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Load flow is an important tool used by power engineers for planning, to determine the best operation for a power system and exchange of power between utility companies. In order to have an efficient operating power system, it is necessary to determine which method is suitable and efficient for the system’s load flow analysis. A power flow analysis method may take a long time and therefore prevent achieving an accurate result to a power flow solution because of continuous changes in power demand and generations. This paper presents analysis of the load flow problem in power system planning studies. The numerical methods: Gauss-Seidel, Newton-Raphson and Fast Decoupled methods were compared for a power flow analysis solution. Simulation is carried out using Matlab for test cases of IEEE 9-Bus, IEEE 30-Bus and IEEE 57-Bus system. The simulation results were compared for number of iteration, computational time, tolerance value and convergence. The compared results show that Newton-Raphson is the most reliable method because it has the least number of iteration and converges faster.

In a power system, power flows from generating station to the load through different branches of the network. The flow of active and reactive power is known as load flow or power flow. Load flow analysis is an important tool used by power engineers for planning and determining the steady state operation of a power system. Power flow studies provide a systematic mathematical approach to determine the various bus voltages, phase angles, active and reactive power flows through different branches, generators, transformer settings and load under steady state conditions. The power system is modeled by an electric circuit which consists of generators, transmission network and distribution network [

The main information obtained from the load flow or power flow analysis comprises magnitudes and phase angles of load bus voltages, reactive powers and voltage phase angles at generator buses, real and reactive power flows on transmission lines together with power at the reference bus; other variables being specified [

For the past three decades, various numerical analysis methods have been applied in solving load flow analysis problems. The most commonly used iterative methods are the Gauss-Seidel, the Newton-Raphson and Fast Decoupled method [

Hand calculations are suitable for the estimation of the operating characteristics of a few individual circuits, but accurate calculations of load flows or short circuits analysis’ would be impractical without the use of computer programs. The use of digital computers to calculate load flow started from mid 1950s. There have been different methods used for load flow calculation. The development of these methods is mainly led by the basic requirement of load flow calculation such as convergence properties, computing efficiency, memory requirement, convenience and flexibility of the implementation [

This paper compares numerical methods: Gauss-Seidel, Newton-Raphson and Fast Decoupled methods use for load flow analysis; for test cases of IEEE 9-Bus, IEEE 30-Bus and IEEE 57-Bus system to determine which of the method is best for power system planning studies.

A bus is a point or node in which one or many transmission lines, loads and generators are connected. In a power system study, every bus is associated with 4 quantities, such as magnitude of voltage (|V|), phase angle of voltage (δ), active power (P) and reactive power (Q) [

This is used as a reference bus in order to meet the power balance condition. Slack bus is usually a generating unit that can be adjusted to take up whatever is needed to ensure power balanced [

This is a voltage control bus. The bus is connected to a generator unit in which output power generated by this bus can be controlled by adjusting the prime mover and the voltage can be controlled by adjusting the excitation

No. | Type of Bus | Variables | |||
---|---|---|---|---|---|

P | Q | |V| | δ | ||

1 | Slack Bus | Unknown | Unknown | Known | Known |

2 | Generator Bus (PV) | Known | Unknown | Known | Unknown |

3 | Load Bus (PQ) | Known | Known | Unknown | Unknown |

of the generator. Often, limits are given to the values of the reactive power depending upon the characteristics of individual machine. The known variable in this bus is P and |V| and the unknown is Q and δ [

This is a non-generator bus which can be obtained from historical data records, measurement or forecast. The real and reactive power supply to a power system are defined to be positive, while the power consumed in a power system are defined to be negative. The consumer power is met at this bus. The known variable for this bus is P and Q and the unknown variable is |V| and δ [

The numerical analysis involving the solution of algebraic simultaneous equations forms the basis for solution of the performance equations in computer aided electrical power system analyses e.g. for load flow analysis [

The nodal equation can be written in a generalized form for an n bus system.

The complex power delivered to bus i is

Substituting for I_{i} in terms of

The above equation uses iterative techniques to solve load flow problems. Hence, it is necessary to review the general forms of the various solution methods; Gauss-Seidel, Newton Raphson and Fast decoupled load flow.

This method is developed based on the Gauss method. It is an iterative method used for solving set of nonlinear algebraic equations [

This is an iterative method which is used to solve Equation (5) for the value of V_{i}, and the iterative sequence becomes

Using Kirchhoff current law, it is assumed that the current injected into bus i is positive, then the real and the reactive powers supply into the buses, such as generator buses, _{i} nd Q_{i} are solved from Equation (5) which gives

The power flow equation is usually expressed in terms of the bus admittance matrix, using the diagonal elements of the bus admittance and the non-diagonal elements of the matrix, then the Equation (6) becomes,

and

The admittance to the ground of line charging susceptance and other fixed admittance to ground are included into the diagonal element of the matrix.

This method was named after Isaac Newton and Joseph Raphson. The origin and formulation of Newton-Ra- phson method was dated back to late 1960s [

The admittance matrix is used to write equations for currents entering a power system.

Equation (2) is expressed in a polar form, in which j includes bus i

The real and reactive power at bus i is

Substituting for I_{i} in Equation (12) from Equation (13)

The real and imaginary parts are separated:

The above Equation (15) and (16) constitute a set of non-linear algebraic equations in terms of |V| in per unit and δ in radians. Equation (15) and (16) are expanded in Taylor’s series about the initial estimate and neglecting all higher order terms, the following set of linear equations are obtained.

In the above equation, the element of the slack bus variable voltage magnitude and angle are omitted because they are already known. The element of the Jacobian matrix are obtained after partial derivatives of Equations (15) and (16) are expressed which gives linearized relationship between small changes in voltage magnitude and voltage angle. The equation can be written in matrix form as:

J_{1}, J_{2}, J_{3}, J_{4} are the elements of the Jacobian matrix.

The difference between the schedule and calculated values known as power residuals for the terms

The new estimates for bus voltage are

The Fast Decoupled Power Flow Method is one of the improved methods, which is based on a simplification of the Newton-Raphson method and reported by Stott and Alsac in 1974 [

This method is a modification of Newton-Raphson, which takes the advantage of the weak coupling between _{2} and J_{3}. Equation (17) is simplified as:

Expanding Equation (22) gives two separate matrixes,

B' and B'' are the imaginary parts of the bus admittance. It is better to ignore all shunt connected elements, as to make the formation of J_{1} and J_{4} simple. This will allow for only one single matrix than performing repeated inversion .The successive and voltage magnitude and phase angle changes are

The simulation for Gauss-Seidel, Newton-Raphson and Fast Decouple is carried out using Matlab for test cases of IEEE 9. The base mva, selected valve for iteration (tolerance), and maximum numbers of iterations is specified.

IEEE 9 bus system represented in

IEEE 9-bus system consist of eleven line data as represented in

LOAD DATA | |||||||
---|---|---|---|---|---|---|---|

Bus | Type of Bus | Voltage | Load | Generation | |||

÷V÷ (P.U) | d (q) | P (MW) | Q (Mvar) | P (MW) | Q (Mvar) | ||

1 | Slack | 1.0300 | 0 | 0 | 0 | ||

2 | PQ | 1.0000 | 0 | 10 | 5 | ||

3 | PQ | 1.0000 | 0 | 25 | 15 | ||

4 | PQ | 1.0000 | 0 | 60 | 40 | ||

5 | PQ | 1.0600 | 0 | 10 | 5 | 80 | |

6 | PV | 1.0000 | 0 | 100 | 80 | ||

7 | PQ | 1.0000 | 0 | 80 | 60 | ||

8 | PV | 1.0100 | 0 | 40 | 20 | 120 | |

9 | PQ | 1.0000 | 0 | 20 | 10 |

LINE DATA | |||||
---|---|---|---|---|---|

Bus No. | Bus No. | R, PU | X, PU | 1/2 B, PU | Transformer Tap |

1 | 2 | 0.0180 | 0.0540 | 0.0045 | 1 |

1 | 4 | 0.0150 | 0.0450 | 0.0038 | 1 |

2 | 3 | 0.0180 | 0.0560 | 0 | 1 |

3 | 9 | 0.0200 | 0.0600 | 0 | 1 |

4 | 5 | 0.0130 | 0.0360 | 0.0030 | 1 |

4 | 6 | 0.0200 | 0.0660 | 0 | 1 |

5 | 6 | 0.0600 | 0.030 | 0.0028 | 1 |

5 | 7 | 0.0140 | 0.0360 | 0.0030 | 1 |

6 | 9 | 0.0100 | 0.0500 | 0 | 1 |

7 | 8 | 0.0320 | 0.0760 | 0 | 1 |

8 | 9 | 0.0220 | 0.0650 | 0 | 1 |

Gauss-Seidel Method | Newton-Raphson Method | Fast Decouple Method | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

From Bus | To Bus | P | Q | Lines loss | P | Q | Lines loss | P | Q | Lines loss | |||

MW | Mvar | MW | Mvar | MW | Mvar | MW | Mvar | MW | Mvar | MW | Mvar | ||

1 | 2 | 47.024 | 5.514 | 0.381 | 0.199 | 46.912 | 10.350 | 0.393 | 0.238 | 47.411 | 8.677 | 0.396 | 0.244 |

1 | 4 | 103.50 | −25.023 | 1.600 | 3.997 | 103.225 | −10.714 | 1.522 | 3.766 | 104.675 | −37.340 | 1.742 | 4.418 |

2 | 3 | 36.633 | 1.317 | 0.233 | 0.725 | 36.519 | 6.113 | 0.239 | 0.743 | 37.018 | 4.435 | 0.242 | 0.752 |

3 | 9 | 11.390 | −11.405 | 0.051 | 0.152 | 11.280 | −6.631 | 0.034 | 0.101 | 11.775 | −8.317 | 0.041 | 0.123 |

4 | 5 | 11.520 | −70.300 | 0.620 | 1.070 | 11.585 | −59.488 | 0.454 | 0.620 | 12.085 | −84.209 | 0.877 | 1.771 |

4 | 6 | 30.343 | 1.291 | 0.175 | 0.577 | 30.119 | 5.009 | 0.179 | 0.591 | 30.860 | 2.456 | 0.180 | 0.594 |

5 | 7 | 38.216 | 68.414 | 0.786 | 1.376 | 38.188 | 68.302 | 0.798 | 1.422 | 40.374 | 96.919 | 1.382 | 2.903 |

6 | 9 | −27.806 | 13.579 | 0.092 | 0.460 | −27.670 | 12.022 | 0.089 | 0.445 | −29.323 | 34.386 | 0.194 | 0.972 |

7 | 8 | −42.572 | 7.039 | 0.571 | 1.357 | −42.610 | 6.880 | 0.583 | 1.385 | −40.984 | 34.028 | 0.870 | 2.066 |

8 | 9 | 36.846 | 9.327 | 0.300 | 0.885 | 36.806 | 6.024 | 0.294 | 0.869 | 38.138 | −13.924 | 0.355 | 1.050 |

The selected tolerance iteration value used for the simulation is shown in

The computation time for load flow solutions using selected iteration value of 0.001 and 0.1 is shown in

Test System | Gauss-Seidel | Newton-Raphson | Fast Decouple |
---|---|---|---|

IEEE 9 Bus | 0.001/0.1 | 0.001/0.1 | 0.001/0.1 |

IEEE30 Bus | 0.001/0.1 | 0.001/0.1 | 0.001/0.1 |

IEEE57 Bus | 0.001/0.1 | 0.001/0.1 | 0.1 |

Test System | Gauss-Seidel | Newton-Raphson | Fast Decouple |
---|---|---|---|

IEEE 9 Bus | 45 | 7 | 9 |

IEEE 30 Bus | 113 | 9 | 25 |

IEEE 57 Bus | 176 | 10 |

Test System | Gauss-Seidel | Newton-Raphson | Fast Decouple |
---|---|---|---|

IEEE 9 Bus | 12 | 2 | 4 |

IEEE 30 Bus | 36 | 4 | 3 |

IEEE 57 Bus | 17 | 5 | 6 |

Test System | Gauss-Seidel | Newton-Raphson | Fast Decouple |
---|---|---|---|

IEEE 9 Bus | 0.003 | 0.004 | 0.004 |

IEEE 30 Bus | 0.008 | 0.103 | 0.012 |

IEEE 57 Bus | 0.008 | 0.013 |

Test System | Gauss-Seidel | Newton-Raphson | Fast Decouple |
---|---|---|---|

IEEE 9 Bus | 0.038 | 0.091 | 0.074 |

IEEE 30 Bus | 0.205 | 0.213 | 0.243 |

IEEE 57 Bus | 0.367 | 0.500 | 0.455 |

Convergence is used to determine how fast a power flow reaches its solution. The rate of convergence is determined by plotting a graph of maximum power mismatch against the number of iterations. Figures 7(a)-(c) shows the graph for convergence on IEEE-9, IEEE-30 and IEEE-57 Bus System respectively using selected iteration value of 0.001. Figures 8(a)-(c) shows the graph for convergence on IEEE-9, IEEE-30 and IEEE-57 Bus System respectively using selected iteration value of 0.1. The convergence rate for Gauss-Seidel is slow compared to the other methods. Newton-Raphson has the fastest rate of converging among the three numerical methods shown in the graph.

All the simulations were carried out using Mathlab and implemented for IEEE 9-bus, IEEE 30-bus and IEEE 57-bus test cases for Gauss-Seidel, Newton-Raphson and Fast Decouple. In the load flow analysis methods simulated, the tolerance values used for simulation are 0.001 and 0.1 for all the simulation carried out except for the IEEE 57-bus using the fast decouple method, which did not converge with the tolerance values. This explains why the Fast Decouple method is not as accurate as Newton-Raphson method because a lower tolerance value of 0.1 was used to carry out the simulation for the IEEE 57-bus Fast Decouple Method.

The time for iteration in Gauss-Seidel is the longest compared to the other two methods, Newton-Raphson and Fast Decouple. The time for iterations in Gauss-Seidel increases as the number of buses increases. The Gauss-Seidel method increases in arithmetic progression, Newton-Raphson increases in quadratic progression while the fast decouple increases in geometric progression. This explains why it takes longer time for Gauss- Seidel to converge. The computational time for Gauss-Seidel is low compared to the other two methods; Newton-Raphson and fast decouple. Newton-Raphson have more computational time due to the complexity of the Jacobian matrix for each iteration but still converges fast enough because less number of iterations are carried out and required.

The results of this paper suggest that the planning of a power system can be carried out by using Gauss-Seidel method for a small system with less computational complexity due to the good computational characteristics it exhibited. The effective and most reliable amongst the three load flow methods is the Newton-Raphson method because it converges fast and is more accurate.

Olukayode A.Afolabi,Warsame H.Ali,PenroseCofie,JohnFuller,PamelaObiomon,Emmanuel S.Kolawole, (2015) Analysis of the Load Flow Problem in Power System Planning Studies. Energy and Power Engineering,07,509-523. doi: 10.4236/epe.2015.710048