n+1},y_{n+1})\\ y_{n+k_T}=(1-k_T)y_n+k_T{y}_{n+1}\end{array},$$

(2)

Assuming that $y_{n+k_T}$ is the term to be solved, $y_{n+1}$ is an intermediate term, and the step size is changed to a new one, which is $\tilde{h}=k_{T}h$, then the above formulas can be rewritten as the following RK formulas:

$$\{\begin{array}{l}{\mathrm{F}}_1=f(t_n,y_n)\\ {\mathrm{F}}_2=f\left(t_n+{\displaystyle \frac{1}{k_T}}\tilde{h},y_n+{\displaystyle \frac{\tilde{h}}{2k_T}}{\mathrm{F}}_1+{\displaystyle \frac{\tilde{h}}{2k_T}}{\mathrm{F}}_2\right)\\ y_{n+k_T}=y_n+{\displaystyle \frac{\tilde{h}}{2}}{\mathrm{F}}_1+{\displaystyle \frac{\tilde{h}}{2}}{\mathrm{F}}_2\end{array}.$$

(3)

For simplicity, the s-stage RK case is often described using the following Butcher tableau:

$$\begin{array}{cccc}{c}_1& a_{11}& \cdots & a_{1\mathrm{s}}\\ ⋮& ⋮& \ddots & ⋮\\ c_{\mathrm{s}}& a_{\mathrm{s}1}& \cdots & a_{\mathrm{ss}}\\ & b_1& \cdots & b_{\mathrm{s}}\end{array}=\begin{array}{cc}c& A\\ & b^T\end{array}.$$

(4)

The Butcher tableau is a useful tool for the Runge–Kutta method, and the accuracy of the algorithm can be calculated according to rooted trees as Formulas (5) to (6), where e is an s-dimensional column vector whose elements are all 1 and C is a diagonal matrix composed of c elements.

$$\mathrm{First}\text{-}\mathrm{order}\mathrm{accuracy}:b^T·e=1,$$

(5)

$$\mathrm{Second}\text{-}\mathrm{order}\mathrm{accuracy}:b^T·C·e={\displaystyle \frac{1}{2}}.$$

(6)

Formula (2) features the Butcher tableau (7), and it is shown that the Butcher tableau (7) satisfies the condition of (5). Only when $k_T=1$ is the condition of (6) satisfied. Thus, when $k_T\ne 1$, the formula is a first-order accuracy algorithm, and the use of first-order linear interpolation will reduce the computational accuracy.

$$\begin{array}{ccc}0& 0& \\ {\displaystyle \frac{1}{k_T}}& {\displaystyle \frac{1}{2k_T}}& {\displaystyle \frac{1}{2k_T}}\\ & {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\end{array}.$$

(7)

2.3. Linear Interpolation after BE Method

To distinguish the symbols in Section 2.2, m and $k_B$ are used instead, and $m=n+k$, $k_B=1-k_T$. As illustrated in Figure 1, the linear interpolation in stage 4 is often performed after calculating the BE method twice, so there are three values that can be used, $y_m$, $y_{m+0.5}$, and $y_{m+1}$, while the first point $y_n$$y_m$ is a discontinuous point; thereby, we obtain the following formulas:

$$\{\begin{array}{l}{y}_{m+0.5}=y_m+{\displaystyle \frac{h}{2}}f(t_{m+0.5},y_{m+0.5})\\ y_{m+1}=y_{m+0.5}+{\displaystyle \frac{h}{2}}f(t_{m+1},y_{m+1})\\ y_{m+k_B}=2(1-k_B)y_{m+0.5}+(2k_B-1)y_{m+1}\end{array}.$$

(8)

Assuming that $y_{m+k_B}$ is the term to be solved, $y_{m+0.5}$ and $y_{m+1}$ are intermediate values, and the step size is changed to a new one, which is $\tilde{h}=k_{B}h$, then Formula (8) can be rewritten as the RK Formula (9) with the Butcher tableau (10).

$$\{\begin{array}{l}{\mathrm{F}}_1=f\left(t_m+{\displaystyle \frac{1}{2k_B}}\tilde{h},y_m+{\displaystyle \frac{\tilde{h}}{2k_B}}{\mathrm{F}}_1\right)\\ {\mathrm{F}}_2=f\left(t_m+{\displaystyle \frac{1}{k_B}}\tilde{h},y_m+{\displaystyle \frac{\tilde{h}}{2k_B}}{\mathrm{F}}_1+{\displaystyle \frac{\tilde{h}}{2k_B}}{\mathrm{F}}_2\right)\\ y_{m+k_B}=y_m+{\displaystyle \frac{\tilde{h}}{2k_B}}{\mathrm{F}}_1+{\displaystyle \frac{(2k_B-1)\tilde{h}}{2k_B}}{\mathrm{F}}_2\end{array},$$

(9)

$$\begin{array}{ccc}{\displaystyle \frac{1}{2k_B}}& {\displaystyle \frac{1}{2k_B}}& \\ {\displaystyle \frac{1}{k_B}}& {\displaystyle \frac{1}{2k_B}}& {\displaystyle \frac{1}{2k_B}}\\ & {\displaystyle \frac{1}{2k_B}}& 1-{\displaystyle \frac{1}{2k_B}}\end{array}.$$

(10)

Formula (10) satisfies the condition of (5). When $k_B=1\pm 1/\sqrt{2}$, the condition of (6) is satisfied. Thus, when $k_B\ne 1\pm 1/\sqrt{2}$, the formula is a first-order accuracy algorithm.

2.4. Error Analysis of CDA Scheme

The BE method is a first-order accuracy algorithm with a local error of $O(h^2)$. Considering error accumulation, the global error is $O(h)$. The TR and BE methods have different orders of accuracy, resulting in significant differences in their errors. When ignoring the secondary error, under the BE method, a three-phase inductor is equivalent to a parallel three-phase resistor $R_{error}=2L/h$, and a three-phase capacitor under the BE method is equivalent to a series three-phase resistor $R_{error}=h/2C$. Detailed error calculations are provided in Appendix A.

Therefore, when switching between the TR method and BE method, the network also undergoes the operation of connecting or disconnecting resistors. In large-scale power grid calculations, this error propagates throughout the entire grid. In other words, although the fault is local, the algorithm switch causes the fault dynamics of connecting or disconnecting resistors to appear across the entire simulated power grid.

In summary, the linear interpolation algorithm can only maintain the original computational accuracy; the traditional CDA method can maintain the computational accuracy at the second order only with a fixed step using the first-order interpolation algorithm, which renders the interpolation meaningless. In the case of a variable step size, the CDA method will cause the computation accuracy to be reduced to the first order, resulting in large errors.

3. 3S-DIRK-Based Integral Interpolation Scheme

3.1. 3S-DIRK

Based on the characteristics of the Butcher tableau, the algorithms can be categorized into various forms, e.g., FIRK, ERK, DIRK, SDIRK, EDIRK, and FDIRK [6]. To comply with the current form of the EMT-type program, the following characteristics need to be met:

• Diagonally implicit RK formula (DIRK): This enables a separate computation for each stage, without the need to generate a large matrix, which has the advantage of being a multistep method.
• Singly diagonally implicit Runge–Kutta formula (SDIRK): The diagonal elements of the algorithm meeting this characteristic are identical, i.e., all the elements of $a_{ii}$ are identical. When solving this special case of the DIRK algorithm, the Jacobi matrix is guaranteed to remain unchanged at each computational stage, which reduces the computational load of matrix decomposition.
• First-same-as-last formula (FSAL): In this formula, $a_{si}=b_i\left(i=1,\cdots ,s\right)$ and $c_s=1$, which means that the first stage of a step is the same as the last stage from the end of the previous step. This condition ensures that when solving rigid problems, order reduction will not occur in the algorithm [34].

In order to maintain the structure of the EMT-type program, the above three features must meet. Additionally, the three-stage method discussed below is referred to as 3S-DIRK.

3.2. TR-3S-DIRK in Interpolation Process

TR-3S-DIRK is a variable step-size interpolation integration method based on the TR. Assuming that the interpolation step is $\tilde{h}=k_{T}h$ and the new step is $\tilde{h}$, the new DIRK method with second-order accuracy is derived according to Formulas (4) and (5), and its Butcher tableau is as follows:

$$\begin{array}{cccc}0& 0& & \\ {\displaystyle \frac{1}{k_T}}& {\displaystyle \frac{1}{2k_T}}& {\displaystyle \frac{1}{2k_T}}& \\ 1& {\displaystyle \frac{3k_T-1-{k_T}^2}{2k_T}}& {\displaystyle \frac{k_T-1}{2}}& {\displaystyle \frac{1}{2k_T}}\\ & {\displaystyle \frac{3k_T-1-{k_T}^2}{2k_T}}& {\displaystyle \frac{k_T-1}{2}}& {\displaystyle \frac{1}{2k_T}}\end{array}.$$

(11)

The above algorithm is a special case of the 3S-DIRK algorithm. The elements in the first row of the Butcher tableau are all 0, which has no computation in the first stage and only requires integration on the basis of the TR. This method is referred to as the TR-3S-DIRK.

The stages of TR-3S-DIRK are as follows:

$$\mathrm{Stage}0:\mathrm{TR} y_{n+1}=y_n+{\displaystyle \frac{h}{2}}f(t_n,y_n)+{\displaystyle \frac{h}{2}}f(t_{n+1},y_{n+1}),$$

(12)

$$\mathrm{Stage}1:\mathrm{Interpolation} \begin{array}{cc}\hfill y_{n+k_T}& =y_n+{\displaystyle \frac{3k_T-1-{k_T}^2}{2}}hf(t_n,y_n)\hfill \\ \hfill & +{\displaystyle \frac{k_T-1}{2}}{k}_{T}hf(t_{n+1},y_{n+1})+{\displaystyle \frac{h}{2}}f(t_{n+k},y_{n+k_T})\hfill \end{array}.$$

(13)

3.3. BE2-3S-DIRK in Synchronous Process

After events or operations occur, the algorithm switches to the BE method, and the interpolation is BE-based synchronous interpolation. The BE method is itself a low-order algorithm, and the linear interpolation further increases the error. Therefore, it is necessary to improve the computational accuracy using an integration method. Based on two-step BE, similar DIRK methods with second-order accuracy can be derived, and their Butcher tableau is as follows:

$$\begin{array}{cccc}{\displaystyle \frac{1}{2k_B}}& {\displaystyle \frac{1}{2k_B}}& & \\ {\displaystyle \frac{1}{k_B}}& {\displaystyle \frac{1}{2k_B}}& {\displaystyle \frac{1}{2k_B}}& \\ 1& {\displaystyle \frac{3k_B-1-k_B^2}{k_B}}& {\displaystyle \frac{1-4k_B+2k_B^2}{2k_B}}& {\displaystyle \frac{1}{2k_B}}\\ & {\displaystyle \frac{3k_B-1-k_B^2}{k_B}}& {\displaystyle \frac{1-4k_B+2k_B^2}{2k_B}}& {\displaystyle \frac{1}{2k_B}}\end{array}.$$

(14)

The interpolation algorithm (13) has three advantages:

• The variable step size can be used for interpolation synchronization;
• The computational accuracy is improved to the second order, while the first-order error brought by the BE algorithm is corrected;
• It satisfies L-stability, which can eliminate the numerical oscillation in the computation.

This algorithm requires an integral operation on the basis of using the BE method twice. This method is referred to as BE2-3S-DIRK.

The stages of BE2-3S-DIRK are as follows:

$$\mathrm{Stage}2:\mathrm{BE} y_{m+0.5}=y_m+{\displaystyle \frac{h}{2}}f(t_{m+0.5},y_{m+0.5}),$$

(15)

$$\mathrm{Stage}3:\mathrm{BE} y_{m+1}=y_{m+0.5}+{\displaystyle \frac{h}{2}}f(t_{m+1},y_{m+1}),$$

(16)

$$\mathrm{Stage}4:\mathrm{Interpolation} \begin{array}{cc}\hfill y_{m+k_B}& =y_m+(3k_B-1-{k_B}^2)hf(t_{m+0.5},y_{m+0.5})\hfill \\ \hfill & +{\displaystyle \frac{1-4k_B+2{k_B}^2}{2}}hf(t_{m+1},y_{m+1})+{\displaystyle \frac{h}{2}}f(t_{m+k_B},y_{m+k_B})\hfill \end{array}.$$

(17)

4. Application of 3S-DIRK in EMT Program

4.1. Application of 3S-DIRK in DAEs

The proposed algorithm is mainly used for interpolation, and the main algorithm in EMT simulation is the TR algorithm (stage 0). When interpolation is required after a switching operation is detected, a new computational logic is adopted, as shown in Figure 2, with the following stages:

Stage 0: Integrate the TR algorithm, with a normal step, while detecting whether any event occurs; if an event occurs, calculate the coefficient for the interpolation (k) and proceed to stage 1; otherwise, repeat stage 0 until the event occurs or the simulation is finished.

Stage 1: Integrate to pint K using the TR-3S-DIRK algorithm; set the new step coefficient to k_T, as in Formula (12); form a current vector; and solve the network’s voltage.

Stage 2: Integrate using the BE algorithm with a 1/2 step.

Stage 3: Integrate using the BE algorithm with a 1/2 step.

Stage 4: Integrate using the BE2-3S-DIRK algorithm and set the new step size coefficient to k_B, k_T + k_B = 1, as in Formula (16). If no events occur during the process, the steps are illustrated as in Figure 2a. If any events occur during stage 2–4 at the point K’, change the step size and then go back to Stage 2 as in Figure 2b.

This method can ensure that the value at each step point has second-order accuracy, and the algorithm in the switching process is L-stable and capable of suppressing numerical oscillation. Specifically, the BE2-3S-DIRK method is L-stable and can eliminate the first-order error caused by BE.

4.2. Linear Inductors

The electrical components in the EMT-type program are treated as a parallel circuit of an admittance ($G_{equ}$) and a current source ($I_{hist}$), as shown in Figure 3, and the component is converted to the form as:

$$i=G_{equ}u+I_{hist}.$$

(18)

The dynamic equation of the inductor is $L{\displaystyle \frac{d}{dt}}i=u$. The equivalent admittances of the TR and BE methods are both $G_{equ}={\displaystyle \frac{h}{2L}}$.

Assuming that the interpolation coefficients of TR-3S-DIRK and BE2-2S-DIRK are k_T and k_B, respectively, then the formulas of the proposed 3S-DIRK are shown as follows. The derivation processes are explained in Appendix B:

$$\mathrm{TR}\text{-}3\mathrm{s}\text{-}\mathrm{FESDIRK}: I_{hist}=i_n+(3k_T-1-k_T^2)G_{equ}{u}_n+k_T(k_T-1)G_{equ}{u}_{n+1},$$

(19)

$$\mathrm{BE}2\text{-}3\mathrm{s}\text{-}\mathrm{FSDIRK}: I_{hist}=i_m+2(3k_B-1-{k_B}^2)G_{equ}{u}_{m+0.5}+(1-4k_B+2{k_B}^2)G_{equ}{u}_{m+1}.$$

(20)

4.3. Linear Capacitors

The dynamic equation of the capacitor is $C{\displaystyle \frac{d}{dt}}u=i$, and the equivalent admittance is $G_{equ}={\displaystyle \frac{2C}{h}}$ in both the TR and BE methods.

Similarly, the formulas are derived:

$$\mathrm{TR}\text{-}3\mathrm{s}\text{-}\mathrm{FESDIRK}: I_{hist}=G_{equ}{u}_n+(3k_T-1-k_T^2)i_n+k_T(k_T-1)i_{n+1},$$

(21)

$$\mathrm{BE}2\text{-}3\mathrm{s}\text{-}\mathrm{FSDIRK}: I_{hist}=G_{equ}{u}_m+2(3k_B-1-{k_B}^2)i_{m+0.5}+(1-4k_B+2{k_B}^2)i_{m+1}.$$

(22)

In summary, the proposed variable step-size integration algorithm for interpolation can ensure that the equivalent admittances of the elements ($G_{equ}$) are consistent, and, furthermore, the matrix of the entire system does not need to be modified.

5. Case Study

Two case studies—the Series R-L Circuit and the AC/DC Hybrid Power Systems—are presented, demonstrating that the proposed algorithm retains L-stability, is capable of eliminating numerical oscillations, and ensures the maintenance of second-order computational accuracy.

5.1. Case A: Series R-L Circuit

Voltage chatter occurs when a disturbance is applied at a node to which only the inductor is connected, as shown in Figure 4. In this circuit, the voltage source is an ideal source with zero resistance, a magnitude of 132.79 kV, a frequency of 60 Hz, a ramp-up time of 0.05 s, and an initial phase of 0 degrees. The open resistance and closed resistance are $1\times {10}^{12}$ ohm and 0.0005 ohm, respectively. The inductance of the inductor is 0.1 H.

Meanwhile, there are two types of breakers: a thyristor and an insulated-gate bipolar transistor (IGBT). The former is a half-controlled type in which the break waits for the current to be reduced to 0 after receiving the off signal, while the latter is the full-controlled type in which the circuit is turned off immediately when the off signal is received. Therefore, the two cases will be discussed separately. In case A1, the switch does not turn off immediately but waits for the current to drop to 0; in case A2, the switch is forced to turn off, i.e., the turn-off action is implemented immediately when the off signal is received.

5.1.1. Case A1: Breaker of Half-Controlled Type

Assuming that the switch is not forced to turn off, the off signal arrives after 0.1 s, but the current at that time is not 0. When the current reaches 0, the time is 0.104167 s, at which point the switch is turned off, so interpolation is required.

As shown in Figure 5, the conventional TR algorithm will cause numerical oscillation, which can only be eliminated through an algorithm with L-stability.

5.1.2. Case A2: Breaker of Full-Controlled Type

In this case, the switch is forced to turn off when the off signal arrives, at which point the current is zero, and the numerical oscillation is greater.

As shown in Figure 6, both the CDA algorithm and the 3S-DIRK algorithm can eliminate numerical oscillations.

5.2. Case B: AC/DC Hybrid Power Systems

The CDA method will switch algorithms after events occur and the computation accuracy will be degraded to first order, leading to large errors, especially in the computations for power electronics. The proposed interpolation method in this study uses variable-step integration to maintain the calculation accuracy in the second order, and the errors can be controlled within a small range.

To verify the validity of the proposed 3S-DIRK method, the IEEE9 circuit and the line-commutated, converter-based, high-voltage DC (LCC-HVDC) circuit are combined, that is, both circuits are solved simultaneously, but there is no electrical connection between them. The IEEE 9-bus system serves as a standard test case, comprising three generators, three loads, nine buses, and nine transmission lines, which makes it highly suitable for verifying power system analysis and control algorithms. The LCC-HVDC circuit represents a typical configuration in high-voltage direct current transmission systems, consisting of converters, filters, reactors, DC transmission lines, and control and protection systems [35].

The LCC system starts up at 2 s and withdraws at 4 s, and in this time, the CDA method is put into effect. Meanwhile, the potential error sources are as follows: (1) the BE algorithm, (2) the interpolation algorithm, (3) the number of steps in the BE algorithm, and (4) the step size.

Therefore, it is necessary to set up four cases to separately discuss the impact of these factors:

Case B1: CDA without the interpolation method is performed to show the errors caused by the BE algorithm alone;

Case B2: The standard CDA scheme is employed to show the errors caused by linear interpolation and BE methods;

Case B3: To enlarge the errors caused by the BE algorithm, BE is used five times after the switching action;

Case B4: As discussed in Section 2.4, the errors are affected by the time step size, so the step size is changed to 100 μs from 50 μs.

5.2.1. Case B1: Impact of the BE Algorithm

The computation is implemented without interpolation; the BE algorithm is used twice after the switching action; and the step size is set to 50 μs.

Figure 7 shows that while switching the algorithm, the CDA algorithm causes errors in Bus1-BusA’s active power between 2 s and 4 s. Although there are no accidents, a large number of fluctuations occur using CDA method. Meanwhile, during the operation of the LCC, the computational error in the IEEE 9 system is very small using the proposed integral interpolation method.

5.2.2. Case B2: Impact of Interpolation

In this case, interpolation is performed on the basis of the above-described Case B1 example, in which interpolation is performed, and then the BE algorithm is used twice after the switching action, where the step size is set to 50 μs.

Figure 8 shows that in the interpolation process, the CDA error increases. However, the proposed algorithm still performs interpolation after the BE algorithm, but the interpolation coefficient remains at the original step size. The errors caused by the two implementations of the BE algorithm are offset by increasing the computation load, thereby maintaining the computational accuracy in the second order.

5.2.3. Case B3: Impact of the Number of Computations Using the BE Algorithm

Based on the Case B1 example, the number of computations of the BE algorithm is increased to five. Specifically, interpolation is not performed, and then the BE algorithm is used five times after the switching action, in which the step size is set to 50 μs. Because two BE computations are combined into a group in the 3S-DIRK algorithm in this case, three iterations of BE2-3S-DIRK algorithm are performed.

Figure 9 shows that, due to the implementation of multiple rounds of the BE algorithm, the computational error of the CDA is greater. However, the proposed algorithm performs an error correction after every two BE computations, which avoids the increased error derived from the increase in the number of BE computations.

5.2.4. Case B4: Impact of the Simulation Step

In this case, the simulation step is changed to 100 μs on the basis of the Case B1 example, in which interpolation is performed; the BE algorithm is then used twice after the switching action, and the step size is set to 50 μs.

Figure 10 shows that the error caused by the BE method is proportional to the step size. However, the proposed algorithm performs well with different step sizes.

To measure the error, the error indicator is defined as the deviation from the stable value within 2 s to 4 s of the active power of the line between Bus 1 and Bus A. The errors are summarized in

Table 1. Error calculation summary table.

Error (MW)	CDA	3S-DIRK
Case B1	0.0404	0.0002
Case B2	0.4454	0.0003
Case B3	0.0831	0.0005
Case B4	0.1454	0.0032

.

It can be seen that the 3S-DIRK interpolation method proposed in this study can control the errors within a small range, while the conventional CDA method has higher errors, which interferes with the normal computing system due to the power electronics system.

6. Conclusions

In this study, we analyzed the computational accuracy of the CDA method and showed that the interpolation algorithm reduces the computational accuracy to the first order, making it impossible to apply the CDA method to the simulation of large power grids. Using the Butcher tableau, we discussed the accuracy order of the linear interpolation methods and found that linear interpolation algorithms cannot improve the computational accuracy. Therefore, it is impossible to use a linear interpolation algorithm to construct a suitable high-order integration algorithm.

Based on the Runge–Kutta theories, we developed two interpolation algorithms (TR-3S-DIRK and BE2-3S-DIRK) which are used for interpolation after the TR and BE methods. These interpolation algorithms support variable step-size computations while keeping the matrix unchanged and can be used for interpolation and synchronization processes. At the same time, BE2-3S-DIRK can eliminate the first-order error derived from the BE algorithm and increase the computational accuracy to the second order, thereby maintaining the second-order accuracy in the EMT simulations; the BE2-3S-DIRK algorithm is L-stable, which can suppress numerical oscillations.

A new integration scheme was proposed by replacing the linear interpolation of CDA into 3S-DIRK. Then, we derived the formulas for inductors and capacitors and verified that the equivalent admittance of these elements was consistent under different stages, without the need to modify the admittance matrix. We used the Case A examples to verify that the interpolation algorithm retained the L-stability of the original BE algorithm and, thus, is capable of suppressing numerical oscillations. Then, using an example in which the LCC-HVDC system and the IEEE9 system were combined, we verified that the proposed method could maintain second-order computational accuracy throughout the whole computation process.

Moreover, in this study, we provide an algorithm development idea based on the RK architecture, which can be used to further develop new high-order algorithms.