An Effective Method to Determine the Available Transmission Capacity with Variable Frequency Transformer

Available transmission capacity (ATC) is a top priority and lucrative solution for electricity market players. Fast and accurate ATC solutions enable participants to gain a competitive advantage in the marketplace. In order to procure a competitive solution, we propose the diferential power fow equation-based static ATC. Using the proposed method, the entire time-domain trajectory of a dynamic system can be solved along with a diferential equation based on a fctional time. In this paper, a variable frequency transformer (VFT) was used to control and increase the transmission power, which made the calculations more complex. In addition, the proposed method was used in seven diferent systems and 50% of the time was saved on small systems. It also enjoyed a much better performance (up to 90%) on large systems. It can be concluded that the proposed static ATC introduces promising results and is suitable for online applications.


Overview of VFT.
Interconnected power grids are used to maximize energy resources [1]. Synchronous and asynchronous methods can be used to connect power grids. In the synchronous method, cofrequency networks are only connected to each other using AC transmission lines. Te method is simple and economical, but it complicates the operation of the power system. In the event of a severe fault, it may also reduce the stability of the power system. HVDC transmission lines are used for asynchronous or nonfrequency connections. Tis method helps to transfer power in large amounts and the system can be operated in a fexible manner. Nevertheless, the design of the HVDC systems is expensive and quite complex [1].
Power transfer between nonfrequency systems can be optimized through better methods. Te variable frequency transformer (VFT) is a new device that can connect two networks (cofrequency or nonfrequency) and exchange power [2]. Te security level of VFT is very high. In 2004, General Electric successfully installed and tested VFT in Langlois for the frst time in the world. Te invented VFT gave power companies a way to control the power between electrical networks more easily than previous methods [3]. Since then, VFT projects and research have prevailed [1][2][3].
Tis paper assumes that at least one VFT in the network is used for optimal power transmission in determining ATC. Ten, the existing methods and the proposed method were used to calculate the ATC without and with VFT.

Overview of ATC.
Available transmission capacity (ATC) has historically been a signifcant problem in both static [4,5] and dynamic [6,7] power system operations. Recently, there has been a need to calculate and send ATC every hour. In dynamic ATC calculations, voltage stability analysis and transient stability analysis (VSA-TSA) are the most critical dynamic constraints [8].
Te well-known continuation power fow (CPF) [9] is notorious for being an approach to computing only static ATC. As a way to improve CPF computations, the general minimum residual (GMRES) method can still be used [10].
In addition, the NRS (Newton-Raphson-Seydel) method has proved to be capable of calculating VSA and SATC faster and more accurately than Newton-Raphson [11].
Furthermore, by removing trigonometric equations in power fow and combining GMRES/CPF, downhill, and NRS algorithms, we can witness that the accuracy and speed of static ATC calculation greatly improved [12]. In the paper [13], the efect of the use of wind farms on the stability of the static voltage of the power system has been investigated.
You can also obtain DATC [14] by increasing production and consumption and by zeroing certain values. Tis method is lame to be proper for online computations. According to [15], diferential evolutionary algorithms can give us the exact Hopf bifurcation bounds for a dynamic ATC computation.
Te quest for efcient and intelligent optimal power fow (OPF) calculation has led to many new aspects such as the application of artifcial intelligence (AI) techniques into the OPF model and, thus, the ATC solution method [16]. A number of artifcial intelligence (AI) methods were reported in the context of ATC calculation including the bee algorithm, artifcial neural network, genetic algorithm, particle swarm optimization, evolutionary planning, cuckoo search algorithm, and grey wolf optimizer [16,17].
Te frst contingency of TTC (FCTTC) was calculated dynamically as reported in [18]. Te shortcoming of this method was that it took a long time to compute the unstable equilibrium points. Te dynamic ATC was calculated in 2002 [19], using two direct Lyapunov methods. With the peak of the potential energy method, the transient stability and Jacobian matrix determinants of a large network can also be approximated with an acceptable accuracy and speed.
In view of the dynamic ATC solution process, artifcial intelligence methods tended to obtain ATC output using a classical approach, but the cumbersome training time and sophisticated way of dealing with AI methods limited the realistic application of this method in utilities [8].
Dynamic ATC computation was presented using the support vector regression (SVR) method (2012-2017) [20]. Using the PEBS method made it possible to identify the various load patterns of support vector regression (SVR). Eventually, as in [12], with the advent of renewable sources on networks, the probabilistic computations feature in ATC rather than the deterministic. Te probabilistic power fow (PPF) [21] is recommended rather than the conventional power fow algorithm. In [22], the authors proposed a novel PPF approach to access ATC. Furthermore, the results advocated that this method can procure an efective and a more accurate solution in evaluating the static ATC.
Te estimation of total transfer capability (TTC) can be modeled as the optimal power fow issue under transient stability constraints (TSC-OPF) [23]. In addition, power fow parameters should be calculated using the state estimation program.
Te holomorphic embedding load fow (HELF) algorithm [6,24] hardly required any initial guessing or simplifcation, and it applied concepts within a complex analysis. By comparison, the salient shortcoming of HEPF is the long computation time [6].
Te diferential load fow approach has shown that this method is efective in solving transient stability simulation [7,25]. Te researchers in [7] have further shown that the nonlinear AC power fow can be transformed into a set of equivalent linear equations that does not need any iteration. Te linearized AC power fow model provided more accurate, timely, and computationally efcient results than conventional CPF [7].
In the current work, the embedding diferential equation power fow (DEPF) within the proposed method (SATC-DE) [26] was used. Moreover, the proposed model in this paper employed the initial model of DEPF [26] and improved the way of static ATC (SATC) calculation. Te developed model represented consistent performance and less computational overhead for practical and large systems while considering N-1 credible contingency scenarios.

Contribution.
Te following are the salient features of the new approach to static ATC evaluation in the presence of VFT.
Te present paper aimed to develop a diferential equation based on static ATC calculation (SATC-DPF) in the presence of VFT. Tis was not an iteration-based method and, therefore, the computational time could be signifcantly decreased. Te proposed idea used the initial work on the diferential equation load fow algorithm reported in [7,26] which proved efcient in terms of the dynamic power fow. Te internal mechanism of DEPF was adopted in the solution of SATC as an alternative to improve its outputs.
Te structure of this paper is to frst defne DEPF in Section 2. Te proposed method for SATC evaluation in the context of VFT is then discussed in Section 3. Te novel method is presented in Section 4.
In Section 5, the discussion and results are presented. Seven systems for various bilateral-multilateral contracts have all been tested using the proposed method. Finally, in Section 6, the conclusions and references are provided.

Differential Equation Power Flow (DEPF)
In the CPF method, a linear equation is solved in every correction-prediction step and in every iteration, whereas in the proposed method, a linear equation is solved only once per time step. Ten, we review the DEPF approach briefy discussed in [7,25,26].
Te load fow is given in equation (1) where S represents the complex power, Y bus is the matrix of bus admittance, V is the bus voltage, λ is the loading parameter, and b denotes a load direction.
It is also possible to write equation (2) in the general form as equation (3) with N as the number of buses, y � [β 1 , · · · , β N , c 1 , · · · , c N ] T as the rectangular coordinate of voltage, λ as the loading parameter, β � real(y), c � Im(y), and g as the nonlinear vector feld.

(3)
A polynomial of the k th degree is represented by the following equation: Y (k) by equation (5) can be approximated as x (t): It can be seen that Y (k) represents y (t) at the k th order [7,25], as In equation (7), there are fve laws. Diferential transformations of y (t) are defned as Y (k), delta functions in the discrete domain are defned as Δ, and a constant is defned as α.
Te following is a brief description of each step of the DEPF method.
Equations (9) and (10) are shown as the diferential power fow. C i is fxed and v l (t) is the voltage of bus i.

International Transactions on Electrical Energy Systems
A y,PQ � a P,1 a Q,1 · · · a P,M a Q,M T , Step 4. We can directly obtain the solution curves for power fow equations as shown by the following equation: In steps 1 to 4, we will use the basic idea outlined to solve Λ(k), X(k), and Y(k) until we get the desired value. Tere is no need to iterate numerically to fnd these coefcients directly (see the DEPF algorithm in Table 1). λ(t) and y(t) can be obtained through equation (16) when the solution curves are demonstrated. In the CPF method, a linear equation is solved in every correctionprediction step and in every iteration, whereas, in the proposed method, a linear equation is solved only once per time step. Te main characteristics of the DEPF model allowed us to fnd a fast and an accurate SATC solution in a user-friendly and simple implementation scheme. Taking into account the normal and N-1 contingency scenarios, the proposed method for SATC assessment using VFT is described in Section 3. Figure 1 shows how VFT can be interconnected to the network and Figure 2 shows the simplifed circuit model of VFT. Figure 1 shows the relationship between the powers as follows:

VFT Model
where P L is the output power of the stator, P d is the mechanical power of the drive system, and P g is the input power of the rotor. VFT's current and voltage equations can easily be calculated by using the following bipolar equations: Now, we can apply equation (18) to the load fow equation and control the transfer power between the stator Decide the initial values: y(0), x(0), and λ(0) Output: Λ(k), X(k), Y(k), k: 0 to K Steps: Find A gy , A gλ , A hy , and A hλ using equation (15) Find A −1 gy if formulation I given in equation (9) or if formulation-II in equation (10) While k : 1 to K Find B g and B h using equation (15) If Te next section describes the determination algorithm of static ATC using VFT.

The Novel Method of SATC with VFT
Te new power fow is assessed using the DEPF approach with an increase and decrease of the loading parameter at the updated contract. Te static ATC for normal and the loss or failure of a small part of the power system have been determined. Te decisions to be made in terms of a decrease or increase of the contract value can be adjusted based on the power fow solution. Figure 3 shows the suggested algorithm for static ATC assessment using DEPF with VFT.
Te static ATC assessment after a generator failure is similar to that of Section 2. If the generator is disconnected, the bus type changes to the load type. Load type is applied to this type of bus since the reactive power would not be sufcient to control the voltage magnitude. As a further method of evaluating static ATC, it is possible to simulate a line outage by adding a parallel line to the line with a negative equal impedance.

Discussion and Results
To simulate the proposed static ATC model, both MATLAB and DIgSILENT power factory were used. A computation machine clicking at Intel Core i7, 2 core (s), 2.3 GHz with 8 GB RAM was employed to study the developed mode performance. Te proposed SATC was tested on seven systems such as 39 [27], 118 [27], 145 [28], 300 [27], 1150, 4440 [19], and 6067 [29] buses in order to demonstrate the efcacy and processing time performance of the static ATC. Te simulation results of the proposed model are compared with six well-known methods including CPF [9], CPF-GMRES [10], improved NRS [11], the old static ATC [19], HEPF [6], and the proposed method.
Te initial DEPF technique benchmarked in this paper as reported earlier [7] was studied on the IEEE 39-bus system, and promising results were outlined. Terefore, Table 2 summarizes the results obtained by solving the six methods over the IEEE 39-bus system. According to Table 2, the power fow outputs are very identical for all the contenders.
To further challenge the proposed SATC-DPF calculations and verify the strength of the model, the real-world operation regimes were implemented. For each of the seven test systems, the static ATC was computed for 4 bilateralmultilateral contracts. In Table 3, the detailed nominated contracts are shown. In this sense, in contractij, (i) stands for system no. and (j) points to the contract number. Table 4 shows the static ATC solutions calculated using the 6 methods for 7 systems for bilateral-multilateral contracts. Te root mean square error (RMSE) and calculate relative speed (CRS) of the proposed method were compared with other conventional approaches.
Te number of buses is not directly correlated with the CRS in Figure 4. Te innovative method in this work outperformed other methods in terms of the execution time required to fulfll the static ATC calculation. According to the fndings, on average, the proposed method consumed almost half of the processing time versus the other methods.
Similarly, Figures 4 and 5 show that the proposed method is still quicker than the customary ways even if there is a VFT. It can be seen, also, that the proposed method is faster despite the VFT.
As shown in Figure 6, we can see that the static ATC solution accuracy found by all methods presents an error scale of 0.08% up to 1.98% compared with ATC calculated from the CPF method.
As in Table 4, the proposed method takes almost 45% to 65% less time to calculate static ATC as compared to the conventional methods 1 to 5, and 8% of the time is saved as compared to the HEPF method.
In the same vein, a similar time-performance analysis can be found for a larger system of the 118-bus IEEE system. As it can be seen, 60% to 89% of the time can be saved compared to the contender methods (1 to 5), as well as a 14% more time is saved than the HEPF method.
To validate the proposed model over a practical and large-scale system, Table 4 outlines the computational time elapsed for each method over several test systems including 300-bus, 1150-bus, 4440-bus, and 6067-bus systems. It can be concluded that the proposed method procures a faster solution to almost 20% to 90% computational-efcient static ATC calculations which reiterates the fact that the developed model is practically useful and suited for online purposes.
Table (4) shows that, as in Figure 6, the calculation error of the proposed approach despite VFT is less than that of the other existing methods.

International Transactions on Electrical Energy Systems
According to the fndings, we can easily understand that the proposed method outperforms other traditional ATC algorithms, and this superiority is even more apparent in the case of the large-scale and real-world utility of the Khorasan distribution system with 4,440 buses. Te simulation results afrm that the most accurate approach belongs to the between the loads and generators. 5. Using DEPF (section 2) at this contract (i) fnd out maximum loading parameter (Lambda).

. Conclusion
We created a new algorithm based on diferential power fow equations to determine the static available transmission capacity. Tis paper also employs a power controller known as a variable frequency transformer to increase power. Calculations are complicated by the presence of this controller. Te static ATC model was developed to fnd a quicker and a more accurate static ATC solution quality. In order to verify the efectiveness of the proposed approach, a wide variety of test systems were examined, from standard IEEE systems to practical and large-scale utility cases. Te proposed approach exhibits an increase in calculation speed and a decrease in calculation error when compared to the current methods with VFT. Shortly, the proposed static ATC method outperformed other methods in terms of the CPU time required to end the execution of the algorithm while providing accurate results. Tis allows us to conclude that the proposed method is suie for online applications in large distribution and transmission networks with VFT.    International Transactions on Electrical Energy Systems

Data Availability
All data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.