Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 10.17485/IJST/v13i27.916 Power Flow Analysis and Optimization in Ring Distribution Network of Bahawalpur using Newton Raphson Method Shakil Muhammad 1 Rashid Zeeshan zeeshan.rashid@iub.edu.pk 1 Hussain Ghulam Amjad 2 Umer Farhana 1 Department of Electrical Engineering, The Islamia University of Bahawalpur 63100, +92-62-925-5576 Pakistan Department of Electrical Engineering, College of Arts and Sciences, American University of Kuwait Safat, 10002 Kuwait 13 27 2020 Abstract

Background: The primary motive of an electrical power system is to generate and supply electric power efficiently and reliably to the consumer-end. Transmission losses, system instability and increasing cost in proportion to demand are the main challenges faced in this process. Power flow analysis is required to robustly predict the active / reactive power within the buses, voltage magnitude / phase angles at each bus, cost of transmission and losses well before the practical installation of the power network. Methods/Statistical Analysis: In this paper, we employ power flow analysis using Newton Raphson method and Fast Decoupled method to minimize cost of the electricity and finding optimum active and reactive powers without affecting the voltage regulation. The power flow algorithms are applied for solving the afore-mentioned load flow problem for ring distribution network of Bahawalpur. We carried out the modeling by obtaining realistic data for constructing bus admittance matrix and specifications of generation units and loads which are connected at the buses. Findings: As a result, optimum flow of power along with the voltage values among different regions of Bahawalpur is obtained. The results from both the algorithms successfully converge and there is an absolute match to validate the accuracy. Novelty: These novel results are of paramount importance since the proposed architecture of Bahawalpur is ring distribution network to replace the existing radial network for improved performance. Furthermore, this research will pave the way for power system planning of Bahawalpur region where all the electrical parameters are known beforehand to design the components according to the requirement.

Keywords Electric power system voltage magnitude phase angle power flow analysis Newton Raphson method ring distribution network active power reactive power None
Introduction

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.

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.

Literature Review

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.

2.1 A Comparison between Newton-Raphson and Fast-Decoupled Methods

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.

Proposed Methodology

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:

3.1 Computational procedure

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

Ybus 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.

ΔPi(r)<ϵ, ΔQi(r)<ϵ

is tolerance level.

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

3.2 The Bus admittance matrix Y<sub id="s-33d20ca8c850">bus</sub>

Consider a small-scale power system network (Figure 1 ) containing three lines, two generators, one load and a static capacitor attached to load bus 3. We will suppose that network is symmetrical and operating under balanced conditions.

Power system network.

We are concerned about the steady state solution of the network shown in Figure 1 consisting of a three-node network. We can write the bus voltage equations as

I1=y12+y31V1-y12V2-y31V3 I2=-y12V1+y12+y23V2-y23V3 -I3=-y31V1-y23V2+(y31+y23+y30)V3

Where y12=1z12, y23=1z23, y31=1z31.

In matrix form

I1I2-I3=y12+y31-y12-y31-y12y12+y23-y23-y31-y23(y31+y23+y30)V1V2V3

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

I1I2-I3=Y11Y12Y13Y21Y22Y23Y31Y32Y33V1V2V3

Where Y11=y12+y31; Y22=y12+y23; Y33=y31+y23+y30 and Y12=Y21=-y12; Y23=Y32=-y23; Y31=Y13=-y31

The elements Y11, Y22, Y33 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 Ypp can be calculated by adding all the admittances attached at bus p. i.e.,

Ypp=yp1+yp2++ypn

Where Ypq is the admittance of the element attached between the buses p and q.

The elements Y12, Y13, Y21, Y23, Y31, Y32 making the off-diagonal entries are termed as mutual admittances.

Y12=Y21=-y12;Y23=Y32=-y23;Y31=Y13=-y31

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.,

Ypq=-ypq

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

I1=Y11V1+Y12V2++Y1nVn I2=Y21V1+Y22V2++Y2nVn In=Yn1V1+Yn2V2++YnnVn

In matrix form as

I1I2In=Y11Y12Y1nY21Y22Y2nY31Y32YnnV1V2Vn

In a more compact form

Ibus=YbusVbus  Where Ibus= bus current vector =I1I2In Vbus=bus voltage vector=V1V2Vn Ybus=bus admittance matrix=Y11Y12Y1nY21Y22Y2nY31Y32Ynn

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, Ybus 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 Ybus is formulated by inspection method.

3.3 The Newton–Raphson algorithm

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:

f1x1,x2,,xN=0f2x1,x2,,xN=0·  ·fNx1,x2,,xN=0   or   F(X)=0

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:

FX=FX0+JX0X-X0+higher-order terms

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,

f1X(1)f2X(1)···fnX(1)f1X(0)f2X(0)··fnX(0)+f1(X)x1f1(X)x2f1(X)xnf2(X)x1f2(X)x2f2(X)xn            fn(X)x1fn(X)x2fn(X)xnX1(1)-X1(0)X2(1)-X2(0)...Xn(1)-Xn(0)

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

FX(i) FX(i-1)+JX(i-1)X(i)-X(i-1)

Where i = 1, 2 . . . . Moreover in case, this has been supposed that Xi is adequately near to result X(*) then FXiFX*=0. Thus, Eq. (13) transforms to

FXi-1+JXi-1Xi-Xi-1=0

Furthermore, resolving for X(i),

X(i)=X(i-1)-J-1X(i-1) FX(i-1)

Iterative result may formulate to be the function for correction vector X(i)=Xi-X(i-1),

ΔX(i)=-J-1X(i-1)FX(i-1)

and initial assumptions are upgraded employing the equation below:

X(i)=Xi-1+X(i)

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:

PQ=-PθPVVQθQVVθΔVV

Various matrices of Jacobian can comprise up to (nb-1)×(nb-1) entries:

Pkθm,PkVmVmQkθm,QkVmVm

and k = 1, . . . , nb, and m = 1, . . . , nb 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 (Pk/Vm)Vm and (Qk/Vm)Vm are multiplied by Vm, it should be indicated that correction expressions ΔVm have been divided by Vm. This assumption leads towards very helpful and simple results as represented in the derivative expressions shown below.

Let an lth element be linked with bus k and bus m, for which mutual and self Jacobian expressions have been shown beneath:

For k ≠ m:

Pk,lθm,l=VkVmGkmsin(θk-θm)-Bkmcos(θk-θm) Pk,lVm,lVm,l=VkVmGkmcos(θk-θm)-Bkmsin(θk-θm) Qk,lθm,l=-Pk,lVm,lVm,l Qk,lVm,lVm,l=Pk,lθm,l

For k=m:

Pk,lθm,l=-Qkcal-Vk2Bkk Pk,lVm,lVk,l=Pkcal+Vk2Gkk Qk,lθm,l=Pkcal-Vk2Gkk Qk,lVm,lVk,l=Qkcal-Vk2Bkk

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

Pkθk=l=1nPk,lθk,l PkVkVk=l=1nPk,lVk,lVk,l Qkθk=l=1nQk,lθk,l QkVkVk=l=1nQk,lVk,lVk,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 Pslack and Qslack 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).

3.4 Newton Raphson Method for power flow solution

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:

Si=Pi+jQi=Vik=1nYik*Vk* Si=k=1n(ViVkYik)(δi-δk-θik) Pi=k=1n(ViVkYik)cos(δi-δk-θik) Qi=k=1n(ViVkYik)sin(δi-δk-θik)

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

Pi=ViViYiicosθii+k=1kinViVkYikcosδi-δk-θik Qi=ViViYiicosθii+k=1kinViVkYiksinδi-δk-θik

And f=JX

If            ΔPi=Pi(sp)-Pi(cal)

Then i=1,2,,n    islack

If                ΔQi=Qi(sp)-Qi(cal)

Then i=1,2,,n    islack    iPVbus

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

PQ=HNMLδV

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

HikPiδk=ViVkYiksinδi-δk-θik,   ik

While for diagonal entries:

HiiPiδi=-Vik=1kinYikVksinδi-δk-θi

From Eq. (40)

Hii=-Qi-BiiVi2

Similarly, the diagonal expressions for N will be:

PiVi=2ViYiicosθii+k=1kinYikVkcosδi-δk-θik

and off-diagonal elements

PiVk=ViVkcosδi-δk-θik

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

Qiδk=-ViVkYikcosδi-δk-θik,   ik Qiδi=k=1kinViVkYikcosδi-δk-θik

And off-diagonal and diagonal expressions for L will be

QiVk=ViYik sinδi-δk-θik,   ik QiVi=-2ViYiisinθii+k=1kinVkYiksinδi-δk-θik

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

Vi(r+1)=Vi(r)+Vi(r) δi(r+1)=δi(r)+δi(r)

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

Results and discussion

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 Five-bus Network in Bahawalpur Area 4.1 Power flow data

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.

4.1.1 Bus data

The five buses in system illustrated inFigure 2 are numbered for identification in load flow algorithm. The type of each bus is mentioned next which is followed by the voltages (magnitude and angle) of the respective buses. Voltage magnitude is expressed in per unit (p.u.) and phase angle is given in degrees. The Muzaffargarh generator bus is chosen as the slack bus and voltage magnitude Vslack and phase angle θslack, are specified. The phase angle θslack 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 inTable 2 .

<bold id="strong-ace184cebdaa4676871fc8d30153bc14"/>Bus-data for the network under consideration.
 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°
4.1.2 Generator data

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 inTable 3.

<bold id="strong-ae722ead327b4786976f57e686c0e705"/>Generator-data for the network under consideration.
 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
4.1.3 Transmission line data

The relation between the transmission line number and the bus number is elucidated below as shown in Figure 2 also. It is elucidated that the transmission line 1 connects buses 1 and 2, transmission line 2 connects buses 1 and 3 and similarly, the rest of the transmission lines are attached as tabularized below. This is followed by the impedances of the respective transmission lines, here Z stands for the series impedance which is the sum of series resistance R and series reactance X. In the last column, admittances of the corresponding transmission lines are represented, where the shunt admittance Y is the sum of shunt conductance G and shunt susceptance B. The data is expressed inTable 4.

<bold id="strong-2bca1391c44d4c279c4a1647cb98d424"/>Line-data for the network under consideration.
 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 PLOAD and scheduled reactive power QLOAD consumed at load buses will be as shown inTable 5.

The network employed to show load flow results provided through NR and FD approaches, is from Bahawalpur area as shown in the Figure 2. It represents the five-bus system consisting of two generator uses and three load buses including seven transmission lines. One generator bus is from Muzaffargarh and the other bus is situated in Kot Addu around Bahawalpur. The three load buses are at Cantt. Area, BWP Main and BWP Baghdad respectively. The data at length is given in the following section and is feasible to utilize either with NR or FD programs.

4.1.5 Bus voltage results

The bus voltage magnitude is expressed in per unit (p.u.) and phase angle in degrees against their respective buses. The results are elaborated inTable 6 .

Bus voltage for the network under consideration.
 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
4.1.6 Power flow results

The sending end real power PSEND and reactive power QSEND have been catalogued below followed by the receiving end real power PREC and reactive power QREC against their respective transmission lines. The buses associated with lines are also tabularized. These power flow results have been presented inTable 7 .

Power flow results for the network under consideration.
 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
Conclusion

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.