Metaphor-less Rao-3 and artificial neural network with parallel computing-based wheeling pricing in competitive power market

Abstract

Fast and accurate wheeling pricing has emerged as an important issue in the recent competitive power market. Embedded cost-based wheeling pricing is well accepted by power market, because it is based on actual flow of power wheeled by them. It also recovers fully the fixed cost of wheeling facility installation and operation. In this article, metaphor-less Rao-3-based ACOPF, MVA-mile method and Bialek tracing has been employed to compute wheeling prices across various generators and loads. In actual power market due to continuously varying load conditions, the computation of wheeling prices is quite a time taking process. Because for computing wheeling prices, the optimal power flow (OPF) program has to be run each time for every loading condition. In this scenario, the artificial neural network (ANN) approach has been found to be very useful, to estimate wheeling prices instantly and accurately for any unseen loading scenario. Here, a number of ANNs have been developed under parallel computing environment. This article presents a metaphor-less Rao-3-based approach to project wheeling prices in the competitive power market by developing a new radial basis function neural network (RBFNN). The present work of wheeling pricing has been demonstrated and examined on IEEE 30-bus system.

Keywords:

• Bialek tracing
• radial basis function neural network (RBFNN)
• MVA-mile method
• parallel computing
• wheeling pricing
• metaphor-Less rao-3 algorithm

Reviewing Editor:

• Qingsong Ai, Senior Editor, Wuhan University of Technology, CHINA

Subjects:

• Technology
• Engineering & Technology
• Electrical & Electronic Engineering
• Engineering Economics
• Industrial Engineering & Manufacturing

1. Introduction

The recent trend of deregulation in electric supply industry has renewed interest in wheeling pricing. Because in this new electricity commodity business, all market participants (suppliers, buyers and transmission system operators [TSO]) are keen to know the transmission cost as well as wheeling prices in order to accommodate their existing and future transactions. This also helps in taking right economic decisions while performing and promoting various wheeling services. Assessment of wheeling prices also ensures efficient use of wheeling facilities by various users (Lee et al., 2001).

Wheeling pricing has become an important issue in terms of use, operation and expansion of the existing wheeling facilities in the current competitive power market environment. Transmission pricing describes how to share the complete cost of a transmission system among all network users while Wheeling pricing describes how to determine the wheeling facility cost incurred by bilateral and multilateral power transactions over the power grid (Duan et al., 2005).

Fixed transmission cost includes initial investment in installing transmission lines, operation and maintenance expenses. Besides this wheeling facility cost involves increment in fuel cost incurred due to repeated generation re-scheduling, cost incurred in accommodating different bilateral and multilateral transactions due to various transmission constraints and any other expenses involved to avail specific wheeling facility. In order to recover these charges from transmission and wheeling facilities, embedded cost-based methods are very effective (Happ, 1994) and hence used in this work.

In this work, power flow execution-based embedded cost method of wheeling pricing has been adopted which is Mega volt-Amp (MVA)-mile method. MVA-mile method of wheeling pricing is basically an extension of original MW-mile method which is based on only DC flow. In the present MVA-mile method, AC-optimal power flow (OPF) has been implemented to consider real as well as reactive flows which seems to be more realistic (Su & Liaw, 2001; Kumar et al., 2011).

To obtain real and reactive power flows (AC power flows), OPF can be run either by conventional optimization methods or by evolutionary computing-based optimization methods. However conventional optimization methods are now outmoded due to their sluggish response, unwanted complex structure and high dependency on data availability. In the present work metaphor-less Rao-3-based AC-OPF (Shyamkant Halve et al., 2023) has been implemented to determine apparent power flows over different transmission lines.

To establish the contribution of each generator and each load on power flows over whole transmission network, tracing technique has been implemented. In this work, Bialek tracing has been employed to allocated flows across each system user (Bialek, 1998).

In actual power market, loading scenario is continuously changing with time, hence OPF has to be run every time repeatedly to obtain apparent power flows for each loading condition. The OPF is iteratively executed for every loading condition and each requested transaction, which results into time intensive solution. Metaphor-less RAO-3 algorithm-based AC-OPF evaluates the fitness function iteratively for each loading scenario, which makes them slow. Moreover, evolutionary computing-based OPF techniques counter problems like selection of exploration and exploitation parameters, population size, and number of iterations, etc. (Aote et al., 2013). On the other hand, artificial neural network (ANN) based Artificial Intelligence (AI) techniques can overcome all these problems by providing tolerance for imprecision, uncertainty and partial truth. Thus presents robustness. Once an appropriate ANN model has been developed, accurate and instant prediction can be obtained for any unseen load pattern, even for missing or partially corrupted data (Pandey et al., 2010). ANN is found to have excellent capability of mapping complex and highly nonlinear input-output patterns in real power market (Chawla et al., 2005; Hagh et al., 2005).

Though ANN has been found its application in predicting spot pricing like locational marginal prices, nodal congestion prices, etc. (Pandey et al., 2008a, 2008b). However, it has not been found yet in the literature for predicting wheeling prices. In this article, an ANN based on radial basis function algorithm named as radial basis function neural network (RBFNN) (Canizares et al., 2004; Demuth & Beale, 2002) has been proposed for predicting wheeling prices. RBFNN is very much effective in function approximation and time series prediction (Bi et al., 2000).

For large power markets, it has been found that wheeling prices obtained through wheeling pricing methods after having suitable traced flows are lie in a wide range for varying loading scenario. A single ANN developed for estimating these prices will not provide satisfactory results. To overcome this problem wheeling prices lie in a closed range are grouped to form different clusters (Ahmed et al., 2020). Then for each cluster a separate ANN has been developed (Pandey et al., 2008a, 2008b) which provides accurate and instant estimation of wheeling prices. Further parallel computing has also been employed to solve this computationally intensive problem by dividing a task into different subtasks (Varshney et al., 2012; Parallel Computing Toolbox, 2023). The proposed approach of estimating wheeling pricing employing RBFNN and parallel computing has been demonstrated and examined on IEEE 30-bus system (Zaeim-Kohan et al., 2018).

The organization of this article is as follows. Section 2 addresses the problem formulation. Section 3 presents RBFNN architecture, clustering and parallel computing. Section 4 describes research methodology. Section 5 presents solution algorithm. Section 6 discusses and analyses results. The conclusion and future scope of the proposed work have been given in Section 7.

2. Problem formulation

In this article, problem of wheeling price estimation employing ANN has been formulated through AC-OPF, flow tracing and computation of wheeling prices using MVA-mile method. All these formulations are given below except flow tracing which is given in Appendix A1.

2.1. AC-OPF using metaphor-less Rao-3 algorithm

Metaphor-less Rao-3 has been implemented for running OPF program by using Matlab sim power toolbox. Mathematical formulation for AC-OPF (Shyamkant Halve et al., 2023) is given as follows: (1) $$\mathit{\text{Min }}f\left(x,u\right)$$(1)

Subject to (2) $$g\left(x,u\right)=0\mathit{\text{and }}h\left(x,u\right)\le 0$$(2) where x and u denote dependent and control variables. In this work, dependent variables are real power on slack bus ($P_{\mathit{\text{sl}}}$), voltage on load bus ($V_L$), reactive power output of generators ($Q_g$) and transmission line flows ($S_l$). Apart from it control variables are real power generation on generator buses ($P_g$) except slack bus (1 number bus is slack bus), voltage on generator buses ($V_G$), transformer tap setting (T) and shunt VAR compensation (VAR).

Here in this work, total number of transmission lines is taken as N, total number of load buses are taken as N_L, total number of generators buses are taken as N_G, total number of regulating transformers are RT and total number of shunt compensators are SC. (3) $$x^T=[P_{\mathit{\text{sl}}},V_{L_1}\dots .V_{L_{N_L}},Q_{g_1}\dots \dots Q_{g_{N_G}},S_{l_1}\dots ..S_{l_N}]$$(3) (4) $$u^T=[\mathrm{ }{P}_{g_2}\dots \dots .P_{g_{N_G}},\mathrm{ }\mathrm{ }{V}_{G_1}\dots \dots .V_{G_{N_G}},T_1\dots \dots T_{\mathit{\text{RT}}},{\mathit{\text{VAR}}}_1\dots \dots {\mathit{\text{VAR}}}_{\mathit{\text{SC}}}]$$(4)

In Equation (2)(2) $$g\left(x,u\right)=0\mathit{\text{and }}h\left(x,u\right)\le 0$$(2) , g represents equality constraint which is the standard load flow equation. All inequality constraints are represented by h. These are

(a) Generation Constraints: Due to inadequate of Generator Voltages, actual power outputs and reactive power outputs their upper and lower limits are $$V_{G_i}^{\mathrm{\text{min}}}\le V_{G_i}\le V_{G_i}^{\mathrm{\text{max}}},\hspace{1em}i=1,2,\dots \dots ..N_G$$ $$P_{G_i}^{\mathrm{\text{min}}}\le P_{G_i}\le P_{G_i}^{\mathrm{\text{max}}}\mathrm{ },\mathrm{ }\hspace{1em}i=1,2,\dots .N_G$$ $$Q_{G_i}^{\mathrm{\text{min}}}\le Q_{G_i}\le Q_{G_i}^{\mathrm{\text{max}}},\hspace{1em}i=1,2,\dots \dots .N_G$$

With that Transformer constraints (5) $$T_i^{\mathrm{\text{min}}}\le T_i\le T_i^{\mathrm{\text{max}}},\hspace{1em}c=1,\dots \dots \mathit{\text{RT}}$$(5)

Shunt VAR constraints (6) $$Q_c^{\mathrm{\text{min}}}\le Q_c\le Q_c^{\mathrm{\text{max}}},\hspace{1em}c=1,\dots ..\mathit{\text{SC}}$$(6) and (7) $$V_{\mathit{\text{Li}}}^{\mathrm{\text{min}}}\le V_{\mathit{\text{Li}}}\le V_{\mathit{\text{Li}}}^{\mathrm{\text{max}}},\hspace{1em}c=1,\dots \dots N_L$$(7) (8) $$S_l\le S_l^{\mathrm{\text{max}}},\hspace{1em}c=1,\dots \dots .N_L{}_{}$$(8)

Because of controlled variables are self-constrained then the objective function can be augmented as follows (9) $$f_{\mathit{\text{avg}}}=f+{\lambda }_p{\left(P_{\mathit{\text{sl}}}-P_{\mathit{\text{sl}}}^{\mathrm{\text{lim}}}\right)}^2+{\lambda }_v\sum _{c=1}^{N_L}{\left(V_{\mathit{\text{LC}}}-V_{\mathit{\text{LC}}}^{\mathrm{\text{lim}}}\right)}^2+{\lambda }_Q\sum _{c=1}^{N_G}{\left(Q_{\mathit{\text{GC}}}-Q_{\mathit{\text{GC}}}^{\mathrm{\text{lim}}}\right)}^2+{\lambda }_s\sum _{c=1}^{N_L}{\left(S_{N_L}-S_{{}_{N_L}}^{\mathrm{\text{max}}}\right)}^2$$(9) where ${\lambda }_p,{\lambda }_v,{\lambda }_Q,{\lambda }_s$ are penalty factors and '$x$' is the limit value of dependent variable '$x$'. So (10) $$x^{\mathrm{\text{lim}}}=\{\begin{array}{l}{x}^{\mathrm{\text{max}}},\hspace{1em}x\ge x^{\mathrm{\text{max}}}\\ x^{\mathrm{\text{min}}},\hspace{1em}x<x^{\mathrm{\text{min}}}\end{array}$$(10)

In this paper, the well-known metaphor-less Rao-3 optimization algorithm, which is an evolutionary computing-based algorithm, has been implemented with AC-OPF.

2.2. MVA-mile method for wheeling pricing

In this work, wheeling prices have computed using MVA-mile method, which considers the real and reactive both flows, thus presents realistic economic signals among market participants (Su & Liaw, 2001; Kumar et al., 2011). In this article, wheeling prices are computed across generators and loads using flow tracing. The wheeling prices for generators and loads can be given as follows (11) $$W{P}_G={\mathit{\text{TTC}}}_G\frac{\sum _{k\in N}{C}_k{L}_k{S}_{G,k}}{\sum _{G\in N_G}\sum _{k\in N}{C}_k{L}_k{S}_{G,k}}$$(11) where $W{P}_G$ is the wheeling price allotted to Gth generator after having flow tracing, TTC_G is the total transmission cost share for generators, $C_k$ is the cost of kth wheeling facility per MVA and per unit length of line, $L_k$ is length of kth transmission line and $S_{G,k}$ is the apparent power flow over kth transmission line due to G^th generator. N and $N_G$ are total number of transmission lines and number of generators. (12) $$W{P}_L={\mathit{\text{TTC}}}_L\frac{\sum _{k\in N}{C}_k{L}_k{S}_{L,k}}{\sum _{L\in N_L}\sum _{k\in N}{C}_k{L}_k{S}_{L,k}}$$(12) where $W{P}_L$ is the wheeling price allotted to Lth load after having flow tracing, TTC_L is the total transmission cost share for loads, $C_k$ is the cost of kth wheeling facility per MVA and per unit length of line, $L_k$ is length of kth transmission line and $S_{L,k}$ is the apparent power flow over kth transmission line due to Lth load. $N_L$ is the total number of loads.

2.3. Bilateral and multilateral transactions

Bilateral and multilateral transactions represent trade and price contracts between generators and loads without violating the transaction wheeling constraints (Varshney et al., 2012). Simultaneously, the viability of these transactions depends on the security breach of the wheeling system. Mathematically bilateral transactions, can be modeled as follows: (12) $$S_{G_i}^k-S_{L_j}^k=0\hspace{1em}\mathit{\text{for }}k=1,2,3\dots \dots \dots ..\mathit{\text{BT}}$$(12) where $S_{G_i}$ and $S_{L_j}$ are the apparent power generation at ith generator bus and power consumption at jth load bus, respectively, BT is the total number of bilateral transactions. In case of multilateral transaction, generators can inject power to multiple buses and loads can also draw power from multiple buses. However, there must a clear balance between total power injection and total power withdrawal. Mathematically, a multilateral transaction involving more than one generator and load can be expressed as: (13) $$\sum _i{S}_{\mathit{\text{Gi}}}^k-\sum _j{S}_{\mathit{\text{Lj}}}^k=0\hspace{1em}\mathit{\text{for }}k=1,2,3\dots \dots \dots \dots \mathit{\text{MT}}$$(13) where $S_{G_i}$ and $S_{L_j}$ are the apparent power generation at ith generator bus and power consumption at jth load bus respectively, MT is the total number of multilateral transactions.

3. Radial basis function neural network

In this work, wheeling pricing estimation has been performed through developing an appropriate ANN employing radial basis function algorithm termed as RBFNN. With enough RBF neurons, the RBFNN can realize almost zero error to all the training samples (Bi et al., 2000; Demuth & Beale, 2002). RBFNN consists of three layers namely input layer, hidden layer and output layer. The hidden layer composed of RBF neurons.

The RBFNN can be designed in a fraction of time as compared with other design approaches in multilayer feed forward neural networks. For a chosen error goal, different values of RBF spread widths have been investigated to find the acceptable results.

The popular discussed model has been developed and trained for prediction of wheeling pricing using RBFNN algorithm using MATLAB deep learning toolbox (Bi et al., 2000). In the proposed RBFNN, all non-zero real power loads and total apparent power demand in a power network are taken as inputs, while wheeling prices for network users are taken as outputs as in Figure 1 is shown.

Figure 1. RBFNN configuration.

In this work, inputs of the RBFNN are real and reactive powers at all the non-zero load bus, total real power demand and total reactive power demand.

3.1. Formation of clusters

For wheeling pricing through ANN, the input–output pairs form training and testing data set. Inputs are real and reactive loads at all the non-zero load buses and total real power load and total reactive power load. Outputs are wheeling prices obtained through metaphor-less Rao-3 algorithm-OPF, Bialek tracing and MVA-mile method of wheeling pricing. The wheeling prices thus obtained lie in a wide range depending on loading scenario of the power market. It has been observed that a single ANN developed for instant and accurate estimation of such heterogeneous wheeling prices will not give satisfactory performance. To solve this problem, all the generators and loads in a given large power market, have been grouped into different clusters such that in each cluster wheeling prices lie in a closed range. Clusters have been formed by employing k-means clustering algorithm, which is based on unsupervised learning and provides k sets from a multi-dimensional data (Zaeim-Kohan et al., 2018; MaraggonLima, 1996).

3.2. Parallel computing

In parallel computing, a computationally intensive task has been divided into different sub-tasks and then executed simultaneously by exploiting the processing capacity of multi-processor computer. Thus the execution of time costly task is speed-up by assigning these tasks to different processor of same computer. In this article, parallel computing (Kumar et al., 2004; Sarwar et al., 2021) has been employed to develop a number of ANNs simultaneously for estimating wheeling prices across different generators and loads. In parallel computing the assignment of tasks to different processors must be wisely done such that each processor may finish the task in almost same time as far as possible. In this way, a significant speed gain can be achieved by using parallel computing over sequential computing.

4. Research methodology

Research methodology of the proposed work has been shown in Figure 2. A large number of load patterns have been generated within ±10% variation of base case load. For each load pattern various bilateral and multi-lateral transactions have been implemented. Then AC-OPF is run to determine real and reactive power flows over all the transmission lines of the given power market. Consequently, Bialek tracing has been employed to determine contribution of each generator and load in the line flows. Then wheeling prices have been computed for each load pattern using MVA-mile method. For practical power markets, these wheeling prices vary in a vast range. k-means clustering has been applied to form different clusters by grouping closed-range wheeling prices in one cluster. For each cluster, a separate RBFNN has been developed. All these ANNs then trained and tested under parallel computing environment to speed up the process.

Figure 2. Block diagram of research methodology.

5. Solution algorithm

The complete solution algorithm for RBFNN-based wheeling pricing estimation using parallel computing has been presented as follows.

1. Generate a large number of load patterns by randomly varying the load at each bus in a wide range (± 10% of base case load)

2. For each loading scenario

   1. Solve metaphor-less Rao-3 algorithm based AC-OPF to obtain real and reactive power flows in each transmission line of the power market.

   2. Apply various transactions (bilateral and multi-lateral both) on given power market.

   3. Again run AC-OPF to obtain apparent flows.

   4. Apply Bialek tracing to find the contribution of each generator and load in all the line flows.

   5. Compute wheeling prices for each generator and load using MVA-mile method.

3. Present input-output patterns as data set for developing RBFNN. Consider all non-zero real and reactive loads and total real and reactive load as inputs and wheeling prices for different generators and loads as outputs.

4. For large power markets, if wheeling prices vary in a wide range then go to step 5 otherwise

   1. Train and validate the ANN using radial basis function algorithm to estimate accurate wheeling pricing for all the users.

   2. Test the trained ANN for previously unseen load patterns.

   3. If percentage absolute error found within acceptable accuracy limit then stop, otherwise go to Step 1.

5. For large power system in which wheeling prices lie in a wide spread range,

   1. Apply k-means clustering to group similar outputs i.e. wheeling prices lie in closed range, in same cluster.

   2. For each cluster develop separate RBFNN to estimate wheeling prices instantly and accurately.

   3. Apply parallel computing for simultaneous development of all the ANNs in order to obtain speed gain as compared to sequential development.

6. Results and discussion

In this work, an attempt has been made to obtain the output using RBFNNs to make an accurate prediction of the wheeling prices. It has been demonstrated along with the well-known IEEE-30 bus system to demonstrate the effectiveness of the suggested method. In this article, a model of a feed forward neural network (RBFNN) which is part of the MATLAB deep learning toolbox has been used to obtain accurate wheeling prices with AC power flow. Radial basis function (RBF) back propagation has been applied to provide training to the neural network.

The IEEE-30 bus systems used have 41 transmission tracks, 20 load buses and 6 generator buses with total annualized cost for wheeling price allocation is 8851.84k$. Out of this the total contribution of generator is taken as 30% (2655.6k$) and the total contribution of load is taken as 70% (6196.3k$). Particle warm optimization algorithm is used to find the power flow across all the lines to get the target prices.

After this, when ANN was applied, it was found that wheeling price prediction is very high in large size power system while the error scope is high due to different range of load. For this purpose, 03 clusters were developed for generator and 07 for load using K-means clustering in IEEE-30 bus systems. So that same and more accurate results can be obtained for each cluster.

Developing an ANN requires multiple loading patterns to include the maximum possible loading conditions and to simulate the operation of any real electricity market. To obtain the desired result, 501 loading scenarios have been generated by varying the load by 10% of the base condition on each load bus and implementing various bilateral and multilateral transactions in this paper. These generated 501 loading patterns (input-output) are split into two and are simultaneously used for training and testing the proposed RBFNNs. For training, 78% of patterns were used, while the remaining 22% of unseen loading scenarios were used for testing. Considering the different architectures of RBFNNs, various tests were carried out here, and the best results obtained from testing the developed RBFNNs are displayed in this article.

6.1. Wheeling pricing for generators

There are a total of 06 generators in the IEEE-30 bus system which accounts for 30% of the total annualized cost in the present work. When to calculate wheeling pricing and to achieve acceptable accuracy all generators vary over a wide range, they were divided into 03 clusters to get desirable accuracy. In Table 1, generators at bus no. 1 in cluster 1, generators 3, 4 and 6 in cluster 2 and generators 2 and 5 in cluster 3 have been displayed for wheeling pricing prediction.

Table 1. Clustering details for generators in IEEE-30 bus system.

Gen. unit	Minimum WPG	Maximum WPG	Cluster
Gen1	2029.5800	2080.9030	1
Gen 3	2.3927	2.8869	2
Gen 4	3.8424	4.2474
Gen 6	12.3575	14.8160
Gen 2	330.1468	365.9250	3
Gen 5	190.2849	276.3295

After taking several trials, the optimal structure of cluster 1 was obtained on 21-05-2. Table 2 shows the testing performance of only 10 patterns due to limited space while Figure 3 shows the entire 100 testing patterns. MAPE 0.0847, maximum error of 0.4574 and elapsed time 0.302690 seconds of trained ANN has been obtained about generator 1. Also, the hidden layer size for cluster 1 is 5.

Figure 3. Testing performance of ANN in IEEE-30 bus system (cluster 1).

Table 2. Wheeling Price prediction for Generators in IEEE-30 bus system (cluster 1).

TP	Target	ANN output	Error	%Abs. error
1	2060.499	2063.326	−2.82654	0.13718
2	2074.638	2075.824	−1.18622	0.05718
3	2076.977	2074.403	2.5737	0.12392
4	2075.963	2072.842	3.12022	0.1503
5	2073.804	2075.878	−2.07462	0.10004
6	2059.285	2060.802	−1.5177	0.0737
7	2077.226	2077.37	−0.14354	0.00691
8	2077.065	2076.242	0.82268	0.03961
9	2073.275	2075.094	−1.81866	0.08772
10	2077.175	2073.636	3.53877	0.17036

Like Cluster 1, Cluster 2 is made up of generators 3, 4 and 6. With this, the optimal structure of Cluster 2 21-24-3 was obtained. Table 3 shows the same testing performance of 10 patterns as in the previous table, while Figure 4 shows the entire 100 testing patterns. MAPE 0.26320% of the trained ANN, maximum error 1.1976, 0.2702 and 1.3738 have been obtained against generators 3,4 and 6 respectively. Also the hidden layer size for cluster 2 is 24.

Figure 4. Testing performance of ANN in IEEE-30 bus system (cluster 2).

Table 3. Wheeling price prediction for generators in IEEE-30 bus system (cluster 2).

TP	Target	ANN output	Error	%Abs. error
1	2.5447	2.53514	0.00956	0.37581
	4.0881	4.08916	−0.00106	0.02597
	12.8513	12.93493	−0.08363	0.65072
2	2.6846	2.67998	0.00462	0.17199
	4.1255	4.12475	0.00075	0.01829
	14.1465	14.13598	0.01052	0.07439
3	2.733	2.73748	−0.00448	0.16377
	3.9514	3.95407	−0.00267	0.0676
	13.1808	13.12628	0.05452	0.41363
4	2.7497	2.73854	0.01116	0.40585
	4.119	4.11642	0.00258	0.06268
	12.9712	12.98253	−0.01133	0.08734
5	2.6359	2.62742	0.00848	0.32155
	4.1174	4.11558	0.00182	0.04425
	14.5426	14.44993	0.09267	0.63724
6	2.4298	2.44502	−0.01522	0.62633
	4.1735	4.16532	0.00818	0.19594
	12.8143	12.82971	−0.01541	0.12027
7	2.5976	2.59572	0.00188	0.07229
	4.031	4.03396	−0.00296	0.07344
	14.0455	14.01909	0.02641	0.18802
8	2.7471	2.72654	0.02056	0.74835
	4.0022	4.00146	0.00074	0.01853
	14.0998	14.07801	0.02179	0.15456
9	2.7417	2.72683	0.01487	0.54225
	4.1531	4.15392	−0.00082	0.01966
	13.9346	13.97905	−0.04445	0.31901
10	2.6955	2.7057	−0.0102	0.37834
	4.0815	4.07959	0.00191	0.04674
	12.7665	12.8702	−0.1037	0.8123

Like Cluster 1 and 2, Cluster 3 is made up of generators 2 and 5. With this, the optimal structure of Cluster 3 21-18-2 was obtained. Table 4 shows the same testing performance of 10 patterns as in the previous table, while Figure 5 shows the entire 100 testing patterns. MAPE 0.0257703% of the trained ANN, maximum error 0.6503 and 1.7218 have been obtained against generators 2 and 5, respectively. Also the hidden layer size for cluster 3 is 18.

Figure 5. Testing performance of ANN in IEEE-30 bus system (cluster 3).

Table 4. Wheeling price prediction for generators in IEEE-30 bus system (cluster 3).

TP	Target	ANN output	Error	%Abs. error
1	360.4243	360.4664	−0.04209	0.01168
	215.1436	215.6731	−0.52951	0.24612
2	363.6351	363.3066	0.3285	0.09034
	196.3219	196.2921	0.02982	0.01519
3	363.2951	363.4313	−0.13621	0.03749
	195.4143	194.8778	0.53648	0.27453
4	362.5051	362.7772	−0.27207	0.07505
	197.2437	198.0433	−0.79959	0.40538
5	363.815	363.9876	−0.17255	0.04743
	196.6369	195.8526	0.78432	0.39887
6	359.2349	358.1034	1.13155	0.31499
	217.6143	220.3002	−2.68593	1.23426
7	362.9344	363.3285	−0.39407	0.10858
	194.7168	193.4927	1.22408	0.62865
8	364.4147	364.3246	0.09014	0.02474
	193.2224	191.72	1.50243	0.77756
9	362.6886	363.1815	−0.49292	0.13591
	198.7581	197.6085	1.14962	0.5784
10	362.8096	363.011	−0.20144	0.05552
	196.0231	194.1773	1.84584	0.94164

Table 4. Summary of wheeling price prediction for generators.

Max.% error	Wheeling pricing to generators	Maximum % Abs. error
0.4574	WPg1	0.4574
1.3738	WPg3	1.1976
	WPg4	0.2702
	WPg6	1.3738
1.7218	WPg2	0.6503
	WPg5	1.7218

Research has found that when more than one ANN is developed as compared to developing one ANN, then the prediction accuracy gets improved as per Table 4.

6.2. Wheeling pricing for loads

Like generators, 07 clusters with 20 loads were developed by k-means clustering to make better price prediction in load also. The details of load clusters are shown in Table 5 in which maximum and minimum value of wheeling pricing across load is shown for all the clusters.

Table 5. Clustering details for wheeling pricing of loads in IEEE-30 bus system.

Cluster	loads	Minimum WPL	Maximum WPL	No. of loads
4	Load 1	86.2330	105.9420	5
	Load 5	54.9831	75.0552
	Load 7	56.7352	69.6743
	Load 15	117.5005	158.6143
	Load 19	40.6804	74.2029
5	Load 10	496.6136	576.3927	1
6	Load 8	828.1529	954.9922	2
	Load 17	799.6099	890.8661
7	Load 18	2.5551e + 03	2.8686e + 03	1
8	Load 3	199.1852	234.8157	3
	Load 14	243.4587	300.0052
	Load 16	219.6578	295.5389
9	Load 2	7.9566	9.4750	7
	Load 9	11.0151	13.2156
	Load 11	0.6609	0.8734
	Load 12	8.5682	10.3800
	Load 14	6.4819	7.6414
	Load 20	1.3846	1.7437
	Load 21	2.7808	3.7862
10	Load 6	0.0647	0.0868	2
	Load 13	0.1456	0.2007

After taking various trials, the optimal structure for Cluster 4 is 21-13-5. Table 6 shows only 10 testing patterns due to space limitation. However, Figure 6 shows all 100 testing patterns for all five loads. Displays trained ANN provided MAPE 0.27678722% and Maximum Error for all five loads are 0.3684, 1.5245, 0.5574, 0.5833 and 2.1097, respectively. Also, the hidden layer size for cluster 4 is 13.

Figure 6. Testing performance of ANN in IEEE-30 bus system (cluster 4).

Table 6. Wheeling price prediction for loads in IEEE-30 bus system (cluster 4).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 1	97.3975	97.41715	−0.01965	0.02017
	Load 5	63.1311	63.35419	−0.22309	0.35337
	Load 7	66.7269	66.61158	0.11532	0.17282
	Load 15	141.7752	141.6811	0.09408	0.06636
	Load 19	60.1611	59.88105	0.28005	0.46551
2	Load 1	95.1963	95.31335	−0.11705	0.12295
	Load 5	67.5094	67.71864	−0.20924	0.30994
	Load 7	63.0117	63.0675	−0.0558	0.08855
	Load 15	131.6742	131.8802	−0.20602	0.15646
	Load 19	52.7794	52.46224	0.31716	0.60091
3	Load 1	94.7325	94.90091	−0.16841	0.17777
	Load 5	59.3125	59.04034	0.27216	0.45886
	Load 7	62.2885	62.37831	−0.08981	0.14419
	Load 15	127.1874	127.371	−0.18363	0.14438
	Load 19	65.5475	65.32955	0.21795	0.3325
4	Load 1	94.7809	94.81192	−0.03102	0.03273
	Load 5	60.9723	61.31825	−0.34595	0.56739
	Load 7	61.8565	61.89685	−0.04035	0.06523
	Load 15	127.9781	127.7797	0.19842	0.15504
	Load 19	50.6288	50.70876	−0.07996	0.15794
5	Load 1	91.656	91.51953	0.13647	0.1489
	Load 5	64.3088	64.04123	0.26757	0.41607
	Load 7	61.1691	61.17463	−0.00553	0.00905
	Load 15	132.3305	132.5911	−0.26055	0.19689
	Load 19	61.0408	61.57866	−0.53786	0.88115
6	Load 1	93.5406	93.4631	0.0775	0.08285
	Load 5	63.883	63.94308	−0.06008	0.09405
	Load 7	62.7669	62.64866	0.11824	0.18838
	Load 15	135.6104	135.694	−0.08359	0.06164
	Load 19	58.468	58.66214	−0.19414	0.33204
7	Load 1	101.4214	101.3095	0.11188	0.11031
	Load 5	61.6651	61.97001	−0.30491	0.49446
	Load 7	66.0963	66.11143	−0.01513	0.02289
	Load 15	142.3908	142.5997	−0.20893	0.14673
	Load 19	55.8901	56.20489	−0.31479	0.56323
8	Load 1	97.65	97.56238	0.08762	0.08973
	Load 5	69.9671	69.32873	0.63837	0.91238
	Load 7	63.224	63.37892	−0.15492	0.24503
	Load 15	135.2215	135.1489	0.07259	0.05368
	Load 19	58.3844	58.91883	−0.53443	0.91537
9	Load 1	96.7454	96.81826	−0.07286	0.07531
	Load 5	66.5995	66.7518	−0.1523	0.22868
	Load 7	63.8858	63.81031	0.07549	0.11817
	Load 15	139.7	139.6909	0.00907	0.00649
	Load 19	63.5943	63.72054	−0.12624	0.19851
10	Load 1	91.3039	91.50364	−0.19974	0.21877
	Load 5	56.0679	56.24598	−0.17808	0.31761
	Load 7	59.933	59.93982	−0.00682	0.01139
	Load 15	122.1012	121.8171	0.28414	0.23271
	Load 19	55.8616	55.26195	0.59965	1.07346

The optimal structure for Cluster 5 is 21-06-1. Table 7 shows only 10 testing patterns as similar as last cluster. However, Figure 7 shows all 100 testing patterns for all five loads. Displays trained ANN provided MAPE 0.1079% and Maximum Error for one load is 0.9759. Also, the hidden layer size for cluster 5 is 6.

Figure 7. Testing performance of ANN in IEEE-30 bus system (cluster 5).

Table 7. Wheeling price prediction for loads in IEEE-30 bus system (cluster 5).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load10	534.5354	534.4787	0.05668	0.0106
2	Load10	565.9721	565.6724	0.29972	0.05296
3	Load10	502.2149	502.9685	−0.75358	0.15005
4	Load10	525.1114	525.1031	0.00832	0.00158
5	Load10	549.887	549.7689	0.11813	0.02148
6	Load10	535.1343	534.7164	0.41792	0.0781
7	Load10	533.2243	532.761	0.4633	0.08689
8	Load10	547.0462	547.0206	0.02564	0.00469
9	Load10	542.7897	542.1032	0.68651	0.12648
10	Load10	513.9104	512.7632	1.14723	0.22324

The optimal structure for Cluster 6 is 21-16-2. Table 8 shows only 10 testing patterns as similar as last cluster. However, Figure 8 shows all 100 testing patterns for all five loads. Displays trained ANN provided MAPE 0.2317% and Maximum Error for both of loads are 1.6708 and 0.5346. Also the hidden layer size for cluster 6 is 16.

Figure 8. Testing performance of ANN in IEEE-30 bus system (cluster 6).

Table 8. Wheeling price prediction for loads in IEEE-30 bus system (cluster 6).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 8	905.0864	908.3151	−3.22873	0.35673
	Load 17	834.8993	836.0191	−1.11978	0.13412
2	Load 8	924.2664	926.4069	−2.14053	0.23159
	Load 17	864.1005	864.7056	−0.60513	0.07003
3	Load 8	858.5029	854.7518	3.75109	0.43693
	Load 17	835.4308	834.6363	0.79453	0.0951
4	Load 8	866.0712	864.9901	1.08114	0.12483
	Load 17	876.914	875.9036	1.01045	0.11523
5	Load 8	910.3861	916.5783	−6.19216	0.68017
	Load 17	848.361	849.2545	−0.89345	0.10532
6	Load 8	899.8849	904.333	−4.44807	0.49429
	Load 17	840.1935	841.8232	−1.62974	0.19397
7	Load 8	899.2019	907.5365	−8.33461	0.92689
	Load 17	827.8247	829.2495	−1.4248	0.17211
8	Load 8	937.5637	929.1352	8.4285	0.89898
	Load 17	837.3153	835.1869	2.12844	0.2542
9	Load 8	909.0777	910.9553	−1.8776	0.20654
	Load 17	846.6879	847.367	−0.67907	0.0802
10	Load 8	871.9805	868.3061	3.67445	0.42139
	Load 17	863.3833	863.4959	−0.11262	0.01304

The optimal structure for Cluster 7 is 21-09-1. Table 9 shows only 10 testing patterns as similar as last cluster. However, Figure 9 shows all 100 testing patterns for all five loads. Displays trained ANN provided MAPE 0.1093% and Maximum Error for one load is 0.8658. Also, the hidden layer size for cluster 7 is 9.

Figure 9. Testing performance of ANN in IEEE-30 bus system (cluster 7).

Table 9. Wheeling price prediction for loads in IEEE-30 bus system (cluster 7).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 18	2687.263	2686.868	0.39445	0.01468
2	Load 18	2628.69	2626.749	1.94156	0.07386
3	Load 18	2831.243	2828.492	2.75111	0.09717
4	Load 18	2767.434	2766.515	0.91935	0.03322
5	Load 18	2690.577	2686.553	4.02427	0.14957
6	Load 18	2733.885	2736.392	−2.50737	0.09171
7	Load 18	2703.194	2697.214	5.98041	0.22123
8	Load 18	2636.52	2642.798	−6.27841	0.23813
9	Load 18	2645.863	2641.183	4.67969	0.17687
10	Load 18	2829.604	2833.344	−3.74033	0.13219

The optimal structure for Cluster 8 is 21-06-3. Table 10 shows only 10 testing patterns as similar as last cluster. However, Figure 10 shows all 100 testing patterns for all five loads. Displays trained ANN provided MAPE 0.1734% and Maximum Error for all three loads are 0.3557, 1.5788 and 1.6594. Also the hidden layer size for cluster 8 is 6.

Figure 10. Testing performance of ANN in IEEE-30 bus system (cluster 8).

Table 10. Wheeling price prediction for loads in IEEE-30 bus system (cluster 8).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 3	222.5147	222.5147	0.00004	0.00002
	Load 4	272.4497	271.6368	0.81295	0.29839
	Load 16	265.8845	265.6295	0.25499	0.0959
2	Load 3	224.1407	224.2819	−0.14118	0.06299
	Load 4	287.2671	287.7477	−0.48063	0.16731
	Load 16	247.8932	248.1445	−0.25134	0.10139
3	Load 3	210.8981	211.1644	−0.26628	0.12626
	Load 4	256.9002	257.6621	−0.76193	0.29659
	Load 16	249.2333	250.0515	−0.81815	0.32827
4	Load 3	215.0707	215.1757	−0.105	0.04882
	Load 4	265.8582	266.8068	−0.94857	0.3568
	Load 16	240.4355	240.2206	0.21493	0.08939
5	Load 3	216.9313	216.8269	0.10443	0.04814
	Load 4	276.1049	275.9128	0.1921	0.06958
	Load 16	251.3874	251.8444	−0.45697	0.18178
6	Load 3	212.8329	212.7731	0.05984	0.02812
	Load 4	267.8874	267.5693	0.31814	0.11876
	Load 16	249.6989	250.2546	−0.55565	0.22253
7	Load 3	225.803	225.934	−0.13099	0.05801
	Load 4	270.9744	271.2442	−0.26984	0.09958
	Load 16	262.2987	263.2267	−0.92803	0.35381
8	Load 3	227.4087	227.747	−0.33831	0.14877
	Load 4	293.1941	294.0226	−0.8285	0.28258
	Load 16	248.6463	249.9775	−1.33121	0.53538
9	Load 3	224.5741	224.5737	0.00045	0.0002
	Load 4	284.2265	284.6816	−0.45512	0.16013
	Load 16	268.9299	269.3301	−0.40016	0.1488
10	Load 3	202.6425	202.736	−0.09352	0.04615
	Load 4	245.6922	246.0944	−0.40217	0.16369
	Load 16	242.1368	242.3852	−0.24842	0.10259

The optimal structure for Cluster 9 is 21-22-7. Table 11 shows only 10 testing patterns as similar as last cluster. However, Figure 11 shows all 100 testing patterns for all seven loads. Displays trained ANN provided MAPE 0.1421584% and Maximum Error for all seven loads are 0.9006, 0.4731, 1.7118, 1.2530, 1.1632, 1.3736 and 1.1139, respectively. Also the hidden layer size for cluster 9 is 22.

Figure 11. Testing performance of ANN in IEEE-30 bus system (cluster 9).

Table 11. Wheeling price prediction for loads in IEEE-30 bus system (cluster 9).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 2	8.8841	8.88453	−0.00043	0.00484
	Load 9	12.5262	12.51913	0.00707	0.05644
	Load 11	0.8132	0.81085	0.00235	0.28894
	Load 12	9.5413	9.54437	−0.00307	0.03221
	Load 14	7.249	7.254	−0.005	0.06893
	Load 20	1.6461	1.64523	0.00087	0.05309
	Load 21	3.546	3.54967	−0.00367	0.10349
2	Load 2	8.7169	8.73871	−0.02181	0.25022
	Load 9	12.3269	12.35378	−0.02688	0.21807
	Load 11	0.7165	0.71371	0.00279	0.38967
	Load 12	9.6158	9.65952	−0.04372	0.45463
	Load 14	7.1512	7.17794	−0.02674	0.37386
	Load 20	1.651	1.65518	−0.00418	0.25294
	Load 21	3.3482	3.35513	−0.00693	0.20698
3	Load 2	8.5755	8.58102	−0.00552	0.06441
	Load 9	12.0104	12.03173	−0.02133	0.1776
	Load 11	0.7313	0.73301	−0.00171	0.23336
	Load 12	9.4813	9.48574	−0.00444	0.04678
	Load 14	7.0246	7.0266	−0.002	0.02842
	Load 20	1.5922	1.59375	−0.00155	0.09711
	Load 21	3.1272	3.12317	0.00403	0.12888
4	Load 2	8.5817	8.57759	0.00411	0.04786
	Load 9	12.0623	12.02504	0.03726	0.30891
	Load 11	0.7265	0.72763	−0.00113	0.15602
	Load 12	9.424	9.42476	−0.00076	0.00804
	Load 14	7.0128	7.00568	0.00712	0.10154
	Load 20	1.6332	1.63555	−0.00235	0.14382
	Load 21	3.4793	3.48602	−0.00672	0.19316
5	Load 2	8.4232	8.42894	−0.00574	0.0681
	Load 9	11.9234	11.92428	−0.00088	0.00734
	Load 11	0.7011	0.70417	−0.00307	0.43787
	Load 12	9.2358	9.23662	−0.00082	0.00889
	Load 14	6.8925	6.89886	−0.00636	0.09221
	Load 20	1.5397	1.53995	−0.00025	0.01632
	Load 21	3.1916	3.19024	0.00136	0.04254
6	Load 2	8.5567	8.55676	−0.00006	0.00075
	Load 9	12.0687	12.07093	−0.00223	0.01844
	Load 11	0.7658	0.76392	0.00188	0.24562
	Load 12	9.2398	9.24662	−0.00682	0.07378
	Load 14	7.0498	7.05435	−0.00455	0.0645
	Load 20	1.5564	1.55669	−0.00029	0.01856
	Load 21	3.0036	2.99965	0.00395	0.13143
7	Load 2	9.1842	9.18576	−0.00156	0.01704
	Load 9	13.025	13.00071	0.02429	0.18646
	Load 11	0.7769	0.78124	−0.00434	0.55851
	Load 12	10.0979	10.09652	0.00138	0.01368
	Load 14	7.5516	7.55476	−0.00316	0.04189
	Load 20	1.6975	1.69837	−0.00087	0.05108
	Load 21	3.6943	3.68746	0.00684	0.18513
8	Load 2	8.9498	8.92309	0.02671	0.29843
	Load 9	12.5462	12.5654	−0.0192	0.15301
	Load 11	0.756	0.75184	0.00416	0.55067
	Load 12	9.721	9.68666	0.03434	0.35321
	Load 14	7.1833	7.15995	0.02335	0.32508
	Load 20	1.6071	1.60282	0.00428	0.26601
	Load 21	3.1469	3.13351	0.01339	0.42565
9	Load 2	8.8021	8.79131	0.01079	0.12261
	Load 9	12.3881	12.37874	0.00936	0.07558
	Load 11	0.7454	0.74081	0.00459	0.6158
	Load 12	9.5578	9.55887	−0.00107	0.01117
	Load 14	7.1504	7.14726	0.00314	0.04396
	Load 20	1.5765	1.57202	0.00448	0.28439
	Load 21	3.1542	3.1459	0.0083	0.2631
10	Load 2	8.257	8.24852	0.00848	0.10267
	Load 9	11.7296	11.71747	0.01213	0.10342
	Load 11	0.7079	0.69832	0.00958	1.3535
	Load 12	9.0964	9.12093	−0.02453	0.26963
	Load 14	6.714	6.71697	−0.00297	0.04424
	Load 20	1.5658	1.56743	−0.00163	0.10425
	Load 21	3.3716	3.38259	−0.01099	0.3261

The optimal structure for Cluster 10 is 21-06-2. Table 12 shows only 10 testing patterns as similar as last cluster. However, Figure 12 shows all 100 testing patterns for both of the loads. Displays trained ANN provided MAPE 0.2644% and Maximum Error for both of the loads are 1.2820 and 1.4526, respectively. Also the hidden layer size for cluster 9 is 6.

Figure 12. Testing performance of ANN in IEEE-30 bus system (cluster 10).

Table 12. Wheeling price prediction for loads in IEEE-30 bus system (cluster 10).

TP	Load unit	Target	ANN output	Error	% Abs. error
1	Load 6	0.0761	0.07617	−0.00007	0.09459
	Load 13	0.1797	0.17958	0.00012	0.06931
2	Load 6	0.0743	0.07415	0.00015	0.20666
	Load 13	0.1843	0.18494	−0.00064	0.34688
3	Load 6	0.0803	0.08063	−0.00033	0.41597
	Load 13	0.1721	0.17242	−0.00032	0.1849
4	Load 6	0.0775	0.07771	−0.00021	0.26535
	Load 13	0.1773	0.1776	−0.0003	0.17147
5	Load 6	0.0675	0.06731	0.00019	0.27804
	Load 13	0.1715	0.17174	−0.00024	0.14036
6	Load 6	0.0795	0.07948	0.00002	0.03133
	Load 13	0.1806	0.18021	0.00039	0.21574
7	Load 6	0.0795	0.07983	−0.00033	0.41683
	Load 13	0.1943	0.19357	0.00073	0.37765
8	Load 6	0.0754	0.07443	0.00097	1.28197
	Load 13	0.1597	0.16168	−0.00198	1.24162
9	Load 6	0.0733	0.07323	0.00007	0.08947
	Load 13	0.165	0.16519	−0.00019	0.11289
10	Load 6	0.0753	0.07496	0.00034	0.45479
	Load 13	0.152	0.15403	−0.00203	1.33472

Parallel computing has been used in this article to reduce the time taken to develop 10 ANN developed for all 10 clusters. In parallel computing, due to the tasks assigned to all the workers, each processor completes its task at the same time. The computational work was carried out using MATLAB 2020a version on a personal computer having a 1.8 GHz Intel Processor, 8GB RAM, Core i5, and 64-bit operating system with 4 processing cores.

Table 13 shows the elapsed time required by each worker while ten ANNs were developed simultaneously by 4 workers using parallel computing. However, the total elapsed time in this case was 10.879545 s. To evaluate the effectiveness of this approach, all ten ANNs were developed sequentially. In this, the total time elapsed by each worker was 34.471610 s. And thus the total speed gain is 3.1685.

Table 13. Parallel computing details for wheeling pricing in IEEE 30-bus system.

Worker	ANNs developed	Structure of ANN	Elapsed time (s)	Speed gain
1	Cluster 1	21-05-2	6.5915	3.1685
	Cluster 2	21-24-3
	Cluster 3	21-18-2
2	Cluster 4	21-13-5	5.5087
	Cluster 5	21-06-1
3	Cluster 6	21-16-2	4.2717
	Cluster 7	21-09-1
	Cluster 8	21-06-3
	Cluster 10	21-06-2
4	Cluster 9	21-22-7	10.6956

7. Conclusion

The assessment and allocation of wheeling prices have emerged as an important issue in the deregulated power market. Wheeling price estimation and allocation can be done by various optimum power flow based methods and evolutionary techniques also. Achieving low accuracy and long-time wheeling price predictions for a specified range by various search-based optimization techniques poses a problem. Apart from this, getting accurate wheeling prices in less time is also a big problem in large size power system. In this article, Wheeling Price Prediction has been successfully obtained accurately and in a short time by developing multiple clusters by k-means Clustering and Parallel Computing. In addition, better results have been obtained in multi-layer perceptron RBFNNs as compared to normal ANN applications. For this the proposed approach can prove to be highly beneficial in wheeling pricing with actual power market. This technique can also be successfully used with large size power systems, which shows the novelty of the work.

Availability of data and material

The datasets generated and/or analyzed during the current study are not publicly available due [confidential] but are available from the corresponding author on reasonable request.

Acknowledgment

The authors sincerely acknowledge the Director, MITS, Gwalior, India, Deputy Director (Legal Metrology), Regional Reference Standard Laboratory, Ahmedabad (RRSL), Director, M/s GSL Technology and Services, Kiran Garden, Uttam Nagar, New Delhi, Principal, BRA Polytechnic College, Gwalior and Management ITM University, Gwalior, for providing facilities to carry out research work.

Disclosure statement

The authors declare that they have no competing interests.

Additional information

Notes on contributors

Abhishek Saxena

Abhishek Saxena received the B.E. degree in electrical engineering from Maharana Pratap College of Technology, Gwalior and the M.E. degree in industrial system and drives from Madhav Institute of Technology and Science, Gwalior, Madhya Pradesh, India, in 2006 and 2009, respectively.

Seema N. Pandey

Dr. Seema N. Pandey received the B.E. degree in electrical engineering and the M.E. degree in power systems from the Madhav Institute of Tech- nology and Science, Gwalior, India, in 1998 and 2003, respectively, and the Ph.D. degree from ABVIIITM, Gwalior, in 2010.

Shishir Dixit

Dr. Shishir Dixit, Associate Professor, teaches at the Madhav Institute of Technology & Science in Gwalior. As of July 2003, he was a Lecturer at MITS. He received his Master's in Design and Production of H.E.E. from MANIT, Bhopal in 2003, his Ph.D. in Electrical Engineering Stream from Maulana Azad National Institute of Technology, Bhopal in 2014.

Appendix A1

* Bialek tracing technique

The only requirement of the Bialek method is that it should satisfy Kirchhoff’s current law completely. For this, this method is equally applicable for real and reactive power flows. In Bialek tracing, the allocation to each user is done through topology analysis. The Bialek tracing method is divided into two algorithms ‘upstream’ and ‘downstream’ based on the proposed sharing principle, which performs tracing for each user to calculate wheeling pricing. The upstream algorithm displays how much of the total current flowing to a node on a particular generator bus is present (Bialek, 1998). The total apparent power flow S_i through the node given all the currents can be expressed as follows: (13) $$S_i=\sum _{j\in v_i^{\left(u\right)}}\left|S_{\mathit{\text{ij}}}\right|+S_{\mathit{\text{Gi}}}\hspace{1em}\mathit{\text{for }}i=1,2,3,\dots n$$(13) where $v_i^{(u)}$ is the set of buses supplying directly to bus i. $S_{\mathit{\text{ij}}}$ shows the apparent power flow from node $i$ to line $i-j$ and $S_{\mathit{\text{Gi}}}$ is the generation at node $i.$ Along with this '$n$' is the total number of buses.

Since each outflow down the line from a node is dependent only on the voltage gradient and the impedance of the line, according to the principle of proportional sharing, each current that leads to the node has the same ratio of current to the total nodal flow S_i. The line flow may be written in terms of nodal flow at bus j as $\left|S_{\mathit{\text{ij}}}\right|=x_{\mathit{\text{ji}}}{S}_j$ Thus (14) $$S_i=\sum _{j\in V_i^{\left(u\right)}}{x}_{\mathit{\text{ji}}}{S}_j+S_{\mathit{\text{Gi}}}$$(14) (15) $$S_{\mathit{\text{Gi}}}=S_i-\sum _{j\in V_i^{\left(u\right)}}{x}_{\mathit{\text{ji}}}{S}_j$$(15) or (16) $$A_{u}S=S_G$$(16) where $A_u$ is the upstream distribution matrix, S is the total vector nodal flow and $S_G$ is the vector of bus generation. Here $A_u$ is (17) $${\left[A_u\right]}_i{}_j=\{\begin{array}{cc}1& \mathit{\text{for }}i=j\\ -x_{\mathit{\text{ji}}}& \mathit{\text{for }}j\in v_i^{(u)}\\ 0& \mathit{\text{otherwise}}\end{array}$$(17) if $A_u^{-1}$ exist then (18) $$S=A_u^{-1}{S}_G$$(18)

The ith element of S will be (19) $$S_i=\sum _{m=1}^n{\left[A_u^{-1}\right]}_{\mathit{\text{im}}}{S}_{\mathit{\text{Gm}}}\hspace{1em}\mathit{\text{for }}i=1,2,3\dots .n$$(19)

It is worth noting that the 'm^th’ generator’s share in the 'i^th’ bus power is $S_i=\sum _{m=1}^n{\left[A_u^{-1}\right]}_{\mathit{\text{im}}}{S}_{\mathit{\text{GM}}}.$ Here, the bus power at bus i, S_i is also equal to the sum of power flows in the lines leaving the bus i and the load demand S_Di. By using Proportional Sharing Principle (20) $$\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|=\frac{\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|}{S_i}.S_i=\frac{\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|}{S_i}.\sum _{m=1}^n{\left[A_u^{-1}\right]}_{\mathit{\text{im}}}{S}_{\mathit{\text{Gm}}}$$(20) or (21) $$\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|=\sum _n{D}_{\mathit{\text{ik}},m}^G{S}_{\mathit{\text{Gm}}}\hspace{1em}\mathit{\text{for all}}k\in V_i^{\left(D\right)}$$(21)

Where $D_{\mathit{\text{ik}},m}^G=\left|{(S_{\mathit{\text{ij}}})}_{\mathit{\text{ik}}}\right|{\left[A_u^{-1}\right]}_{\mathit{\text{im}}}/S_i$ and $v_i^{(D)}$ is the set of buses, which are directly supplied from bus i and $D_{\mathit{\text{ik}},m}^G$ is topological distribution factor, which is the portion of the generation due to mth generator that flows in the line ik.

The downstream algorithm calculates the share of the individual load in the individual line flow.when apparent power of bus S_i is expressed as the sum of outflows (22) $$S_i=\sum _{l\in V_i^{\left(d\right)}}\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|+S_{\mathit{\text{Di}}}=\sum _{l\in V_i^{\left(d\right)}}{x}_{\mathit{\text{li}}}{S}_l+S_{\mathit{\text{Di}}}\hspace{1em}\mathit{\text{for }}i=1,2,3,\dots n$$(22) where $V_i^{(d)}$ is the group of buses which directly supplied bus i and (23) $$x_{\mathit{\text{li}}}=\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{li}}}\right|/S_l.$$(23)

Equation (23)(23) $$x_{\mathit{\text{li}}}=\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{li}}}\right|/S_l.$$(23) may be written as (24) $$S_i-\sum _{l\in V_i^{\left(d\right)}}{x}_{\mathit{\text{li}}}{S}_l=S_{\mathit{\text{Di}}}\mathit{\text{\hspace{1em}or\hspace{1em}}}{A}_{D}S=S_D$$(24) where $A_D$ is the downstream distribution matrix $S_D$ is the vector of bus load demand. the il^th element of the matrix A_D is (25) $${\left[A_D\right]}_{\mathit{\text{il}}}=\{\begin{array}{l}1\hspace{1em}\mathit{\text{for }}i=j\\ -x_{\mathit{\text{li}}}\hspace{1em}=-\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{il}}}\right|/S_l\mathit{\text{for }}l\in v_i^{(d)}\\ 0\hspace{1em}\mathit{\text{otherwise}}\end{array}$$(25)

If $A_D^{-1}$ exists then $P=A_D^{-1}{S}_D,$ and its i^th element is equal to (26) $$S_i=\sum _{k=1}^n{\left[A_D^{-1}\right]}_{\mathit{\text{ik}}}{S}_{\mathit{\text{Dk}}}\hspace{1em}\mathit{\text{for }}i=1,2,3,\dots .n$$(26)

With that, the inflow to node i from line i-j can be calculated using the proportional sharing principle as (27) $$\left|{\left(S_{\mathit{\text{ij}}}\right)}_{}\right|=\frac{\left|{\left(S_{\mathit{\text{ij}}}\right)}_{}\right|}{S_i}.S_i=\frac{\left|{\left(S_{\mathit{\text{ij}}}\right)}_{}\right|}{S_i}.\sum _{k=1}^n{\left[A_D^{-1}\right]}_{\mathit{\text{im}}}{S}_{\mathit{\text{Dk}}}$$(27)

Or (28) $$\left|{\left(S_{\mathit{\text{ij}}}\right)}_{\mathit{\text{ik}}}\right|=\sum _{k=1}^n{D}_{\mathit{\text{ij}},k}^L{S}_{\mathit{\text{Dk}}}\hspace{1em}\mathit{\text{for all }}k\in V_i^{\left(D\right)}$$(28) where $D_{\mathit{\text{ij}},k}^L=\left|{(S_{\mathit{\text{ij}}})}_{\mathit{\text{ik}}}\right|{\left[A_D^{-1}\right]}_{\mathit{\text{im}}}/S_{\mathit{\text{Dk}}}$ is the is topological load distribution factor, which gives the portion of the kth load demand that flows in the line ik.