The prime objective of the contemporary electrical power network is continued fulfillment of the power demand utilized by all the users. It becomes a complex issue in power transmission study. Fluctuations in load, generation, lines or interconnections give rise to problems such as system instability and system losses.

By reverting to the load flow analysis and related studies, many of these critical aspects in power system applications can be evaluated to a great degree quite efficiently. Load study proves to be a critical control task in steady and effective functioning of power grids. The principle aim of the load study becomes ascertaining the steady-state working specifications in the existing electric network as well as scheduling and designing the future expansion of the network. We can find steady-state by calculating angles and magnitude of voltage, as well as the flow of reactive and active power in the system for the given loading conditions set. Here, we codify Newton Raphson technique for an optimum load flow analysis employing ring distribution system in and around the Bahawalpur area. It employs five buses, two generating and three load buses.

We have the following sections in this paper: Literature review has been discussed in section 2 in which we have presented different solution techniques for power flow problem, starting from the earliest techniques to the recent ones. This section is followed by the proposed methodology i.e. Newton Raphson algorithm in section 3. Results and conclusions have been presented in sections 4 and 5 respectively.

For sorting out load flow equations, different techniques are presented over the years. The numerical methodologies and loop equations employing Gauss-type solutions were basis of the early techniques ^{1}. The system engineers had to designate the network loops in advance so, this method proved burdensome. Upgraded approaches spotted the development of the nodal analysis in favor of the loop analysis that lead towards significantly reducing the data formation. Nonetheless, accuracy to the convergence still remained primary issue. The acceleration factors in the Gauss Seidel method were developed with the further progression in the field. The features like easy to understand, least storage requirements, easy to codify in format of computer programs are main attraction of the present generation of load flow methodologies. When applied to the solution of reasonable size of systems, these algorithms show poor convergence properties resulting in the only limitation of the method. Load flow solutions that were based upon the bus impedance matrix were shortly tested but at that time, the issues with speed and computer storage came out to be unconquerable problems. In the early 1970s, the Newton Raphson methodology and derived formulations were introduced to overcome such constraints, and it has since become very popular technique all over the power system industry ^{2}.

Stott and Alsac's later work rendered Newton's Fast Decoupled methodology ^{3}. This approach provides extremely fast, simple and very reliable power flow solution. The convergence performance of this method is poor for most of the distribution networks, although it worked well for transmission networks. This is because of its large R/X ratio that degrades Jacobian matrix's diagonal dominance.

Numerous other forms of approaches were proposed for this purpose. For example, Modified Newton Method and Modified Gauss-Seidel algorithm ^{4, 5, 6, 7, 8} etc. Tripathy ^{9} et al. introduced a technique resembling to Newton’s methodology of sorting out unconditioned power networks but could not be used effectively to measure maximal power flow. Baran and Wu suggested a technique in solution of radial power flow to evaluate optimum problem of capacitor sizing ^{10}. The computational efficiency is increased in this process. Chiang et al. suggested three techniques in the resolution of radial system based on the Baran and Wu system ^{11}. Goswami and Basu suggested the direct methodology in solving networks of radial and meshed distribution ^{12}. It also works well for heavy loads. A new power-flow method was proposed by Jasmon and Lee to obtain the solution of radial systems ^{13}. Using the power convergence technique, Das et al. suggested the new method of power-flow analysis by calculation of net reactive and real powers supplied by any bus for the solution of radial distribution systems ^{14}. To solve radial distribution networks, primary assumption was put forward as the zero initial power loss. It saves the computer storage to a great extent. Haque et al. introduced another effective methodology with more than one feeding node for resolving both meshed and radial networks ^{15}. This approach has excellent radial network convergence. In radial distribution networks, Eminoglu and Hocaoglu et al. suggested the simple and efficient technique for solution of load flow problem, taking into account static load voltage dependence and line charging efficiency ^{16}. Prasad et al. suggested the branch-current calculation technique in radial network ^{17}. It takes advantage of a network's tree-structure property. Ghosh and Sherpa et al. developed a system for radial distribution network power-flow solution with minimal data planning ^{18}. Sivanagaraju et al. suggested a distinctive technique for power-flow solution that is employed to analyze weakly meshed system using branch current matrix ^{19}. Kumar and Arvindhababu et al. proposed a power flow solution approach for achieving efficient convergence in distribution systems ^{20}. For study of weakly meshed or radial systems, Augugliaro et al. suggested a technique providing the voltage dependent demand ^{21}. At each step, impedances simulate the loads and furthermore, the solution procedure is iterative in nature. For the solution of radial distribution systems, Kaur et al. has suggested a new power-flow technique using sequential numbering scheme ^{22}. Suggesting a technique, with less computation and reducing data preparation, is the aim of this methodology. Sharma et al. has investigated a couple of new network reconfiguration schemes in introducing two new effective power flow techniques, and presenting simulation test results ^{23}. Parasher et al. showed a method for radial distribution network power-flow solution ^{24}. The author endeavored to reduce the data preparation in some cases, and for some other situations, the author employed data as it is without reducing its preparation.

We have implemented data in both Newton Raphson (NR) and Fast Decoupled (FD) algorithms in MATLAB. The results obtained thus are the same hence, the choice of a single methodology between the two might become challenging as both techniques have some advantages and disadvantages over the other. However, this selection of method might become easier if certain priority points are already kept in mind and the comparison is then drawn between the two. Following is a brief comparison between NR and FD load flow algorithms:

Newton-Raphson | Fast-decoupled |

For both small and large networks, highly accurate solution can be obtained in just two to three iterations. | The number of iterations is always greater as compared to NR method. |

The overall cost in solving power flow problem is less. | The cost is greater than that of NR methodology. |

Large computer memory is required. | Computer storage requirements are smaller. |

Fewer number of iterations result in less computational time. | Computational time is larger. |

Convergence is slower as compared to FD method. | Convergence is faster. |

The NR algorithm executes results in only 6 iterations while FD algorithm takes 27 iterations. In this study, the factors of our concern are mainly economy and computational time. As the overall cost in NR algorithm is smaller than FD method and it has less number of iterations as compared to FD method, so we have preferred NR method to FD method.

Newton Raphson algorithm is utilized in this paper for the solution of load flow problem in ring distribution network. It has been elaborated in depth below:

For polar coordinates, the Newton Raphson Computational procedure in power flow analysis is as follows:

Y_{bus} is formed using series line impedances or admittances.

Initial values for phase angles and bus voltages will be supposed for all load buses, and phase angles for generator buses. Usually, these assumed values will be maintained equal to slack, reference or swing bus voltage magnitude and angle.

For each load bus, P and Q will be computed employing, respectively, Eqs. (38) and (39).

Eqs. (42) and (43) are used to compute scheduled errors ΔP and ΔQ. Although the exact value of Q is not specified in case of PV buses, its limits are known. So, there are two situations before us; if calculated value of Q is within limits, there is no need to calculate ΔQ. But in case value of Q exceeds the limit, we set an appropriate limit and subtract the calculated value of Q from it and the bus is treated as PQ bus.

Jacobian matrix is formed by computing its elements one by one.

ΔV and Δθ are calculated using Eq. (44).

Voltage angles and magnitudes are updated using values of ΔV and Δθ found in step 6, Eqs (54) and (55) are employed in this computation. Next iteration is started from step 2, utilizing the updated values of V and θ.

This process is continued until scheduled errors ΔP and ΔQ are within specified tolerance for all the load buses.

Lastly, line power flows and slack bus power is calculated.

Consider a small-scale power system network (

We are concerned about the steady state solution of the network shown in

Where

In matrix form

It is notable that the negative sign indicates that the current is extracted while the positive sign indicates that the currents injected. The above equations can be written as

Where

The elements Y_{11}, Y_{22}, Y_{33} making the diagonal entries are called self admittances. The self admittance of a bus x can be found by adding all the admittances at that bus. As a general rule in bus admittance matrix, the diagonal element Y_{pp}_{ }can be calculated by adding all the admittances attached at bus p. i.e.,

Where Y_{pq} is the admittance of the element attached between the buses

The elements Y_{12}, Y_{13}, Y_{21}, Y_{23}, Y_{31}, Y_{32} making the off-diagonal entries are termed as

It is notable that all mutual admittance entries have a negative sign. As a general rule in bus admittance matrix, the off-diagonal element equals negative of the admittance attached at the nodes p and q. i.e.,

For a network having n buses excluding ground, a set of following equations, one for each bus, might be expressed as

In matrix form as

In a more compact form

With reference to ground, we calculate the nodal voltages, while Eq. (10) is a vector equation comprising n scalar equations. It is called the nodal current equation. If the power system elements have mutual coupling, the bus admittance matrix cannot be developed directly by examining the one line diagram. In existence of mutual coupling between power system elements, the inspection method flops. In such a case, Y_{bus} can be formed from graph theoretic approach. However, the mutual coupling between power system elements is only present in the case of transmission lines going parallel for a long distance. But this coupling is also weak. Therefore, the mutual coupling can be disregarded for all practical purposes and Y_{bus} is formulated by inspection method.

Owing to its powerful convergence characteristics, the Newton Raphson (NR) methodology has come out as the most fruitful technique in the large-scale load flow analysis. Iteration approach is used by this technique to resolve a set of nonlinear-algebraic equations, as shown below:

While X is vector for n unknown state variables, and F stands for the set of n nonlinear equations.

In principle, through executing the expansion of Taylor series for F(X) around the initial assumption X(0), the vector of state variables X can be determined. This is in brief what the methodology comprises of. Taylor expansion is shown below:

J(X^{(0)}) will be called Jacobian matrix. This represents the matrix of first-order partial derivatives for F(X) with respect to X, calculated at X=X^{(0)}.

By presuming that X^{(1)} represents the value calculated through method for iteration 1 and it will be adequately near to primary guess X^{(0)}, the expansion leads towards an appropriate calculation to compute the vector of state variables X. Based upon this proposition, all high-order derivative expressions in Eq. (15) can be ignored. Thus,

In case for iteration (i), we generalize the above expression in compact form as,

Where i = 1, 2 . . . . Moreover in case, this has been supposed that ^{(*)} then

Furthermore, resolving for X^{(}^{i}^{)},

Iterative result may formulate to be the function for correction vector

and initial assumptions are upgraded employing the equation below:

Employing the most recent values of X in Eq. (20), the calculations are performed again and again as many times as needed. It is executed till mismatches ΔX are inside the specified small tolerance (10^{-12}).

For finding solution of load flow problem and applying NR methodology, Eq. (20) represents all the related equations, where X stands for the set of unknown phase angles and bus voltage magnitudes. The expansions about the base term (θ(0),V(0)) is performed for mismatch equations ΔP and ΔQ and, thus, the following relationship is formed for the load flow Newton Raphson method:

Various matrices of Jacobian can comprise up to (n_{b}-1)×(n_{b}-1) entries:

and k = 1, . . . , n_{b}, and m = 1, . . . , n_{b} however, eliminating slack bus elements.

Voltage magnitude V and reactive power Q are also omitted for PV buses for the related rows and columns. Moreover, the related k–m element in Jacobian matrix is zero in case bus k and bus m don’t directly associate through a transmission element. Jacobians of power-flows are largely scarce because of slow rate connection that is common for power networks. An extra characteristic is that they are symmetrical in shape but not in value.

In order to recompense for the feature that Jacobian expressions _{m}, it should be indicated that correction expressions ΔV_{m} have been divided by V_{m}. This assumption leads towards very helpful and simple results as represented in the derivative expressions shown below.

Let an l^{th} element be linked with bus k and bus m, for which mutual and self Jacobian expressions have been shown beneath:

For k ≠ m:

For

Generally, the bus self-elements transform to the following format for the bus k comprising n transmission elements l:

Whether there are n transmission elements or one transmission element finishing on bus k, mutual elements presented in Eqs. (24) and (27) stand the same. The reactive and active load-flows all over transmission network will be calculated very simply after the phase angles and voltage magnitudes have been computed by iteration. A significant point to keep in mind is that the yet to be found variables P_{slack} and Q_{slack} will be calculated after network load-flows and the mismatch expressions ΔP and ΔQ for swing bus haven’t been added to Eq. (22) since the computations are carried out for n-1 buses excluding the swing or slack bus.. Also, to find out whether the generators are inside reactive power limits, QG for PV nodes will be computed during every cycle. But mismatch reactive power expressions ΔQ of PV nodes won’t be added to Eq. (22).

In this technique, we have two solution techniques available for power flow result. One is rectangular coordinate method and the other is polar coordinate method. We will use polar coordinate method as it is the most widely used method.

The complex power in bus i can be expressed through Eq. (36) while Eqs. (38) and (39) give active and reactive powers:

We can also write Eqs. (38) and (39) as:

And

Then

Then

In Eq. (42), the subscripts sp and cal represent respectively, specified and calculated values. And

H, N, M and L are sub-matrices which are found by differentiating with respect to δ and |V| Eqs. (38) and (39). We calculate off-diagonal elements of H by

While for diagonal entries:

From Eq. (40)

Similarly, the diagonal expressions for N will be:

and off-diagonal elements

Likewise, off-diagonal and diagonal expressions for M will be

And off-diagonal and diagonal expressions for L will be

Updated voltage magnitude and angle for next iteration are computed as below:

Where subscript r represents values from previous iteration and subscript r+1 represents the next iteration values.

This section is divided into two subsections. First subsection describes the data while the second one gives the results. The network is as shown in figure below:

The data of the above network is classified into four subsections. It constitutes bus specifications, generator power limits, transmission line impedances and admittances and lastly, the load descriptions at various buses. Literature and field survey is conducted to collect power flow data of the selected area. All the aforementioned parameters will be expounded in detail one by one.

The five buses in system illustrated in_{slack} and phase angle θ_{slack}, are specified. The phase angle θ_{slack} is selected as reference against the rest of the angles. It is evident that the voltage at each bus is approximately the same. All of these bus parameters are shown in

Bus | Number | Type | Voltage |

Muzaffargarh | 1 | Slack | 1.00∠0° |

Kot Addu | 2 | Generator | 1.00∠0° |

Cantt. Area | 3 | Load | 1.00∠0° |

BWP Main | 4 | Load | 1.00∠0° |

BWP Baghdad | 5 | Load | 1.00∠0° |

The scheduled active power PGEN and reactive power QGEN contributed by the two generator buses are shown below along with the upper and lower limits of reactive power QMAX and QMIN respectively. This data is shown in

Bus | Power Generated PQGEN | Reactive Power Limit | |

Upper Limit QMAX | Lower Limit QMIN | ||

Muzaffargarh | 0+0i | 8 | -8 |

Kot Addu | 0.16+0i | 5 | -5 |

The relation between the transmission line number and the bus number is elucidated below as shown in

Bus Number | Line Number | Series Impedance (Z=R+X) | Shunt Admittance (Y=G+B) |

1-2 | 1 | 0.020579+0.052057 | 0+0.06i |

1-3 | 2 | 0.0198017+0.1294047i | 0+0.05i |

2-3 | 3 | 0.006332+0.016017i | 0+0.04i |

2-4 | 4 | 0.006901+0.045095i | 0+0.04i |

2-5 | 5 | 0.0216018+0.14116i | 0+0.03i |

3-4 | 6 | 0.052241+0.132146i | 0+0.02i |

4-5 | 7 | 0.020579+0.052057i | 0+0.05i |

The scheduled real power P_{LOAD} and scheduled reactive power Q_{LOAD} consumed at load buses will be as shown in

Bus | Power Consumed PQLOAD |

Kot Addu | 0.226+0.1094i |

Cantt. Area | 0.168+0.0812i |

BWP Main | 0.189+0.091i |

Baghdad | 0.104+0.0508i |

The network employed to show load flow results provided through NR and FD approaches, is from Bahawalpur area as shown in the

The bus voltage magnitude is expressed in per unit (p.u.) and phase angle in degrees against their respective buses. The results are elaborated in

Bus | Voltage | |

Magnitude | Angle | |

Muzaffargarh | 1.00 | 0.00 |

Kot Addu | 1.00 | -1.2425 |

Cantt. Area | 0.9990 | -1.2735 |

BWP Main | 0.9970 | -1.6759 |

BWP Baghdad | 0.9964 | -1.7847 |

The sending end real power P_{SEND} and reactive power Q_{SEND} have been catalogued below followed by the receiving end real power P_{REC} and reactive power Q_{REC} against their respective transmission lines. The buses associated with lines are also tabularized. These power flow results have been presented in

Bus Number | Line Number | Sending End Power PQSEND | Receiving End Power PQREC |

1-2 | 1 | 0.3618-0.1685i | -0.3587+0.1163i |

1-3 | 2 | 0.1691-0.0414i | -0.1685-0.0049i |

2-3 | 3 | 0.0502+0.0216i | -0.0502-0.0615i |

2-4 | 4 | 0.1734+0.0204i | -0.1732-0.0588i |

2-5 | 5 | 0.0691-0.0001i | -0.0690-0.0291i |

3-4 | 6 | 0.0510-0.0148i | -0.0509-0.0048i |

4-5 | 7 | 0.0351-0.0279i | -0.0350-0.0217i |

In this research, Newton Raphson MATLAB algorithm is developed for a ring distribution network of five-buses which is feeding different areas of Bahawalpur to find the solution for load flow problem. Power flow solutions provide systematic approach to calculate voltage magnitude, phase angle, active and reactive power before they occur in generation, load, transmission lines or interconnections ^{25}. Furthermore, power flow study is essentially important to estimate the reactive power which is a salient feature of power systems with widespread penetration of inductive loads and their compensation using FACTS devices ^{26}. Literature and field survey is conducted to collect power flow data of the selected area and the load flow equations of primarily Newton Raphson algorithm and Fast-decoupled method secondarily. Consequently, this data is utilized in the above mentioned algorithms to find out the optimum solution, best operation of the existing system and planning for future expansion to be at par with the growing demand.