Power Flow Analysis Using Numerical Computational Methods on a Standard IEEE 9-Bus Test System

Power flow analysis, also known as load flow analysis, is the foundation of voltage-current system analysis and design. The study of power flow analysis is essential to understanding problems in voltage-current system operation and distribution. It is also a fundamental technique in electrical engineering that enables the planning, operation, and optimization of power systems by determining steady-state conditions, ensuring reliability, efficiency, and the integration of renewable sources, and facilitating informed decision-making for grid stability and resource utilization. The goal of such an analysis is to determine how safely the system can work, i.e., whether there are installation overloads or extremely high or extremely low voltages nodes. It also provides more information about all of the system's components, ensuring that each generator operates at its optimal performing level that consumer’s needs are achieved without overloading the facilities, and that plans for maintenance can be carried out without jeopardizing the system's network. Load flow investigations also provide a mathematical methodology for calculating different bus voltages, phase angles, real power (A) and reactive power flow (B) via various nodes under steady-state conditions [1]. It also aids in determining the impact of a single generating station or transmission path failure on the load system. Moreover, it provides a balanced steady-state operation state of the load system while ignoring a non-steady process system. This means that any load flow problem's mathematical formulation is a nonlinear algebraic equation system without differential equations. As a result, using prepared algorithms and programs for load flow analysis is critical for performing recent load system analysis.

Over the years, different methods have been used for calculating load flow. The use of these methods is primarily driven by the fundamental requirements of power flow calculation, which include its iteration properties, computing accuracy and storage requirements, as well as its convenience and flexibility of implementation [2-5].

The power flow problem is a system of nonlinear algebraic equations that must be solved mathematically. Its answer will almost always require some iteration. As a result, the most crucial condition for a load flow calculation method is reliable convergence. The dimension of load flow equations grows increasingly large as the size of the load system expands. Thus, the power flow analysis is becoming increasingly important for equations with such high dimensions.

In this work, power flow analysis using the Newton Raphson method and Gauss-Seidel on an IEEE standard 9 bus test system is compared. The 9-bus test system is a simplified model of an electrical power network used for analyzing power system techniques. It comprises 9 buses, including a slack bus with known values, load buses with demand, a generator bus with both load and generation, and branches representing connections between buses. Loads and generation are specified at certain buses, and the system is used to study power flow, voltage profiles, and stability under various conditions.

In the realm of power system analysis, methods for solving the power flow equations play a pivotal role in ensuring the efficient and reliable operation of electrical networks. Two widely employed techniques for solving these equations are the Newton-Raphson and Gauss-Seidel methods. This study aims to compare and evaluate the performance of these methods in the context of power flow analysis. By systematically assessing their convergence characteristics, computational efficiency, and accuracy, we seek to gain insights into their respective strengths and limitations. This study compares the Newton-Raphson and Gauss-Seidel methods for power flow analysis on the IEEE 9-bus test system, focusing on convergence, accuracy, and efficiency. The simulation setup involves replicating the system, implementing methods with defined criteria, and running analyses. Through this analysis, we aim to provide valuable guidance for selecting the most suitable method based on specific application scenarios, contributing to the advancement of power system analysis techniques and the optimization of electrical grid operation.

Understanding the analysis in the solution of nonlinear algebraic simultaneous equation serves as the foundation in solving nonlinear equations in digital power system flow analysis [3]. The main idea in power flow study is to get the X-bus admittance matrix using the transmission and input data transformer. The system’s equation for a power with an X bus is:

$I=X_{\text {BUS }} V$     (1)

In a generalized pattern for n number of bus:

$I_i=V_i \sum_{i=0}^n X_{i j}-\sum_{j=1}^n X_{i j} V_j \quad \text{For} \,\mathrm{i}=1,2,3 \ldots n$   (2)

At BUS i, real and reactive power is:

$A_i+B_i=I_i^* V_i$   (3)

Or

$\frac{A_{\mathrm{i}}+B_{\mathrm{i}}}{\mathrm{V}_{\mathrm{i}}^*}=\mathrm{I}_{\mathrm{i}}$     (4)

Substituting for I in terms of A_i & B_i  in the equation gives:

$\frac{A_{\mathrm{i}}+B_{\mathrm{i}}}{\mathrm{V}_{\mathrm{i}}^*}=V_i \sum_{j=0}^n X_{i j}-\sum_{j=1}^n X_{i j} V_j$    (5)

The mathematical formulation of the load flow problem, derived from the above equation, yields a set of non-linear algebraic equations that must be solved iteratively. As a result, the Newton Raphson and Gauss Seidel solution methods must be reviewed.

3.1 Newton-Raphson method

The method described above was named after Isaac Newton and Joseph Raphson. This numerical method can be traced back to the 1960s [4]. Taylor's series is used in this method to approximate a series of nonlinear algebraic equations to a series of linear algebraic equations. This method has a more powerful convergence characteristic than any other alternative process and has also proven reliable because, unlike any other iterative process, it can solve a case of divergence [6]. The number of iterations needed to reach a solution is unaffected by the size of the system, making it more robust and completely efficient.

Eq. (1) gives the current flowing into the bus I for a notable bus system. Thus, the bus admittance matrix, is as follows:

$I_i=\sum_{j=1}^n X_{i j} V_j$    (6)

Bus i is included in the above equation as j. When we express this equation in polar form, we get:

$I_i=\sum_{j=0}^n\left|X_{i j}\right|\left|V_j\right| \angle \theta_{i j}+\delta_j$      (7)

Then complex load at bus i is:

$A_i-j B_i=V_i^* I_i$           (8)

Substituting for I_i in Eq. (7) from Eq. (8), we have

$A_i-j B_i=\left|V_i\right| \angle-\delta_i \sum_{j=1}^n\left|X_{i j}\right|\left|V_j\right| \angle \theta_{i j}+\delta_j$          (9)

Separating the real and imaginary parts,

$A_i=\sum_{j=1}^n\left|V_i\right|\left|V_j\right|\left|X_{i j}\right| \cos \left(\theta_{i j}-\delta_i+\delta_j\right)$       (10)

$B_i=\sum_{j=1}^n\left|V_i\right|\left|V_j\right|\left|X_{i j}\right| \sin \left(\theta_{i j}-\delta_i+\delta_j\right)$   (11)

Eqs. (10) and (11) form a set of nonlinear algebraic equations in terms of |V| per unit and in radians. For each load bus, use the two equations from Eqs. (10) and (11). Also, the controlled-voltage bus is solved by Eq. (10). The following set of linear equations is derived by enlarging Eqs. (10) and (11) in Taylor's series about the initial guess and leaving out all terms in higher order.

$\left[\begin{array}{c}\Delta A_2^{(k)} \\ \cdot \\ \cdot \\ \Delta A_n^{(k)} \\ \hline \Delta B_2^{(k)} \\ \cdot \\ \cdot \\ \Delta B_n^{(k)}\end{array}\right]=\left[\begin{array}{ccc}\left(\begin{array}{ccc}\frac{\partial A_2^{(k)}}{\partial \delta_2} & \cdots & \frac{\partial A_2^{(k)}}{\partial \delta_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial A_n^{(k)}}{\partial \delta_2} & \cdots & \frac{\partial A_n^{(k)}}{\partial \delta_n}\end{array}\right) & \left(\begin{array}{ccc}\frac{\partial A_2^{(k)}}{\partial\left|V_2\right|} & \cdots & \frac{\partial A_2^{(k)}}{\partial\left|V_n\right|} \\ \vdots & \ddots & \vdots \\ \frac{\partial A_n^{(k)}}{\partial\left|V_2\right|} & \cdots & \frac{\partial A_2^{(k)}}{\partial\left|V_n\right|}\end{array}\right) \\ \left(\begin{array}{cccc}\frac{\partial B_2^{(k)}}{\partial \delta_2} & \cdots & \frac{\partial B_2^{(k)}}{\partial \delta_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial B_n^{(k)}}{\partial \delta_2} & \cdots & \frac{\partial B_n^{(k)}}{\partial \delta_n}\end{array}\right) & \left(\begin{array}{ccc}\frac{\partial B_2^{(k)}}{\partial\left|V_2\right|} & \cdots & \frac{\partial B_2^{(k)}}{\partial\left|V_n\right|} \\ \vdots & \ddots & \vdots \\ \frac{\partial B_2^{(k)}}{\partial\left|V_2\right|} & \cdots & \frac{\partial B_n^{(k)}}{\partial\left|V_n\right|}\end{array}\right)\end{array}\right]\left[\begin{array}{c}\Delta \delta_2^{(k)} \\ \cdot \\ \cdot \\ \Delta \delta_n^{(k)} \\ \hline \Delta\left|V_2^{(k)}\right| \\ \cdot \\ \cdot \\ \Delta\left|V_n^{(k)}\right|\end{array}\right]$         (12)

Because the swing bus variable |V| and < are already known, they are omitted from Eq. (12). After expressing the partial derivatives of Eqs. (10) and (11) that give a linearized relationship between small changes in |V| and <, the element of the Jacobian matrix is derived. In matrix form, the equation is as follows:

$\left[\begin{array}{l}\Delta A \\ \Delta B\end{array}\right]=\left[\begin{array}{ll}J_1 & J_2 \\ J_3 & J_4\end{array}\right]\left[\begin{array}{c}\Delta \delta \\ \Delta|V|\end{array}\right]$    (13)

where, J_{1 …} J₄ are the elements of the matrix.

The terms $\Delta A_i^{(k)}$ and $\Delta B_i^{(k)}$ are the difference between the scheduled and calculated values. This is called the power residuals, given by

$\Delta A_i^{(k)}=A_i^{s c h}-A_i^{(k)}$   （14）

$\Delta B_i^{(k)}=B_i^{s c h}-B_i^{(k)}$      (15)

The new estimates for bus voltages are:

$\delta^{(k+1)}=\delta_i^{(k)}+\Delta \delta_i^{(k)}$       (16)

$\left|V_i^{(k+1)}\right|=\left|V_i^{(k)}\right|+\Delta\left|V_i^{(k)}\right|$           (17)

The proposed Newton-Raphson method procedure for power flow is presented as follows:

(1) For load buses, where $A_i^{s c h}$ and $B_i^{s c h}$ are known, magnitudes of the voltage and phase angles are set the same as the swing bus values, or 1.0 and 0.0 , i.e., $\left|V_i^0\right|=1.0$ and $\delta_i^{(0)}=$ 0.0 . For voltage-regulated buses, where $\left|V_i\right|$ and $P_i^{s c h}$ are known, phase angles are set the same as the swing bus angle, or 0 i.e., $\delta_i^{(0)}=0$.

(2) For load buses, $A_i^{(k)}$ and $B_i^{(k)}$ are solved by (10) and (11). While $\Delta A_i^{(k)}$ and $\Delta B_i^{(k)}$ are solved by (14) and (15) respectively.

(3) Controlled-voltage buses, $A_i^{(k)}$ and $\Delta A_i^{(k)}$ are solved using Eqs. (10) and (11) respectively.

(4) Elements of the Jacobian matrix $\left(\mathbf{J}_1, \mathbf{J}_2, \mathbf{J}_3\right.$, and $\left.\mathbf{J}_4\right)$ are solved.

(5) The linear simultaneous equation in Eq. (13) is calculated directly by optimally ordered triangular factorization and Gaussian elimination.

(6) The new $\left|V^{(k+1)}\right|$ and phase angles are calculated using Eqs. (16) and (17).

(7) The procedure is repeated until the residuals $\Delta A_i^{(k)}$ and $\Delta B_i^{(k)}$ are less than the specified accuracy, i.e.,

$\left|\Delta A_i^{(k)}\right| \leq \epsilon$     (18)

$\left|\Delta B_i^{(k)}\right| \leq \epsilon$     (19)

3.2 Gauss seidel method

This is a method for solving a set of nonlinear algebraic equations that is iterative [7]. The method employs an initial guess for the value of voltage to get a derived value of a specific parameter. A derived value replaces the initial guess value. After that, the process is repeated until the convergence of the iteration [8-12]. The initial guess has a large effect on the convergence time. However, the method has a poor convergence characteristic [8, 13-16].

This is an iterative method used to solve (5) for the value of V_i, and the iterative series becomes:

$V_i^{(k+1)}=\frac{\frac{A_i^{s c h}-j B_i^{s c h}}{V_i^*}+\sum X_{i j} V_j^{(k)}}{\sum X_{i j}} j \neq i$     (20)

Kirchhoff's Current Law suggested that current entering the bus will be positive. Thus, for buses where real and reactive powers are inserted into the bus, such as generator buses, $A_i^{s c h}$ and $B_i^{s c h}$ are positive. For load buses where real and reactive powers are flowing out from the bus, $A_i^{s c h}$ and $B_i^{s c h}$ are negative. Solving the power flow equation in (5) for $A_i$ and $B_i$, we have:

$A_i^{(k+1)}=\Re\left\{V_i^{*^{(k)}}\left|V_i^{(k)} \sum_{j=0}^n X_{i j}-\sum_{j=1}^n X_{i j} V_j^{(k)}\right|\right\} j \neq i$   (21)

$B_i^{(k+1)}=-\Im\left\{V_i^{*(k)}\left[V_i^{(k)} \sum_{j=0}^n X_{i j}-\sum_{j=1}^n X_{i j} V_j^{(k)}\right]\right\} j \neq i$       (22)

The power flow expression is typically expressed in terms of the bus admittance matrix elements. Since the off-diagonal elements of the bus admittance matrix, $X_{b u s}$, shown by uppercase letters, are $X_{i j}=-X_{i j}$, and the diagonal $X_{i i}=\sum X_{i j}$, Eq. (20) gives

$V_i^{(k+1)}=\frac{\frac{A_i^{s c h}-j B_i^{s c h}}{V_i^*}+\sum_{j \neq i} X_{i j} V_j^{(k)}}{X_{i i}}$    $j \neq i$  (23)

$A_i^{(k+1)}=\Re\left\{V_i^{*^{(k)}}\left[V_i^{(k)} X_{i i}+\sum_{j=1}^n X_{i j} V_i^{(k)}\right]\right\}$     $j \neq i$  (24)

$B_i^{(k+1)}=-\Im\left\{V_i^{*^{(k)}}\left[V_i^{(k)} X_{i i}+\sum_{j=1}^n X_{i j} V_j^{(k)}\right]\right\}$ $j \neq i$          (25)