A Quick and Comprehensive Method for Determining Static ATC with NRS and VFT

: Electricity market players prioritize available transfer capability (ATC) as an attractive solution. Market participants can gain a financial advantage through accurate and fast ATC solutions. In order to provide a suitable and valuable solution, we use differential load flow equations. A dynamic system's entire time-domain trajectory can be solved by this method, along with a fictional time-domain differential equation. This article uses Newton-Raphson-Seydel instead of Newton-Raphson, which can also be used to determine voltage stability. A variable frequency transformer (VFT) was used in this study to increase and control transmission power. A 50% time saving on small systems was achieved with the proposed method, which was applied to seven different systems. In addition, it performed better on large systems by more than 90%. This proposal for static ATC presents promising results and can be applied to online applications.


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‫ﻣﺤﻮ‬ ‫اﻟﻤﺘﺎﺣﺔ؛‬ ‫اﻟﺘﺤ��ﻠ�ﺔ‬ ‫ﻟﻘﺪرة‬ ‫اﻟﺘﻔﺎﺿﻠ�ﺔ.‬‫اﻟﻤﻌﺎدﻟﺔ‬ ‫اﻟﺠﻬﺪ؛‬ ‫ار‬ ‫اﺳﺘﻘﺮ‬ ‫؛‬ ‫ي‬ ‫اﻟﻤﺘﻐ�‬ ‫دد‬ ‫ت‬ ‫اﻟ�‬ Corresponding author's e-mail: eidiani@ijesse.net َ ‫ﺗ‬ ABSTRACT: Electricity market players prioritize available transfer capability (ATC) as an attractive solution.Market participants can gain a financial advantage through accurate and fast ATC solutions.In order to provide a suitable and valuable solution, we use differential load flow equations.A dynamic system's entire time-domain trajectory can be solved by this method, along with a fictional time-domain differential equation.This article uses Newton-Raphson-Seydel instead of Newton-Raphson, which can also be used to determine voltage stability.A variable frequency transformer (VFT) was used in this study to increase and control transmission power.A 50% time saving on small systems was achieved with the proposed method, which was applied to seven different systems.In addition, it performed better on large systems by more than 90%.This proposal for static ATC presents promising results and can be applied to online applications.DEPF load change in each direction Fnew: New LF equations gdepf: DEPF general form of LF Jnew: New Jacobian matrix PL: Output power Pd: Mechanical power of the drive system Pg: Input power to the rotor Rs, Rr: Xm, Xs: VFT impedance Vg, VL: Ig, IL: Voltage and current of both sides of VFT Xold: Old network variables Xnew: New general variables XVFT: VFT variables YVFT: VFT admittance matrix ZVFT: VFT impedance matrix y0: NRS initial node injection w: DEPF decompose the voltage λ,υ: Eigenvalues and eigenvectors

INTRODUCTION
Energy sources are maximized by interconnecting power grids with asynchronous and synchronous methods (Khan et al. 2021).Asynchronous and synchronous connections are made with HVAC and HVDC transmission lines, respectively.As the backbone of the future transmission network, HVDC lines are considered in ATC calculations.HVDC transmission lines allow large amounts of power to be transferred, but the design and analysis of HVDC systems are quite complex and expensive (Khan et al. 2021).Power can be exchanged between two asynchronous or synchronous networks using a variable frequency transformer (VFT) (Khan et al. 2021).The first VFT was installed and tested by GE in Langlois in 2004.With the VFT, power between electrical networks can be controlled more easily than before (Merkhouf et al. 2006).Assumed in this paper is that one VFT in the network determines optimal power transmission.It has long been an important issue in power system operation to calculate and send ATC every hour -both static ATC (SATC) (Liu et al. 2020, Eidiani et al. 2010) and dynamic ATC (DATC) (Eidiani 2021).DATC calculations are highly dependent on transient and voltage stability analysis (TSA-VSA) (Mohammed et al. 2019).A general minimum residual method (GMRES) (Eidiani et al. 2010) can still be used to improve continuation power flow (CPF) (Zambroni et al. 2000) computations in SATC.The advantage of Newton-Raphson-Seydel (NRS) over NR is its faster and more accurate calculation of VSA and SATC (Eidiani et al. 2010).ATC solution methods have been improved by incorporating artificial intelligence (AI) techniques into the optimal power flow calculation (OPF) (Lai et al. 1997).The ATC calculation was carried out using several AI methods, including cuckoo search, artificial neural networks, genetic and bee algorithms and particle swarm optimization (Lai et al. 1997).Total transfer capability's (TTC) first contingency was limited by the high calculation time needed to compute unstable equilibrium points.ATC can be approximated with acceptable speed and accuracy using the Jacobian matrix determinant, transient stability and peak of potential energy method (Kim et al. 2009).The DATC calculation with renewable sources on networks was presented using the support vector regression (SVR) method from 2012 to 2020 (Shaban 2018).A probabilistic power flow (PPF) approach for accessing SATC was proposed in (Karuppasamypandiyan et al. 2020).According to the results, this method provides a more accurate and effective evaluation of SATC.An optimal power flow problem under transient stability constraints (TSC-OPF) can be used to estimate total transfer capability (Zhang et al. 2020).The state estimation program should also be used to calculate load flow (LF) parameters (Eidiani 2021).There is no simplification or initial guessing required with the holomorphic embedding power flow (HEPF) algorithm (Eidiani 2021), although it has a long computation time.Using the differential LF approach, transient stability simulations can be solved effectively (Eidiani 2021).A numerical algorithm that does not require iteration is developed by the researchers in (Eidiani 2021) in order to solve nonlinear AC load flows.An embedding of differential equation power flow (DEPF) into the proposed method (SATC-DE) was used in the current work.Additionally, this study used DEPF (Eidiani 2021)'s initial model and improved the method of calculating Static ATC (SATC).With the developed model, large and practical systems can be analyzed with less computational overhead and consistent performance.
We describe here the key characteristics of the new SATC evaluation method with VFT.We developed a differential equation-based SATC calculation in the presence of VFT in this study.The new algorithm is based on the initial work on the differential equation LF algorithm that was published in (Eidiani 2021), which showed efficiency in terms of dynamic LF.In section 2, DEPF with NRS and VFT model is defined, and in section 3, the proposed approach for SATC assessment with VFT is discussed.The fourth section presents the results and discussions.
Lastly, the proposed method is tested on several bi-and multilateral contract systems.

DEPF WITH NRS AND VFT MODEL
An illustrated network connection for VFT can be seen in Figure 1, and a simplified circuit diagram for VFT can be seen in Figure 2. The relationship between Pg, Pd, and PL is shown in Figure 1.PL is the output power, Pd is the mechanical power of the drive system, and Pg represents the input power to the rotor.

Pg+Pd=PL
(1) The following are bipolar equations that can be used to calculate VFT's voltage and current equations (Figure 2).
[ ] [ ] Equation ( 2) can be applied to the LF equation, and the VFT angle (δ) can be varied to adjust the transfer power between the rotor and stator.
Increased ATC can be achieved by controlling power at the VFT, which then controls power at the transmission line.
It should be noted that in a VFT, where torque is applied to the rotor, the induced Emf in the coil of the rotor remains constant, but its phase angle changes by θrs.The following real equations are obtained by simplifying the LF equations in Figure 2.
The main LF equations can be easily modified by deriving LF flow equations (3).We can replace old network variables ( ) with new general variables ( ) by defining VFT variables as ( ).It is now possible to calculate the new LF equations and the new Jacobian matrix of the network.

(
) NRS load flow can now be used to solve equation (6) (Eidiani et al. 2010) [ ] And: ( ) The NRS equation can now be combined with the DEPF equation.The proposed method solves a linear equation only once per time step, whereas the CPF method solves it in every correctionprediction step.We now briefly review the DEPF approach (Eidiani 2021).DEPF calculations are based on the concept of converting continuoustime parameters into discrete variables like (w(t)� �W(k)).The normal LF is depicted in ( 9), where (b) is the load change in each direction and ψ is the loading parameter.
In order to decompose the voltage, we need to do the following: It is now possible to write ( 10) as ( 11) in the general form.
First, the DEPF method expands the algebraic equation ( 11) by adding a state variable (x) to a set of DAEs.We obtain the following linear equation by linearizing all the equations using ( 12).As a result, we can obtain the Jacobian matrix of the method as follows (Eidiani 2021): The nonlinear LF equation ( 14) is solved using (A1A2A3A4) matrices in a differential transformation (Eidiani 2021).We have:

W k k B A I A A A A A A A B A A A A
DEPF model's key characteristics helped us find a simple and user-friendly SATC solution in an accurate and fact manner.

THE PROPOSED METHOD OF SATC
LFs are calculated using the DEPF approach at the contract update with a decrease or increase in loading parameters.SATCs have been determined for normal and contingency cases.An algorithm for assessing SATC using DEPF, NRS, and VFT can be seen in the following.
1-All network information, controllers, and sales contracts should be entered.2-For each bi/multilateral contract (i) between seller and buyer, change the vector of load direction in (9) accordingly.3-Utilize the DEPF method developed in the previous section of this contract to find the maximum load parameter.4-In this contract, calculate the ATC for normal and contingencies.5-Using the previous steps, determine the maximum allowable transmission power between buyer and seller.6-At each contract, evaluate SATC.7-If the type of contract is changed, the algorithm goes back to step 1 until the final contract.
It is similar to the previous section in the SATC calculation after the generator is lost or disconnected, except the bus type changes to PQ.A parallel line with negative impedance can also simulate a line disconnected.Lines and transformers have a maximum thermal load limit of 100%, active power of 0 to 90%, a reactive power of 50% to -50%, and a maximum voltage difference of 5%.This method shows high speed and accuracy in several networks based on the above conditions.

Figure 3 -
Figure 3-Comparing the proposed method with other methods with and without VFT 0 5 10 15 20