Load Flow Analysis

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Load Flow Analysis

Consider a four bus system.  The complex power in each bus is given by ‘S’.

 S_i = P_i + jQ_i = V_iI_i*

Consider bus 2                                                                                                                        

V₂I₂* = P₂ + jQ₂

I₂ = (P₂ – jQ₂)/V₂*

In terms of self and mutual admittances of the nodes

(P₂ – jQ₂)/V₂* = Y₂₁V₁ + Y₂₂V₂ + Y₂₃V₃ + Y₂₄V₄

Where:

V: stands for Voltage

Y: stands for admittance ie (1/Z)

Z: stands for impedance (the impedance could be resistive(R,r); inductive(X_L, jωL,2πfL); capacitive(X_C, 1/(jωC),1/(j2πfC)))

Solving for V₂ gives

V₂ = 1/Y₂₂[(P₂ – jQ₂)/V₂* - (Y₂₁V₁ + Y₂₂V₂ + Y₂₃V₃ + Y₂₄V₄)]                                {A}

This is basic standard procedure.

From this analytical procedure, one can determine the voltage at any given bus given the voltages at the other buses and the system parameters as well as the system configuration and the loading at the various buses.

Since the voltages at the various buses are not known initially, we start with an educated guess and calculate these voltages iteratively until an acceptable degree of accuracy is attained.

There are two main options used for Load flow analysis. These are the Gauss Seidel Method and the Newton Raphson method. The Gauss Seidel method uses the algebraic expressions derived above iteratively to determine the voltages at the various buses. On the other hand, the Newton Raphson iterative method uses error analysis applied to the Taylor series expansion of the power system showing an expression for complex power about a singularity ignoring higher order derivatives and solving for the associated error and using the error to improve the guessed value or that result after each iteration until an acceptable degree of accuracy is attained. 

In the expression above;

The idea is to calculate the voltage using estimated values of voltages at the other buses or the previously calculated voltages (being an iterative method) in addition to real and reactive power values delivered to the bus from generators or supplied to the load connected to the bus as well as self and mutual admittances of the nodes.

The quantities are used to form the network equation with the aim of finding the voltage.

Equation {A} gives a corrected value of V₂ based upon scheduled P₂ and Q₂

(where V₂ is the voltage at bus 2 and P₂ is the real net power at bus 2 and Q₂ is the imaginary or reactive net power at bus2; net power : the net power is the difference between the generated power and the demand power at any given bus (P₂=PG₂ – PD₂, Q₂ = QG₂ – QG₂ [the unit of P is in Watt (W)[x10⁰ = 1]or kilowatt(kW)[x10³=1000] or Megawatt(MW)[x10⁶ = 1,000,000]. The unit of Q is Vars[10⁰ =1 ] or KiloVars[x10³ = x1000] or MegaVars[x10⁶ = x1,000,000]. [S] stands for complex power so S = P + jQ. The unit of ‘S’ is VoltAamperes (VA)[x10⁰ = x1] or KiloVoltAmperes(KVA)[x10³=x1000] or MegaVoltamperes(MVA)[x10⁶=1,000,000]]))

Where  the values estimated originally are substituted for the voltage expressions; on the right side of the equation.

These improved values of voltages are re-substituted in an iterative way until a desired degree of accuracy is attained.

The Gauss – Seidel Method

In the Gauss-Seidel Method, as the corrected voltage is found at each bus, it is used in calculating the corrected voltage at the next. This calculation is carried out at each bus except the swing bus throughout the network to complete the first iteration. The swing bus is fully specified hence no calculation is carried out on it or required to be carried out. You could however calculate the total power of the system at this bus. Ie. The total power delivered to the system.

This method involves an iterative solving of linear algebraic equations.

Note that it is possible for the voltages to converge upon an erroneous solution if the original voltages are widely different from the correct values.

To prevent this, original values of reasonable magnitude and which do not differ too widely in phase are chosen.

Unwanted solutions can be detected by the experienced engineer easily by inspection of the result since the voltage of the system do not normally have a range in phase wider than 45^o and the difference between nearby buses is less than 10^o and often very small.

For ‘N’ buses, the voltage at any bus k where P_k and Q_k are given is

n

n=1

Σ

 

V_k = (1/Y_kk)([(P_k-jQ_k)/V_k^*]-[        Y_knV_n])

                                                                                                 {AA}

Where  n =/= k

The value of the voltages on the right side of the equation are the most recently calculated values for the corresponding buses (or the estimated voltage if no iteration has yet been made at that particular bus)

Acceleration Factors

These are multipliers designed to improve convergence. They are some constant values that increases the amount of correction to bring the voltage closer to the value it is approaching. Without this, convergence takes a bit longer in a normal Gauss-Seidel method.

The acceleration factor for the real and imaginary values may differ. A wrong choice would obviously have an opposite effect.

At a bus where voltage magnitude rather than reactive power is specified, a value of reactive power is calculated first to help find the real and imaginary components of the voltage for each iteration.

n

k=1

Σ

 

P_k – jQ_k = (Y_kkV_k +      Y_knV_n*)

        

Where   n =/= k                                                                            

If we allow n=k

         P_k – jQ_k = V_k*

n

k=1

Σ

    Y_knV_n

          Q_k = -Im{V_k*

n

n=1

Σ

    Y_knV_n}                                                 {b}

Reactive Power Q_k is evaluated by eqn {b} for the best previous voltage values at the buses, and this value of Q_k is substituted in eqn {AA} to find a new value of V_k. The components of the new V_k are the multiplied by the ratio of the specified constant magnitude of V_k to the magnitude of the V_k found by eqn {AA}. The result is the corrected complex voltage of the specified magnitude.

The reason n =/= k   is because that expression when n =/= k   has been taken out of the Σ sign in one case and expressed separately. In the other case it is used to divide the quantity hence using the Σ sign you have to remove it.

Buses

Buses are also called Busbars. They are connection points in the Electrical Power system. Node in electrical circuit analysis, are abstractions of buses for analytical purposes. There are three main types of buses in the Electrical Power System. These buses are analysed differently. The buses are:

The Slack bus, also called the Reference Bus or the Swing bus.

The Voltage controlled bus or the generator bus

The Load bus

The Voltage controlled Bus

Usually, the voltage controlled bus has a Generator connected to it hence it is also called a Generator Bus. P_Gi and V_i are specified.

P_Gi is constrained b/w limits

P_Gimin ≤ P_Gi ≤ P_Gimax

Q_Gi is also set within limits

Q_Gimin ≤ Q_Gi ≤ Q_Gimax

If Q_Gi falls outside the limit, the limit is taken and V_i becomes the unknown. The calculation is carried out with this understanding and if Q_Gi falls between the limits, V_i is fixed again and the process continued.

The Reference Bus

The Reference Bus has to be a Generator Bus (with a Generator connected to it) or a Tie Bus (for interconnectivity of major system networks) 

Load Bus

This is the most numerous Bus in the Electrical Power System.

All P’s and Q’s are known. The unknowns are the voltage magnitude ‘V’ and the angle ‘δ’ V/δ^o.

Newton Raphson Iterative Method

The Electrical Power System is expressed as a function of ‘δ’ & ‘V’. It is then expressed as a Taylor Series.

It is noted that there are differential quantities in the Taylor Series expression. Second order values of the differential  and higher order values are ignored, because  their effect is  minimal.

 Incremental change ‘ΔV’ & ‘Δδ’ are approximated to error quantities or these differentials.

An iterative method is set in motion such that  the difference between the calculated complex power and the specified Power is as a result of this error quantity. This error is hence calculated and used to improve  the assumed value of the voltages and angles. The exercise is continued until an acceptable degree of accuracy  or a given level of accuracy is established or reached.

Consider a guessed solution x⁽⁰⁾

Let f(x⁽⁰⁾) + Δx⁽⁰⁾ = 0

If we expand f(x) about x⁽⁰⁾ we  obtain

f(x⁽⁰⁾) + Δx⁽⁰⁾(df/dx)⁽⁰⁾ + ½ (Δx⁽⁰⁾)²(d²f/dx²)⁽⁰⁾ + ……..+ = 0

Δx(0) is the error associated with the guess x(0)

If the error is small, the higher order terms can be neglected to give

F(x⁽⁰⁾) + Δx⁽⁰⁾(df/dx)⁽⁰⁾ ≈ 0

The error can be calculated as

Δx⁽⁰⁾ = - f(x⁽⁰⁾)/((df/dx)⁽⁰⁾)

If we add this error to the original guess, we then have an improved guess, such that

X⁽¹⁾ = x⁽⁰⁾ + Δx⁽⁰⁾ = x⁽⁰⁾ – f(x⁽⁰⁾)/(df/dx)⁽⁰⁾

This procedure is repeated until the correct or specified degree of accuracy is attained.

X^(a+1) = x^(a) + Δx^(a) = x^(a) – f(x^(a))/(df/dx)^(a)

Consider ‘n’ dimentional Equations

x1(0) x2(0) . . . xn(0)

x⁽⁰⁾  =  

                                                                                                                                   

Here only the first derivatives are considered.

F₁(x⁽⁰⁾) +(df₁/dx₁)⁽⁰⁾Δx₁ +………………….+ ((df₁/dx_n)⁽⁰⁾) Δx_n ≈ 0

.

.

.

.

F_n(x⁽⁰⁾) +(df_n/dx₁)⁽⁰⁾Δx₁ +………………….+ ((df_n/dx_n)⁽⁰⁾) Δx_n ≈ 0

 

f1(x(0)) . . . fn(x(0))

(df1/dx1)(0)-- -------(df1/dxn)(0)       |                              |       |                              |       |                              | (dfn/dx1)(0)-- -------(dfn/dxn)(0)

Δx1 | | | Δx1

0 | | | 0

 

                                 +                                                                             ≈

Using compact matrix – vector notation

f(x⁽⁰⁾) + J⁽⁰⁾ Δx⁽⁰⁾ ≈ 0

hence

Δx⁽⁰⁾ ≈ -[J⁽⁰⁾]^-1f(x⁽⁰⁾)

This error vector is added to the original guess and the process repeated in an iterative way such that.

x^(a+1) = x^(a)- [J^(a)]^-1f(x^(a))

where ‘a’ is the iteration counter ie counts the number of iterations. (0) stands for guessed values.

J ≡ is an nXn matrix of partial derivatives (df_i/dx_j) called a Jacobian matrix.

In Power Systems Analysis,

δ2(0) δ3(0) . . |vn-1(0)| |vn(0)|

 

x⁽⁰⁾  =

Notice that the reference bus is not included.

This is because it is fully specified. This gives a (2n-2) – dimensional state vector.

By expanding functions for real and imaginary power ‘f_ip’ and ‘f_iq’ in Taylor Series around the initial guess, we have.

P_i ≈ f_ip⁽⁰⁾ + (df_ip/dδ₂).Δδ₂⁽⁰⁾ +………+ (df_ip/d|v_n|)⁽⁰⁾.Δ|v_n⁽⁰⁾|

                                                     For i = 2,…………,n

Q_i ≈ f_iq⁽⁰⁾ + (df_iq/dδ₂).Δδ₂⁽⁰⁾ +………+ (df_iq/d|v_n|)⁽⁰⁾.Δ|v_n⁽⁰⁾|

f_ip⁽⁰⁾ & f_iq⁽⁰⁾ represent real & imaginary (reactive) power leaving bus ’i’ or associated with bus ‘i’ if the bus voltage are set at guessed values.

The difference in power is also called Power Mismatch at a given bus. This is the difference between the measured Power at any given Bus and the calculated power at the same given bus using bus parameters like voltages and admittance and the angle of the voltage ie the power factor. Note that these voltage values and their angles could be guessed values or improved values after each iteration, until an acceptable degree of accuracy is attained. Ideally if the values used for the calculations are correct, there should be no mismatch. This means that there should be no difference. Where there is an error in the values used for the calculation, there will be a mismatch and this is used for calculating the error associated with these parameters and variables. These calculated errors are then used to improve the values of these parameters and these variables used for the calculation subsequently. This procedure is repeated iteratively until an acceptable degree of accuracy is attained.

                                   

ΔP_i⁽⁰⁾  ≈ P_i – f_ip⁽⁰⁾

ΔQ_i⁽⁰⁾ ≈ Q_i – f_iq⁽⁰⁾

                    ≈

ΔP2(0) . . . ΔQn(0)

(df2p/dδ2)(0)-- -----(df2p/d|vn|)(0)       |                              |       |                              |       |                              | (dfnq/dδ2)(0)-- -----(dfnq/d|vn|)(0)

Δδ2(0) | | | Δ|v|(0)x1

 

ΔU⁽⁰⁾ ≈ J⁽⁰⁾.Δx⁽⁰⁾

Procedure For Load Flow Analysis Using Newton Raphson Method

Guess x(0)

Compute ΔU(0)

P_i           =                 

n

k=1

Σ

  |Y_ik||V_i||V_k|cos(δk+δi+ϒ)≈  f_ip                                             

Q_i           =            -    

n

k=1

Σ

  |Y_ik||V_i||V_k|sin(δk+δi+ϒ)≈  f_ip

For i = 1,2,………………n

Note that

Si =

n

k=1

Σ

S_i   =   P_k – jQ_k =     V_k^*     Y_ikV_k

For i = 1, 2, ………………n

Compute the Jacobian Matrix

Solve the voltage error vector ie

Δx⁽⁰⁾ ≈ (J⁽⁰⁾)^-1.ΔU⁽⁰⁾

Add voltage errors to the initial guesses to obtain an upgraded state vector.

The process is repeated.

NB.

G-S – Voltage error used as a convergence measure. Ie the difference b/w subsequent iterative results.

N-R – Power Mismatch, real and reactive used for this purpose.