The power flow problem can also be solved by using Newton-Raphson method. In fact, among the numerous solution methods available for power flow analysis, the Newton-Raphson method is considered to be the most sophisticated and important. Many advantages are attributed to the Newton-Raphson (N-R) approach.
Gauss-Seidel (G-S) is a simple iterative method of solving n number load flow equations by iterative method. It does not require partial derivatives. Newton-Raphson method is based on Taylor’s series and partial derivatives.
The N-R method is recent, needs less number of iterations to reach convergence, takes less computer time hence computation cost is less and the convergence is certain. The N-R method is more accurate, and is insensitive to factors like slack bus selection, regulating transformers etc. and the number of iterations required in this method is almost independent of the system size.
The drawbacks of this method are difficult solution technique, more calculations involved in each iteration resulting in large computer time per iteration and the large requirement of computer memory but the last drawback has been overcome through a compact storage scheme.
Convergence can be considerably speeded up by performing the first iteration through the G-S method and using the values of voltages so obtained for starting the N-R iterations. These voltages are used to compute active power P at every bus except the swing bus and also reactive power Q wherever reactive power is specified.
The difference between the specified and calculated values is used to determine the correction of bus voltages. The process of iteration is continued till the difference in the specified and calculated values of P, Q and V are within the given permissible limit.
Before explaining the application of N-R method to the power flow problems, it is useful to review this method in its general form.
Let the unknown variables be (x1, x2, x3 …, xn) and the specified quantities (y1, y2, y3, …, yn).
These are related by the set of nonlinear equations:
The solution of above nonlinear equations is started with an approximate solution –
the zero subscript defining the zeroth iteration in the process of solving the above nonlinear Eq. (6.82). The noteworthy point is that the initial solution for the nonlinear equations should not be very far from the actual solution, otherwise, there are chances of the solution diverging rather than converging and it may not be possible to have a solution whatever the computer time taken.
At first glance it may appear to be a great drawback for the N-R method but the problem of initial guess for a power system is not at all difficult. A flat voltage profile, i.e., Vi = (1.0 + j 0) for i = 1, 2, 3 … n except the slack bus has been found to be satisfactory for almost all practical systems.
Let Δx10, Δx0, Δx30, ….. Δxn0 be the corrections, which on being added to the initial assumed values, give the actual solution.
Expanding these equations in Taylor’s series around the initial guess, we have –
Partial derivatives of second and higher order are neglected according to N-R method. In fact it is this assumption that requires the initial solution to be close to the final solution.
Linearizing all the equations and arranging them in matrix form, we have:
In vector from above equation can be written as –
B = J.C …(6.86)
where J is the square matrix of the partial derivatives on the RHS and is known as Jacobian matrix. The solution of the equations needs calculation of left hand vector B which is the difference of the specified quantities and calculated quantities at (x01, x02, x03,…,x0n). Similarly J is calculated at this guess.
Solution of the matrix equation provides (Δx01, Δx02, Δx03,………..,Δx0n) and the better estimates of the solution are given by –
Repeating the process of iterations, with these values, we get yet better estimated values. The (Δx1, Δx2, Δx3,………..,Δxn) becomes smaller and smaller with every iteration and finally the iteration process is stopped when (Δx1, Δx2, Δx3,………..,Δ xn) are lesser than pre-specified values.
Newton-Raphson Method Applied to Power Flow Problem:
N-R method can be applied to power flow problems in a number of ways, the most common being those using:
1. Rectangular coordinates and
2. Polar coordinates.
1. N-R Method using Rectangular Coordinates:
In this formulation the quantities are expressed in rectangular form.
The general expression for power is given a –
Where ei and fi are the real and imaginary components of the bus voltage vi, and therefore –
Where Gik and Bik are conductance and susceptance respectively.
Substituting Eqs. (6.89), (6.90) and (6.91) in Eq.(6.88) for power, we have:
Separating the real and imaginary parts, we have-
Separating for ith bus, the power Eqs. (6.93) and (6.94) become –
Thus the above formulation results in a system of nonlinear algebraic equations, two equations (one for Pi and the other for Qi) at each bus. So excluding the slack bus (bus 1) where V and δ are specified and remains fixed throughout, the total number of equations to be solved for n bus system will be (2n – 1) equations.
With the help of the Newton-Raphson method, the above nonlinear algebraic equations of power is transferred into a set of linear algebraic equations inter-relating the changes in power (i.e., error in power) with the change in real and reactive components of bus voltages with the help of jacobian matrix. Thus we have from Eqs. (6.93) and (6.94).
In short form Eq. (6.98) can be written as –
In case the system contains all types of buses, the set of equations can be written as –
The elements of the jacobian matrix can be derived from the three power flow Eqs. (6.96), (6.97) and (6.95).
The off-diagonal elements of J1 are –
And diagonal elements of J1 are –
The off-diagonal elements of J2 are –
And the diagonal elements of J2 are –
The off-diagonal elements of j3 are –
And the diagonal elements of j3 are –
The off-diagonal elements of j4 are –
And the diagonal elements of j4 are –
The off-diagonal and diagonal elements of j5 are –
The off-diagonal and diagonal elements of j6 are –
The steps for solving power flow problem by the N-R method are given below:
1. So, for the load buses where P and Q are given, we assume the bus voltages magnitude and phase angle for all the buses except the slack bus where V and δ are specified. Normally we have the flat voltage start, i.e., we set the assumed bus voltage magnitude and its phase angle (i.e., the real and imaginary components e and f of the bus voltages) equal to the slack bus quantities.
2. Substituting this assumed bus voltages (i.e., e and f) in Eqs. (6.96) and (6.97), we calculate the real and reactive components of power, i.e., Pi and Qi for all the buses i = 2, 3, 4, …, n except the slack bus (bus no. 1).
3. Since Pi and Qi for any bus i is given, i.e., specified, the error in power will be –
where r is an iteration count.
Here Rri and Qri are the power calculated with the latest value of bus voltages at any iteration r.
4. Then the elements of Jacobian matrix (J1, J2, J3 and J4) are determined with the latest bus voltages and calculated power Eqs. (6.96) and (6.97).
5. After this the linear set of Eq. (6.98) is solved by iterative technique or by the method of elimination (normally by Gaussian elimination method) to determine the voltage correction, i.e., Δei and Δfi at any bus i.
6. This value of voltage correction is used to determine the new estimate of bus voltages as follows:
Where r is an iteration count.
7. Now this new estimate of the bus voltage, i.e. eir + 1 and fir + 1 is used in Eqs. (6.96) and (6.97) for power to re-compute the error in power and thus entire algorithm starting from step 3 as listed above is repeated.
Here in each iteration, the elements of Jacobian are computed as these depend upon the latest voltage estimate and calculated power. The process is continued till the error in power becomes very small.
where ϵ is very small number.
2. N-R Method Using Polar Coordinates:
The Newton-Raphson method can also be applied to the solution of power flow problem when the bus voltages are expressed in polar form. In fact, only polar form is used in practice because the use of polar form results in a smaller number of equations than the total number of equations involved in rectangular form.
For any ith bus, we have –
When δ is the phase angle of the bus voltages and θik is and admittance angle.
Then according to Eq. 6.57(b) for any ith bus –
Substituting the values of Vi, Vk and Yik from Eq. (6.114) in Eq. (6.115) we have –
Now the linear equation in polar form becomes –
where J1, J2, J3 and J4 are the elements of Jacobian matrix and can be determined from power Eqs. (6.117) and (6.118) as follows:
The off-diagonal and diagonal elements of J1 are –
The off-diagonal and diagonal elements of J2 are –
The off-diagonal and diagonal elements of J3 are –
The off-diagonal and diagonal elements of J4 are –
The elements of Jacobian matrix are computed with the latest voltage estimate and computed power. However, the procedure (i.e., algorithm, here) is the same as that of the rectangular coordinates. The formulation in the polar coordinates takes less computational efforts and also needs less memory space.
Approximation to N-R Method:
It is well known that the flow of reactive power Q is not much affected by the change in phase angle δ and similarly the flow of active power remains almost in-affected by variation in magnitude of nodal voltage (ΔV).
Keeping these facts in mind and using the polar coordinates, the set of linear power flow equations can be written in matrix form as follows:
where J1 corresponds to the elements ∂P/∂δ which exist.
J2 corresponds to the elements ∂P/∂V which do not exist and, therefore, are zero.
J3 corresponds to the elements ∂Q/∂ δ which do not exist and, therefore, are zero.
J4 corresponds to the elements ∂Q/∂V which exist
This certainly simplifies the calculations and results in small computation time.
Voltage-Controlled or Generation Bus:
Here P and the magnitude of voltage V are given.
Now the real power P for any bus i is given as:
And also for ith bus, we have –
Where Vi is the voltage magnitude and ei and fi are its real and imaginary components.
The matrix equations inter-relating the changes in bus powers and square of the bus voltage magnitude to the changes in the real and imaginary components of voltage are given by Eq.(6.98 b).
Where Δ (Vri)2 = (Vi specified)2 – (Vri)2 and Vri is the calculated bus voltage after the rth iteration.
Elements of jacobian matrix are calculated using Eqs. (6.107)…..(6.110).
Here Vri is the bus voltage computed at the rth iteration and Vi specified is the voltage specified at any bus i as it is the generation bus.
After obtaining bus voltages, power flow and lines losses are calculated using Eqs. (6.77) and (6.78).
The flow-chart for N-R method using polar coordinates is given in Fig. 6.21.