Power Flow Analysis on a 13 bus system

Mohammad Taha Ahmad

School of Science and Engineering

Lahore University of Management Sciences

Prof. Preetham Goli

School of Computing and Engineering

University of Missouri-Kansas City

Abstract

Power flow analysis is fundamental to power system analysis and design. The analysis is essential for

planning and operation of new power distribution networks as well as for the expansion of existing

networks. It determines the total complex power (real and reactive power) that flows through each bus in

a network. This is imperative for successful operation of a power system. For normal operation of a

power system we need to ensure that the total power generated is enough to meet the demand and line

losses, in addition to making certain that no bus or transformer in the transmission system gets

overloaded. Power flow analysis provides tools that assist in keeping track of the voltage levels at each

bus, therefore helping us to rectify and maintain rated voltages and power flow through the network.

Traditionally, power flow analysis is performed using iterative methods such as Gauss-Seidel method and

Newton-Raphson method. However, for more complex networks that involve a large number of buses, it

becomes increasingly difficult to find solutions by hand. For this reason more complicated networks are

analyzed using computer software programs. The objective of this case study is to analyze a 13 bus

network system using Power World Simulator software.

I. Introduction

Power flow in an electric power system is the complex power that flows from the power

generating plants to the loads. The aim of power flow or load flow analysis is to keep track of

the complex power that flows through each bus (node) of the transmission network to ensure

normal operation of the entire power system [1]. The successful operation of a power system

under three phase steady state requires the following pre-requisites [2]:

Real and reactive components of power from the generators are close to the rated power

supply.

Power supplied is enough to meet the demand and losses.

Power through all the transmission lines and transformers is within their rated capacities.

Bus voltages remain close to rated values.

Power flow in a network is determined by obtaining the magnitude and phase of the voltages for

all the busses in the network and the impedances of the lines that connect the buses [3]. Although

networks are composed of components that are (or are approximated as) linear, real and reactive

power flows are actually nonlinear. Since non-linear equations do not lend themselves to close

form solutions, except in their simplest forms, we rely on numerical iterative methods to obtain a

solution. Methods that are typically used in power flow analysis employ Gauss-Seidel method

and Newton-Raphson method. We would briefly review these methods in the paper. Large

networks that consist of a significant number of buses (which is typical of a power system that

serves a city), involve as many nonlinear equations as the number of buses. This makes it too

complex to find a solution by hand. In order to solve such a large number of equations we use

computer programs to aid us in quickly finding numerical solutions. In our case study, we used

Power World Simulator to analyze a network that consisted of 13 busses, 5 generators and 4

loads. [4]

II. Bus Classification

Fig. 1 Classification of buses

A bus in a network system is a node at which one or more lines or loads, or generators are

connected. The state of any bus in a network system is defined by the following four quantities:

The magnitude of the voltage at the bus.

The phase angle of the voltage at the bus.

The real power that is injected into the bus.

The reactive power that is injected into the bus.

Classification of Buses

Load BusesVoltage

Controlled Buses

Slack or Swing Bus

Two of these four quantities are generally known, and the other two are to be determined by the

solution of the power flow equations. Based on the quantities that are known, the buses are

classified as load buses, generator controlled or voltage controlled buses, and swing or slack

buses.

2.1 Load Buses

Buses are classified as load buses if the real and reactive power injected into the buses are

known. This is the most common type of bus also known as PQ bus. These buses are not

connected to the generators, therefore the generated real and reactive power are taken to be zero.

The real and reactive power injected into the buses are taken as negative, where the negative sign

signifies the complex power drawn by the load. In case of load buses the purpose of load flow

analysis is to determine the magnitude and phase angle of the bus voltage.

The upper limit of the bus voltage magnitude is decided by considering the insulation and the

operation requirements of the loads that are connected to the bus. The magnitude of the lower

limit is decided by the thermal capacity of the network components, accepted levels of the power

that is lost and the operation requirements of the loads. It should be noted that a load bus could

be a generator bus if it is voltage controlled.

2.2 Generator/Voltage controlled buses

Buses are classified as voltage controlled buses if the known quantities are the real power

injected into the bus and the magnitude of the bus voltage. These buses are directly connected to

the generators. The real power injected into the bus can be controlled by changing the prime

mover of the generator and the bus voltage can be controlled by changing the excitation current

of the generator.

However, if a load bus is provided with a reactive source e.g. a synchronous compensator, or

with a tap changing transformer which can maintain the magnitude of the bus voltage, then the

bus would be treated as a voltage controlled bus. Moreover, a generator could be a load bus if the

reactive power limits are violated. For a generator bus the upper limit of the reactive power is

defined by the generator’s real and reactive power capability, while the lower limit depends on

the generator’s stability capacity. Both limits could also be affected by the generator excitation.

2.3 Swing or slack buses

Buses with known magnitude and phase angle of the bus voltage are classified as slack buses.

The slack bus is responsible to compensate for any real and reactive power that needs to be

injected in addition to the complex power being injected by other generators such that the power

demand by the load and the line losses are met. The phase angle of all other buses is taken with

reference to this bus, so its angle is assumed to be zero. [5]

III. Bus Admittance matrix

The admittance matrix of the network provides the relationship governing the behavior of the bus

voltages and the currents that are entering into the bus. The admittance matrix also known as

𝑌𝑏𝑢𝑠, has widespread application in determining the network solution and forms an integral part

of most modern day power system analysis. The relationship between the injected current into

the nodes and the node voltages is therefore given as:

𝐼 = 𝑌𝑏𝑢𝑠𝑉 (1)

𝐼 = vector of currents injected into the node

𝑌𝑏𝑢𝑠 = bus admittance matrix

𝑉 = vector of node voltages

After making the line diagram the following steps are required:

i. Label all the nodes from 0 to n, with node 0 as the reference node.

ii. Replace all generators with an equivalent current source in parallel with an admittance.

iii. The bus admittance matrix is then formed by inspection using the following rule of

thumb;

The diagonal entries (𝑌𝑘𝑘) of the admittance matrix, also known as the self-

admittance or the driving point admittance are computed by summing over all the

admittances (𝑌𝑟) connected to that node.

𝑌𝑘𝑘 = ∑ 𝑌𝑟𝑛𝑟=0 (2)

The off diagonal entries (Ykn) in the matrix, also known as mutual admittance is

the negative of the sum of all the admittances (Ykr) connected between node k and

n.

Ykn = − ∑ Ykrnr=0 (3)

IV. Power Flow Equations and Iterative methods

The bus current equation at bus ‘i’ is,

Ii = ∑ YikVk𝑛𝑘=1 (4)

The complex power equation is given by,

Si = Pi + jQi = ViIi∗ (5)

Where, 𝑉𝑖 = |𝑉𝑖|𝑒𝑗𝜃𝑖

Using Equation 1 in the above equation yields,

Si = Vi ∑ Yik∗ Vk

∗nk=1 (6)

Si = ∑ |Vi|ejθi(|Vk|ejθk)

∗Yik

∗nk=1 (7)

𝑆𝑖 = ∑ |𝑉𝑖||𝑉𝑘|𝑒𝑗(𝜃𝑖−𝜃𝑘)𝑌𝑖𝑘∗𝑛

𝑘=1 (8)

𝑃𝑖 = |𝑉𝑖| ∑ |𝑉𝑘||𝑌𝑖𝑘|cos (𝜃𝑖𝑘 − ∅𝑖𝑘)𝑛𝐾=1 (9)

𝑄𝑖 = |𝑉𝑖| ∑ |𝑉𝑘||𝑌𝑖𝑘|sin (𝜃𝑖𝑘 − ∅𝑖𝑘)𝑛𝐾=1 (10)

Where, 𝜃𝑖𝑘 = 𝜃𝑖 − 𝜃𝑘

∅𝑖𝑘, is the phase angle of the admittance 𝑌𝑖𝑘.

The equations that result in the analysis of power flow are non-linear in nature. The solution to

those equations exists in iterative procedures provided that they converge. The following two

methods are typically used in power flow analysis:

4.1 Gauss-Seidel Method

If bus 1 is a slack bus and the rest are all load buses, then for slack bus;

S1 = V1 ∑ Y1k∗ Vk

∗nk=1 (11)

And for bus 2 through n;

Si = Vi ∑ Yik∗ Vk

∗nk=1 (12)

𝑛 − 1 power equations excluding the slack bus,

Si∗ = Vi

∗ ∑ YikVknk=1 (13)

Si∗

Vi∗ = YiiVi + ∑ YikVk

nk=1,k≠i (14)

Hence,

Vi =1

Yii (

Si∗

Vi∗ − ∑ YikVk)n

k=1,k≠i (15)

Where, 𝑖 = 2, … , 𝑛.

This equation can be used to calculate the voltage at any bus. The most recent value of the

voltage is used on the right hand side of the equation to calculate the next value of the voltage on

the left hand side. For the first iteration, an arbitrary value of the voltage is assumed. This

iterative process is repeated until the value converges to within an acceptable tolerance limit (𝜀).

4.2 Newton-Raphson Method

To obtain a solution using this method the following steps are required, assuming a single bus ‘k’

1. For the iterative step i=0; calculate the bus powers Pk.calci , Qk.calc

i , where 𝑘 = 1, … , 𝑛

using the initial values of bus voltage and phase angles.

2. The calculated power is then compared to the specified value; ∆Pki = Pk.spec − Pk.calc

i ,

∆Qki = Qk.spec − Qk.calc

i .

3. If the magnitude of the bus power mismatch is within the accepted tolerance, the iterative

process ends, or else move to the next step.

4. Calculate the Jacobian terms using the bus voltage and phase angle at the iterative step ‘i’

and invert the Jacobian matrix.

5. Calculate the increments in the phase angle and voltage magnitude.

[ [∆δ]

[∆V] ]

i+1

= ( [J]i )−1 [ [∆P] [∆Q]

]i

(16)

6. Update the phase angle and the bus voltage for the next iterative step,

δki+1 = δk

i + ∆δki+1 (17)

Vki+1 = Vk

i + ∆Vki+1 (18)

7. Go to step 1.

V. Case Study

Fig. 2 below shows a single line diagram of a 13 bus system with all the relevant data. The

transmission system operates at 230 kV.

Fig. 2 Single line diagram of a 13 bus network

The network from fig. 2 was modelled in Power World software. Tables 1-4 show data of the

network before running full Newton-Raphson method.

Table 1. Bus Input Data

Table 2. Line and Transformer Input Data

Table 3. Generator Bus Data

Table 4. Admittance Matrix

The network was run thrice for Gauss-Seidel method iteratively. The mismatches after each

iteration are as follows;

Table 5. Iteration 1

Table 6. Iteration 2

Table 7. Iteration 3

The network was then run for full Newton-Raphson method. The following bus data stemmed as

a result:

Table 8. Bus Data for full Newton-Raphson

Fig. 3 shows the state of the network after full Newton-Raphson method is run. Note that the

figure shows a complete picture of how much the lines and transformers are loaded.

Fig. 3 Network state after running full Newton-Raphson method

VI. Discussion

The network in fig. 2 seems to get overloaded when run for full Newton-Raphson method. Fig. 3

shows three overloaded transformers. Transformer connected between bus 3 and 1 is heavily

overloaded, up to 117%. Transformer connected between bus 8 and 9 is overloaded to 110%, and

transformer connected to bus 12 and 13 is overloaded to 109%. Therefore, the network is

operating under overloaded conditions and necessary steps need to be taken to ensure that the

system operates normally. A number of solutions could be proposed to rectify the overloading of

transformers. We suggest the use of shunt capacitors to resolve the issue. Shunt capacitors

improve power factor, therefore, leading to higher power transmission capability and increased

control of power flow. Shunt capacitor banks are the most reasonable solution because they are

easy to install and maintain. They have a compact design and can be optimized as per

requirement. In addition to this, shunt capacitor banks come with all sorts of fuse technologies

which would further protect the network against any faults.

The network in fig. 2 was modified by introducing three shunt capacitors banks at bus 1, 9 and

13. The capacitor banks provide 62.1 Mvar, 22.9 Mvar, and 23.4 Mvar of reactive power

respectively. This helps just enough to avoid the overloading of transformers. To further reduce

the load, capacitor banks that provide even more reactive power could be installed. Fig. 4 shows

the state of the network after the shunt capacitors are connected. It can be seen that the modified

network now operates under normal conditions.

Fig. 4 Network state after installing capacitor banks

VII. Conclusion

Power flow analysis is crucial in planning new power systems as well as for expansion of

existing ones in addition to determining the optimum operation of current system. The principal

information provided by the analysis is the total complex power flowing in the lines and the

magnitude and phase angle of the voltage at each node (bus). This helps in monitoring the power

system and plan for an emergency situation. [6]

References

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Journal of Scientific and Research Publications 2.11 (2012): 1-2. Web.

[2] Overbye, Thomas J. "Power Flows." Power System Analysis and Design. By J. Duncan.

Glover and Mulukutla S. Sarma. 5th ed. Boston: PWS Pub., 1994. N. pag. Print.

[3] Stevenson, William D., Jr., and John J. Grainger. Power System Analysis. N.p.: n.p.,

1994. Print.

[4] Saadat, Hadi. "Power Flow Analysis." Power System Analysis. 3rd ed. N.p.: n.p., 2010.

N. pag. Print.

[5] El-Shatshat, Ramadan. "Power Systems Engineering." (n.d.): n. pag. Web.

[6] D P Kothari and I J Nagrath., “Modern Power System Analysis”, Third Edition, Chapter

No.6, Tata McGraw-Hill Publishers, 2010