Load flow studies 1

LOAD FLOW STUDIES (LFS)

Newton-Raphson Power Flow

SOLUTION OF A SET OF NONLINEAR

EQUATIONS BY

NEWTON-RAPHSON METHOD

In this section we shall discuss the solution of a set of

nonlinear equations through Newton-Raphson method.

Let us consider that we have a set of n nonlinear

equations of a total number of n variables x1, x2, , xn.

Let these equations be given by

nnn

n

n

xxf

xxf

xxf

,,

,,

,,

1

212

111

where f1, , fn are functions of the variables x1, x2,

, xn. We can then define another set of functions g1,

, gn as given below

0,,,,

0,,,,

0,,,,

11

21212

11111

nnnnn

nn

nn

xxfxxg

xxfxxg

xxfxxg

Let us assume that the initial estimates of the n

variables are x1(0), x2

(0), , xn(0). Let us add

corrections x1(0), x2

(0), , xn(0) to these

variables such that we get the correct solution of

these variables defined by

00

0

2

0

22

0

1

0

11

nnn xxx

xxx

xxx

The functions in equation then can be written in

terms of the variables given as

nk

xxxxgxxg nnknk

,,1

,,,,,000

1

0

11

We can then expand the above equation in Taylor’s series around the nominal values of x1

(0), x2(0), ,

xn(0).

Neglecting the second and higher order terms of the

series, the expansion of gk, k = 1, , n is given as

0

0

0

2

0

2

0

1

0

1

00

11 ,,,,

n

kn

k

knknk

x

gx

x

gx

x

gxxxgxxg

00

1

00

12

00

11

0

0

2

0

1

0

21

22212

12111

,,0

,,0

,,0

nn

n

n

nnnnn

n

n

xxg

xxg

xxg

x

x

x

xgxgxg

xgxgxg

xgxgxg

The square matrix of partial derivatives is called the

Jacobian matrix J with J(0) indicating that the matrix is

evaluated for the initial values of x2(0), , xn

(0). We can

then write the solution of as

0

0

2

0

1

10

0

0

2

0

1

nn g

g

g

J

x

x

x

Since the Taylor’s series is truncated by neglecting the

2nd and higher order terms, we cannot expect to

find the correct solution at the end of first iteration.

We shall then have

001

0

2

0

2

1

2

0

1

0

1

1

1

nnn xxx

xxx

xxx

These are then used to find J(1) and gk(1), k = 1, , n.

We can then find x2(1), , xn

(1) from an equation and

subsequently calculate x2(1), , xn

(1).

The process continues till gk, k = 1, , n becomes less

than a small quantity

Newton Raphson Method

• Power flow equations formulated in polar form. For the system in Fig.1, Eqn.2 can be written in terms of bus admittance matrix as

Expressing in polar form;

Note: j also includes i

Substituting for Ii from Eqn.21 in Eqn. 4

Separating the real and imaginary parts,

Expanding Eqns. 23 & 24 in Taylor's series about the initial estimate neglecting h.o.t. we get

The Jacobian matrix gives the linearized relationship between small changes in Δδi(k) and

voltage magnitude Δ[Vik] with the small changes in real and reactive power ΔPi

(k) and ΔQi

(k)

The diagonal and the off-diagonal elements of J1 are:

Similarly we can find the diagonal and off-diagonal elements of J2,J3 and J4

The terms ΔPi(k) and ΔQi

(k) are the difference between the scheduled and calculated values, known as the power residuals.

Procedures: 1. For Load buses (P,Q specified), flat voltage start. For voltage controlled buses

(P,V specified),δ set equal to 0.

2. For Load buses, Pi(k) and Qi

(k) are calculated from Eqns.23 & 24 and ΔPi(k) and

ΔQi(k) are calculated from Eqns. 29 & 30.

3. For voltage controlled buses, and Pi

(k) and ΔPi(k) are calculated from Eqns. 23 &

29 respectively.

4. The elements of the Jacobian matrix are calculated.

5. The linear simultaneous equation 26 is solved directly by optimally ordered triangle factorization and Gaussian elimination.

6. The new voltage magnitudes and phase angles are computed from (31) and (32). 7. The process is continued until the residuals ΔPi

(k) and ΔQi(k) are less than the

specified accuracy i.e.

Fast Decoupled Method • practical power transmission lines have high X/R ratio. •Real power changes are less sensitive to voltage magnitude changes and are most sensitive to changes in phase angle Δδ. •Similarly, reactive power changes are less sensitive to changes in angle and are mainly dependent on changes in voltage magnitude. •Therefore the Jacobian matrix in Eqn.26 can be written as

The diagonal elements of J1 given by Eqn.27 is written as

Replacing the first term of the (37) with –Qi from (28)

Bii = sum of susceptances of all the elements incident to bus i.

In a typical power system, Bii » Qi therefore we may neglect Qi

Furthermore, [Vi]2 ≈ [Vi] . Ultimately

In equation (28) assuming θii-δi+δj ≈ θii, the off diagonal elements of J1 becomes

Assuming [Vj] ≈ 1 we get

Similarly we can simplify the diagonal and off-diagonal elements of J4 as

With these assumptions, equations (35) & (36) can be written in the following form

B’ and B’’ are the imaginary part of the bus admittance matrix Ybus. Since the elements of the matrix are constant, need to be triangularized and inverted only once at the beginning of the iteration.

Since the voltage magnitude at PV buses is fixed there is

no need to explicitly include these voltages in x or write

the reactive power balance equations

–the reactive power output of the generator varies to

maintain the fixed terminal voltage (within limits)

–optionally these variations/equations can be included

by just writing the explicit voltage constraint for the

generator bus

|Vi | – Vi setpoint = 0

For the two bus power system shown below, use the

Newton-Raphson power flow to determine the

voltage magnitude and angle at bus two. Assume

that bus one is the slack and SBase = 100 MVA.

Line Z = 0.1j

One Two 1.000 pu 1.000 pu

200 MW

100 MVR

0 MW

0 MVR

The most difficult computational task is inverting the Jacobian matrix

– inverting a full matrix is an order n3 operation, meaning the amount of computation increases with the cube of the size size

– this amount of computation can be decreased substantially by recognizing that since the Ybus is a sparse matrix, the Jacobian is also a sparse matrix

– using sparse matrix methods results in a computational order of about n1.5.

– this is a substantial savings when solving systems with tens of thousands of buses

Advantages – fast convergence as long as initial guess is close to

solution

– large region of convergence

Disadvantages – each iteration takes much longer than a Gauss-Seidel

iteration

– more complicated to code, particularly when implementing sparse matrix algorithms

Newton-Raphson algorithm is very common in power flow analysis

Since most of the time in the Newton-Raphson iteration is spend calculating the inverse of the Jacobian, one way to speed up the iterations is to only calculate/inverse the Jacobian occasionally

Decoupled Power Flow

The completely Dishonest Newton-Raphson is not used for power flow analysis. However several approximations of the Jacobian matrix are used.

One common method is the decoupled power flow. In this approach approximations are used to decouple the real and reactive power equations.

Fast Decoupled Power Flow

• By continuing with our Jacobian approximations we can actually obtain a reasonable approximation that is independent of the voltage magnitudes/angles.

• This means the Jacobian need only be built/inverted once.

• This approach is known as the fast decoupled power flow (FDPF)

• FDPF uses the same mismatch equations as standard power flow so it should have same solution

• The FDPF is widely used, particularly when we only need an approximate solution

FDPF Approximations

FDPF Three Bus Example

Use the FDPF to solve the following three bus system

Line Z = j0.07

Line Z = j0.05 Line Z = j0.1

One Two

200 MW

100 MVR

Three 1.000 pu

200 MW

100 MVR

END