Power Flow Studies Using MATLABExecutive SummaryThe report aims to show the differences between Newton-Raphson and Gauss-Seidel methods by using them to analyse a power flow system. During the study in a two-bus system, with a slack bus and PQ bus, two convergence tolerances were used. The case study 1 compares the iterative process of the two methods for a voltage tolerance convergence and case study 2 compares the iterative process of the two methods for a power mismatch convergence. The main points that investigated are the results and the speed of convergence. Although, both methods converge to the same values, Newton-Raphson converges exponentially and compared to Gauss-Seidel which converges linearly, needs only around half iterations to return the expected result.Table of ContentsExecutive SummaryIntroductionBackground TheoryNewton-Raphson MethodGauss-Seidel MethodPower Flow Study Results and AnalysisCase Study 1Case Study 2Analysis of the results and convergence characteristics acquired by the Newton-Raphson and Gauss-Seidel methodsConclusionsReferencesIntroductionA load or power flow analysis is a computational procedure required to determine the sinusoidal steady state of the entire power system. The mathematical solution includes a system of nonlinear algebraic equations, which during an iterative process, the values converge to only one solution. The outcome of the power flow analysis is the acquisition of the voltage magnitudes and angles at each bus. Once the voltage magnitudes and angles at each bus have been acquired using power flow analysis, the real and reactive power at each branch of the power system can be computed. Furthermore, losses in particular lines can be computed by investigating the difference between the power flow in the sending and receiving ends and by determining over and under load conditions, the appropriate solution is taken into consideration. In the power industry, load flow analysis is used on a daily basis as it is essential in determining the fault-free, stable and economical operation of existing power systems, as well as for future expansion of power systems.A two-bus system is given in Figure 1 below.Figure 1: Two-bus SystemThe power system consists of a slack bus, which is illustrated as bus 1 and a load bus, which is illustrated as bus 2.  Bus 1 is a slack bus (or swing, or reference) and its main role is to be the phase angle reference of the system. It also balances the active power |P| and reactive power |Q| in a system by injection in circumstances when is required. It is a reference bus for which the voltage magnitude and angle are input data (V1??1) and the active power |P| and reactive power |Q| are the unknown quantities need to be calculated by the power flow analysis. Usually the V1??1 is 1.0?0°, however in this assignment should be considered as 1.06?0°.Bus 2 is a load (P-Q) bus which absorbs power from the system. The active power |P| and reactive power |Q| are quantified and voltage magnitude |V| and angle |?| need to be computed.Background TheoryNewton-Raphson MethodThe Newton-Raphson method is a powerful method for maximising an objective function by using quadratic convergence of approximations [1].A number of nonlinear algebraic equations is given in matrix formatfx=f1xf2x…=y                  (1.1)where x and y are N vectors and fx is an N vector of functions. The aim is to solve for x, accounting that fx and y are given. By rewriting eq. 1.1y–fx=0                        (1.2)Then, by adding Dx to both of the eq. 1.2y–fx+Dx=Dx           (1.3), where D is a square NxN invertible matrix. Then, multiplying by D–1 in both partsx+D–1y–fx=x             (1.4)The values of x on the left side are used to calculate the next values of x on the right side, so thatxi+1=xi+D–1y–fxi          (1.5)For non-linear equations, it is necessary that matrix D must be specified. One of the methods of specifying matrix D called the Newton-Raphson method.The Newton-Raphson method is based on the Taylor series expansion of   fx .y=fx0+dfdx|x=x0x–x0…          (1.6)By neglecting the higher order terms in eq 1.6x=x0+[dfdx|x=x0]–1 y–fx0             (1.7)During the Newton Raphson iteration process, x0 is been replaced by the old value x and x by the new value xi+1. To continue to the next step, the N×N Jacobian matrix is used.Ji=dfdx|x=x(i)=?f1?x1?f1?x2…?f1?xN?f2?x1?f2?x2…?f2?xN…………?fN?x1?fN?x2…?fN?xN           (1.8)where the elements inside the Jacobian matrix are partial derivatives. Therefore, by substituting the Jacobian matrix into eq. 1.7.xi+1=xi+J–1(i) y–fxi          (1.9)Instead of computing the inverse of the Jacobian matrix J–1, it can be rewritten as :Ji?xi= ?yi                (1.10), where       ?xi=xi+1–xi         1.11    and     ?yi= y–fxi         (1.12)The Newton-Raphson method converges to the result in each iteration and depending on the convergence tolerance we can select the precision needed. The steps below show the process during each iteration.Step 1                By using the eq. 1.12, we compute ?yiStep 2  By using the eq. 1.8, we compute JiStep 3 By using Gauss elimination and back substitution, we solve the eq. 1.10 in order to find ?xiStep 4 Finally we compute xi+1 from eq. 1.11. Compare xi+1 with xi to find the convergence tolerance to determine the precision of the result. [2]Power MismatchThe equation for complex power expressed as Si_Si=Pi+jQi=ViIi*            (1.13)From equation 1.13, the current injection into any bus I can be expressed as_Ii=?k=1nYikVk               (1.14)where, Yik terms are admittance matrix elements. Then, we substitute eq. 1.14 into eq. 1.13:Si=Vi?k=1nYikVk*              (1.15)where, Vk is a phasor, with magnitude and angle Vi=|Vi|??i . Moreover, the admittance matrix is complex with real and imaginary parts Yik=Gik+jBik. Then, by rewriting eq. X asSi=Vi?k=1nYik*Vk*=|Vi|??i?k=1n(Gik+jBik)*|Vk??k*            (1.16)   =|Vi|??i?k=1n(Gik–jBik)(|Vk?–?k              (1.17)=?k=1n|Vi|??i(|Vk|?–?k)(Gik–jBik)        (1.18)=?k=1n(|Vi||Vk|??i–?k)(Gik–jBik)        (1.19)A phasor may be also expressed as a function of sinusoids Vi=Vi??i=Vicos?+jsin?. Then by rewriting the eq. 1.19 we get_Si=?k=1nVi|Vkcos??i–?k+jsin?i–?k(Gik–jBik            (1.20)Then by performing the multiplications inside the parenthesis and by splitting the equation into real and imaginary parts, we can find the active (Pi) and reactive (Qi) power.Si=Pi+jQi          (1.21) Pi=ReViIi=?k=1nVi |Vk|Gikcos?ik+Biksin?ik      (1.22)Qi=ImViIi=?k=1nVi |Vk|Giksin?ik+Bikcos?ik         (1.23)These two equations called the power flow equations and are used as fundamental equations for load flow analysis.Jacobian matrix for a two-bus power system:Jacobian Matrix=?P2??2?P2?|V2|?Q2??2?Q2?|V2|        (1.24) ?P2??2=|V1||V2|B21cos??2–?1           (1.25)     ?P2?|V2|=|V1|B21sin?2–?1              (1.26)?Q2?|?2|=V1|V2|B21sin??2–?1           (1.27)?Q2?|V2|=–V1B21cos??2–?1–V2B22            (1.28) Gauss-Seidel MethodThe Gauss-Seidel method used as an iterative process to solve a square system of linear equations. It uses the formxi=fix1,x2,x3,…xN               (2.1)To do a power flow analysis into a power system, a set of linear equations I=YbusV analogous to y=Ax needs to be solved. For every load bus load bus, Ii can be calculated fromIi=Pi–jQi Vi*                   (2.2)The Gauss-Seidel method equation isxin+1=1Aiiyk–?k=1i–1Aikxkn+1–?k=i+1i–1Aikxkn               (2.3)By using eq. 2.3 and applying it to the nodal equationsVin+1=1YiiPi–jQiVi*n–?k=1i–1YikVkn+1–?k=i+1i–1YikVkn               (2.4)Power Flow Study Results and AnalysisCase Study 1This case study will focus on the comparison of the Newton-Raphson and Gauss-Seidel methods by drawing a relationship log10?(max?Vik+1–Vik) vs number of iterations using a voltage convergence tolerance of ?p=10–6: max?Vik+1–Vik