Modified penalized quoted cost method for transmission loss allocation including reactive power demand in deregulated electricity market

Sustainable Energy, Grids and Networks 16 (2018) 370–379

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Modified penalized quoted cost method for transmission loss allocation including reactive power demand in deregulated electricity market K. Shafeeque Ahmed, S. Prabhakar Karthikeyan

∗,1

School of Electrical Engineering, VIT University, Vellore, 632014, Tamil Nadu, India

article

info

Article history: Received 28 June 2018 Accepted 31 October 2018 Available online 5 November 2018 Keywords: Deregulated electricity market Mathematical loss Real loss component Loss contribution fraction Reactive loss component Load sensitivity factor

a b s t r a c t Transmission loss/cost allocation to the individual buses in a fair way is an essential issue in restructured electricity market. This paper splits active power transmission loss into two components namely real and reactive loss component. The real and reactive loss components are due to the real power transaction from the generator to the load and reactive power demand of the load respectively. The active power loss changes due to the reactive power loads and this increase in loss are allocated to the relevant loads based on load sensitivity factor (LSF). The loss allocation using LSF is a valid approach as it is derived from solved load flow solution. The cost allocation of real loss component to a transaction and reactive loss component to loads is based on the loss contribution fraction (LCF) and LSF respectively. The highlight of this paper is to allocate real loss component to the respective buses equally and to allocate the cost of reactive loss component to the respective loads in a fair way. This paper also presents a detailed discussion on impact of mathematical loss and comparison with four existing loss allocation methods. All the methods are discussed by considering with and without reactive power load. The effectiveness of the proposed approach is illustrated using a sample four bus and 30 bus IEEE system. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In vertically integrated system, the total available power generation is dispatched centrally by minimizing the fuel cost of the generator through traditional economic dispatch. In competitive and open access electricity market, a portion of load may be dispatched through bilateral contracts among the generators and the loads. The remaining load is dispatched similar to vertically integrated system. Bilateral contracts are generally a long term contract between a generator and a load [1]. Garnget. al. [2] has proposed a slack bus independent loss allocation method for bilateral market. The generator supplies both power to the loads and its associated losses. Therefore, total load and its losses are shared among all the generators in the network. Kyung-Il Min [3] has presented a new path-integral method which increases the accuracy of loss allocation considering the non-linearity. A reliable graph theory based loss allocation method is presented in [4] which depend on power flow paths. Loss allocation incorporating optimal power flow is presented in [5] which use sensitivity analysis to determine new optimal operating point. ∗ Corresponding author. E-mail addresses: [email protected] (K. Shafeeque Ahmed), [email protected] (S. Prabhakar Karthikeyan). 1 Senior Member, IEEE. https://doi.org/10.1016/j.segan.2018.10.004 2352-4677/© 2018 Elsevier Ltd. All rights reserved.

In [6], Pro-rata procedure allocates losses to the generators and the loads based on the level of real power injection. Marginal cost based approach allocates losses using incremental transmission loss coefficients. The distance from the load to the generator is not considered in this method [7]. Proportional sharing method uses solved load flow solution along with linear proportional sharing principle [8–10]. Visaka et al. [11] proposed a method based on relative electrical distance. The relative location of the load buses are estimated with respect to the generator buses. Transmission loss pricing based on the circulating currents among generators is presented in [12]. Comparison and discussion on different loss allocation methodologies are discussed in [13–17]. Loss allocation based on graph theory is power flow path dependent. In bilateral market generation and loads are fixed but power flow is independent of contracted path. Therefore, graph theory method is reliable as it depends on power flow paths [18,19]. Li et al. proposed a new loss allocation method for pool, bilateral and hybrid power market. It derives a loss equation relating bus power and power flow in a DC power network. This method allocates loss in natural manner [20]. Nikoukar et al. presented a method which uses power invariant matrix. Line flows are expressed in terms of source or demand currents [21]. A new loss allocation method is discussed in [22] where generators and loads are modeled as current injections and impedances respectively. This method uses circuit theory and Aumann–Shapley

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379

method. A new pricing model is proposed for bilateral, pool and reserve market which uses optimal power flow (OPF) [23]. A method based on incremental transmission loss is presented with two models which uses DC and AC load flow equations for loss allocation [24]. In Proportional Generation and Proportional Load (PGPL) method [25], loss factors are applied to allocate active power loss to the respective buses. Cost allocation to the generators and the loads are dependent on generator and load loss factors respectively. Cost allocation to the respective buses purely depends on magnitude of power generation and demand. PGPL method considering reactive loads is discussed in [26]. Mutual inductance between transmission lines and its impact on loss/cost allocation is discussed in [27] and the results are demonstrated with reference to PGPL method. In Zbus method [28], network equations are used and this method is reasonable as it uses current injections for loss allocation to the individual buses. It method allocates unequal costs among the generators and the loads (i.e., equal 50–50 allocation is not maintained) which is a major drawback. Moreover it is dependent on system size i.e., it results in different percentage of cost allocation to the generators and the loads for different systems. In modified Zbus method [29], the individual line power flow is divided among all nodal currents and the cost of individual line is allocated among all generators and loads of the network. This method is independent of slack bus selection but the cost allocation is not fair as 50–50 allocation is not maintained. None of the above existing approaches either consider the reactive power loads while pricing real power loss or share the loss due to real power flow equally among the generators and the loads. This paper presents a pricing methodology for reactive loss component and it is an extended work of pricing real loss component approach as addressed by the authors in [30]. In Penalized Quoted Cost (PQC) method, loss allocation is carried out without considering reactive loads and in modified PQC method the loss allocation problem is addressed including reactive loads (by introducing LSF to price reactive loads). The same pricing approach of reactive loss component is implemented on the existing methods like PGPL method, Zbus and modified Zbus methods. The reactive power loss/cost allocation is beyond the scope of this paper and hence it is not discussed in this work. The entire results presented in this paper are applicable only for normal operation of the grid. Mathematical loss is the loss obtained from the load flow solution without generation and load. This loss is due to the small voltage difference among the nodes which drives small current through the lines contributing to the system loss. In literature, mathematical loss is not taken into account while addressing loss/cost allocation issue in deregulated electricity market. This paper presents a detailed discussion on mathematical loss and its impact on transmission real power loss/cost allocation. Mathematical loss has a major impact on real power transmission loss allocation on large systems and consequently cannot be ignored. For smaller systems, this loss has a minor effect and hence can be ignored. The contribution of this paper is listed as follows

• Loss allocation including reactive power loads • Impact of mathematical loss on loss allocation—with and without reactive loads • Application of modified PQC method (LSF based cost allocation for reactive loads) on other existing methods and its comparison • comparison of modified PQC method with other existing methods including reactive loads A comparative study on the four existing approaches is shown in Table 1.

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2. Proposed approach In deregulated electricity market, for a‘n’ bus system: ‘ng ’ generators, ‘nl ’ demands and ‘nd ’ dummy buses (buses without generation and load) participates in real power loss or cost allocation. The cost allocation of real and reactive loss components are discussed in the following sections. The real and reactive loss component cost is allocated to the transactions and the loads respectively. To calculate the cost associated with reactive loss component, first the total system loss (both real and reactive loss components) are considered and the associated total cost is computed. The total active power loss including the reactive power loads is given by Eq. (1).

LTotal =

ng ng ∑ ∑

Pgi Bij Pgj +

i=1 j=1

ng ∑

Bi0 Pgi + B00

(1)

i=1

where, Pgi - real power generation at bus ‘i’ Bij , Bi0 &B00 are the AC loss coefficients The PQC for the generator ‘i’ is given in Eq. (2) PQCi = Li ∗ QCi

(2)

where, QC i - quoted cost of the generator ‘i’ Li - penalty factor of the generator ‘i’ The penalty factor of the generators is given in Eq. (3) Li =

1 1−

∂ LTotal ∂ Pgi

(3)

The real loss component is computed from Eq. (1) without reactive loads and it is denoted as LTr . The reactive loss component is shown in Eq. (4). LQD = LTotal − LTr

(4)

where, LTr is the real power loss obtained from load flow solution taking transactions alone i.e., neglecting the reactive power loads. The total cost associated with reactive loss component is given by Eq. (5). CQDT = CTotal − CTr

(5)

where, CTr is the cost of real loss component CTotal is the total cost for supplying losses due to real and reactive loss component 2.1. Cost allocation to the transactions The real loss component cost is allocated to the transactions based on LCF. It is derived from the DC load flow equation [31] and it is given in Eq. (6). Ploss−dc = PiT [Bmn ] Pi

(6)

where, Pi - bus power injections Bmn - DC loss coefficients The real power loss of transaction ‘T’ from bus ‘i’ to bus ‘j’ is given in Eq. (7). LT = Li−j =

∂ Ploss−dc ∂ Ploss−dc Pgi − Pdj for T = 1 to ntr ∂ Pi ∂ Pj

(7)

Loss contribution fraction of transaction ‘T’ is given in Eq. (8). LT fT = ∑ntr

T =1

LT

(8)

Cost allocated to transaction ‘T’ is shown in Eq. (9). CTT = fT ∗ CTr

(9)

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K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379 Table 1 Comparison of four existing loss allocation approaches. Method Reference Year

PQC [30] 2016

PGPL [25] 2016

Zbus [28] 2001

Modified Zbus [29] 2007

Loss supplying generators are prioritized based on penalized quoted cost

Two loss factors are used for cost allocation

Uses basic network equations

Line power flow is divided among all nodal currents

Losses are supplied sequentially from generators based on priority

Generator loss factor used for generators

Uses current injections for loss allocation rather than power injections

Uses electrical distance to allocate losses to the individual lines

Transaction loss cost allocation uses LCF

Load loss factor used for loads

Assumptions

All system loads are uniquely shared by bilateral contracts

All system loads are uniquely shared by bilateral contracts

No simplifying assumptions

No simplifying assumptions

Fairness (50–50 sharing)

Yes

Yes

No

No

Negative loss Allocation

No

No

Yes

No

Features

Methodology

Slack bus selection

Independent

Independent

Dependent

Independent

System size

Independent

Independent

Dependent

Dependent

Quantum of generation

Independent

Independent

Dependent

Dependent

Application of load flow solution

Yes

Yes

Yes

Yes

Relative position of the buses

Accounted

Accounted

Accounted

Accounted

Congestion Issue

Not addressed

Not addressed

Not addressed

Not addressed

The active power loss due to reactive power load ‘i’ is given by Eq. (10).

loss/cost due to congestion should be allocated to the responsible parties. For a one sided or unbalanced bilateral transactions (refers to selling or buying power through an agent), slack bus is regarded as the counterpart in loss/cost allocation process.

LQDi = LTotal − LDi

3. Algorithm

2.2. Cost allocation to the loads

for i = 1 to nl

(10)

where, LDi is the active power loss obtained from solved load flow solution with QDi = 0 for i = 1 to nl and QDi is the reactive power demand of load ‘i’. Load sensitivity factor of load ‘i’ is given by Eq. (11). LQ LSFi = ∑nl Di i=1 LQDi

for i = 1 to nl

(11)

Since transmission real power losses are highly non-linear, direct ∑ use of LQDi for cost allocation results in over recovery of cost as LQDi ̸ = LQD . Therefore, load sensitivity factor is used∑ which is reasonable. The sum of load sensitivity factor is one i.e. LSFi = 1. To compute load sensitivity factor, ‘nl ’ load flow solutions are required. Cost allocated to their active loss component to the load ‘i’ is given by Eq. (12). CQDi = LSFi ∗ CQDT

for i = 1 to nl

(12)

Finally, this reactive loss component cost is added to the respective loads along with their cost of real loss component. The proposed approach is restricted for normal operation of the power grid and hence congestion is not considered. The reason behind restriction is during congestion, the proposed method allocates total cost among the generators and the loads equally. However, this cost allocation is not fair as the cost is not allocated to the parties responsible for congestion i.e. the increase or decrease in

The algorithm for the proposed approach is presented in three sections: Computation of total cost of real and reactive loss component, real and reactive loss component cost allocation. 3.1. Computation of total cost of real and reactive loss component Step 1: Perform AC load flow solution with both real and reactive power demand. Compute loss coefficients, penalty factor, quoted cost and penalized quoted cost of the generators. Step 2: Rank ‘ng ’ generators in the ascending order of penalized quoted cost and form an array ‘R’ (1 × ng ). Step 3: Compute total real power loss(LTotal ) by taking first rank generator as slack bus. Step 4: If LTotal(j) ≥ ACR(j) where, AC R(j) - Additional available capacity of the generator R(j) for meeting the real power loss. If yes go to step 5 Else go to step 6 Step 5: Add ACR(j) to generator R (j) and calculate cost of supply of loss by generator ‘j’ as Closs(j) = ACR(j) ∗ QCR(j)

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379

LTotal(j+1) = LTotal(j) − ACR(j) Increment j = j + 1 and go to step 4. Step 6: Add LTotal(j) to generator R (j) and calculate cost of supply of loss by generator ‘j’ as Closs(j) = LTotal(j) ∗ QCIR(j) Step 7: Calculate total cost of supply of real power loss due to both real and reactive power is CTotal =

ng ∑

Closs(j)

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Table 2 Line data for sample four bus system. Line

1-3 1-2 4-2 4-3 1-4

Series impedance (p.u) Resistance R

Reactance X

0.00744 0.01008 0.00744 0.01272 0.00372

0.0372 0.0504 0.0372 0.0636 0.0186

Shunt admittance (p.u)

0.0775 0.1025 0.0775 0.1275 0.1550

3.3. Reactive loss component: Cost allocation

j=1

Flow chart to compute total cost of supply of active power loss is shown in Fig. 1. From the solved load flow solution, loss coefficients are computed to calculate the penalty factor (neglecting reactive loads). This penalty factor is used to compute the PQC of the generator. The real loss component is computed taking first rank generator as slack bus and supplied sequentially from the ranked generators. Finally, the cost of real loss component is computed. 3.2. Real loss component: Computation and cost allocation

The algorithm for reactive loss component cost allocation to the respective loads is presented in the following section Step 1: Compute reactive loss component using Eq. (4) Step 2: Calculate total reactive loss component cost using Eq. (5) Step 3: Compute active power loss due to reactive power load ‘i’ from Eq. (10) Step 4: Compute load sensitivity factor using Eq. (11)

The cost of real loss component is allocated to the respective transaction based on LCF. This cost is further allocated to the generators and the loads equally. The computation of real loss component cost is presented in the following algorithm.

Step 5: Allocate reactive loss component cost to the relevant loads using Eq. (12)

Algorithm

4.1. Sample four bus system

Step 1: Perform AC power flow solution without reactive power demands. Compute loss coefficients, penalty factor and penalized quoted cost of the generators. Step 2: Rank ‘ng ’ generators in the ascending order of penalized quoted cost and form an array ‘RT’. Step 3: Compute real loss component (LTr ) by taking first rank generator as slack bus. Step 4: If LTr(j) ≥ ACRT (j) where, AC RT (j) - Additional available capacity of the generator RT(j) for meeting the real power loss. If yes go to step 5 Else go to step 6 Step 5: Add ACRT (j) to generator RT(j) and calculate the cost of supply of loss by generator ‘j’ as Cploss(j) = ACRT (j) ∗ QCRT (j) LTr(j+1) = LTr (j) − ACRT (j) Increment j = j + 1 and go to step 4. Step 6: Add LTr(j) to generator RT(j) and calculate the cost of supply of loss by generator ‘j’ as Cploss(j) = LTr(j) ∗ QCRT (j) Step 7: Calculate total cost for delivering active power loss by neglecting reactive power demand using CTr =

ng ∑

Cploss(j)

j=1

The flow chart for cost allocation of real loss component is presented in Fig. 2. In this case the loss coefficients and penalty factor are obtained from the solved load flow solution including reactive loads. The total loss (both real and reactive loss component) is supplied from the ranked generators and the total cost is computed. The real loss component cost is allocated to the respective transaction based on LCF (obtained using DC load flow).

4. Simulation and results

A sample four bus system is taken to illustrate the proposed approach. It consists of two generators, two loads and five transmission lines [32,33]. The line data, transaction data and generator quoted cost data is given in Tables 2 and 3 respectively. Additional available capacity of the generator 1 and the generator 2 is assumed to be 4 MW and 5 MW respectively. Consider a bilateral contract of ‘x’ MW from bus ‘i’ to bus ‘j’. If the real power loss due to this contract and its associated cost is ’y’ MW and ‘z’ $/h respectively, then the cost allocated to bus ‘i’ and bus ‘j’ are 0.5*z $/h. Similarly, the cost allocation is carried out for all bilateral contracts and the final cost allocation to the buses is equal to the sum of cost of all bilateral contracts. The real power loss allocation is discussed for three different cases in the following section. PQC method refers to loss/cost allocation without reactive loads and modified PQC refers to loss/cost allocation with reactive loads. Case 1: Comparison of PQC and modified PQC with other methods The active power loss without reactive power loads is 5.983 MW and the associated cost is 2633 $/h. The total system active power loss including the reactive power loads is 8.2614 MW and its associated cost is 3790.4 $/h. This loss is supplied from the ranked generator as shown in Table 4. The increase in real power loss of 2.2784 MW (8.2614–5.983) and the associated cost of 1157.4 $/h (3790.4–2633) has to be allocated to the relevant loads based on LSF. The cost allocation to the buses without and with considering the reactive power loads is given in Tables 5 and 6 respectively. As negative loss allocation is unfair, the results of positive flow direction are alone considered in PQC method [1,30]. The prime focus of this paper is to allocate reactive loss component cost to the respective load buses. In PGPL, Zbus and modified Zbus loss allocation methods, reactive power loads are not considered. The proposed approach of adding reactive loss component cost is applied to the existing real loss allocation methods like PGPL, Zbus and modified Zbus methods i.e., the reactive loss component

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Fig. 1. Flow chart to compute total cost for delivering active power loss. Table 3 Transaction and Quoted cost data for sample four bus system. Generators

1 2

Loads

3,4 3,4

Active power transaction(MW)

182, 100 38, 180

Generator cost coefficients a

b

c

0.0048 0.0040

6.4 8.0

400 500

Quoted cost($/MWh)

406.4 508

Table 4 Total cost for supply of real power loss with and without QD . Bus No.

1 2 Total

PQC ($/MWh)

418.83 517.55

Generator ranking

1 2

Without QD

With QD

Supplied quantity for loss (MW)

Cost of supply of loss ($/h)

Supplied quantity for loss (MW)

Cost of supply of loss ($/h)

4 1.983

1625.6 1007.4

4 4.2614

1625.6 2164.8

5.983

2633

8.2614

3790.4

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379

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Fig. 2. Flow chart for cost allocation to individual transaction. Table 5 Comparison of PQC method with other methods. Bus No.

Cost allocation ($/h) PQC

PGPL

Zbus

Modified Zbus

1 2 3 4

668.10 648.40 683.55 632.95

742.51 573.99 579.26 737.24

439.65 945.99 906.03 341.34

732.37 601.86 569.37 729.40

Total

2633

2633

2633

2633

cost is allocated to the respective load buses along with cost of real loss component. Therefore, it is concluded that the proposed modified PQC method of reactive loss component pricing can be applied to any other existing loss/cost allocation methods. In PQC and PGPL methods, the cost allocation is fair where the generators and the loads are charged equally. To compare these

two approaches, the incremental cost for Zbus method is taken as 440.08 $/MWh (2633 $/h/5.983 MW) and for modified Zbus method, the cost of each line (i.e., ratio of total transmission loss cost and the number of lines) is 526.6 $/h (2633 $/h/5). As reactive loss component cost is allocated to the loads, the generator cost allocation is independent of reactive loss component cost. Therefore, for all the

376

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379 Table 6 Comparison of modified PQC method with other methods. Bus No.

LSF

Reactive loss component cost ($/h)

Cost allocation ($/h) Modified PQC

PGPL

Zbus

Modified Zbus

1 2 3 4

– – 0.5684 0.4316

657.866 499.534

668.10 648.40 1341.42 1132.48

742.51 573.99 1237.126 1236.775

439.65 945.99 1563.896 840.935

732.37 601.86 1227.236 1228.934

Total

1.0

1157.4

3790.4

3790.4

3790.4

3790.4

Table 7 Comparison of Modified PQC method with other methods including QD . Bus No.

Cost allocation ($/h) Modified PQC

PGPL

Zbus

Modified Zbus

1 2 3 4

668.10 648.40 1341.42 1132.48

1068.89 826.31 833.89 1061.31

899.13 1043.17 1343.10 505.00

966.22 957.74 815.35 1051.09

Total

3790.4

3790.4

3790.40

3790.40

Table 8 Transaction and quoted cost data. Generators

1 2 13 22 23 27

Loads

7 3,4,8 12,14,15,16,17 10,21 18,19,20 24,26,29,30

Active power transaction (MW)

22.8 2.4, 7.6, 30.0 11.2,6.2,8.2,3.5,9.0 5.8,17.5 3.2, 9.5, 2.2 8.7,3.5,2.4,10.6

four loss allocation methods, the reactive loss component cost is allocated to bus 3 and bus 4. From Tables 5 and 6, it is clear that the cost allocation to bus 1 and bus 2 in all the four methods are same. For bus 1 it is 668.1 MW, 742.51 MW, 439.65 MW and 732.37 MW for PQC, PGPL, Z bus and modified Zbus methods respectively. Similarly, the cost allocation for bus 2 remains same for all the four methods. In Table 6, modified PQC method i.e., LSF based cost allocation for reactive loads is applied to the existing methods and a comparison is made. By this approach the cost of reactive loads is allocated to the relevant loads based on LSF. Therefore, the generator cost allocation is unchanged i.e., bus 1 and bus 2. Case 2: Apportion of Mathematical Loss The real loss component with four bilateral transaction is 5.983 MW and its cost is 2633 $/h. The mathematical loss for four bus system is 0.0155 MW. The incremental cost for mathematical loss is assumed as 200 $/h. Therefore, the total cost for mathematical loss is 3.1 $/h. This contributes to 0.114% and 0.084% of cost of real loss component and cost of total loss respectively. Mathematical loss is dependent on size of the system. For four bus system the impact of mathematical loss is small but for larger systems this loss cannot be negotiated. The impact of mathematical loss on loss allocation is analyzed with IEEE 30 bus system in Section 4. Case 3: Comparison of modified PQC approach with other methods Modified PQC methods incorporates load sensitivity factor to price the reactive loss component to the respective loads. In PGPL, Zbus and modified Zbus method discussed in [25,27] and [28], the reactive loads are not considered. Therefore, the cost allocation to the buses including reactive loads is compared with modified PQC method and it is given in Table 7. It should be clear that in case 1 (Table 6) the proposed modified PQC method is applied to other existing methods (i.e., the cost of reactive loads at bus 3 and bus 4 is applied to the other methods) where as in case 3 (Table 7)

Generator cost coefficients a

b

c

0.0070 0.0095 0.0090 0.0090 0.0080 0.0075

7.0 10.0 8.5 11.0 10.5 12.0

240 200 220 200 220 190

Quoted cost ($/MWh)

247 210 228.5 211 230.5 202

the reactive loads are included in the existing methods (PGPL, Zbus and modified Zbus ). From Tables 6 and 7, it is clear that the cost of reactive loads is allocated to the respective loads in modified PQC method whereas in other methods it is shared among all the network buses which are not fair. The unique feature of modified PQC method is that the reactive loss component cost allocation approach can be applied to any other existing loss allocation method. In such case the total cost allocation to the buses will be sum of cost of real and reactive loss component. The reactive loss component and its cost is 2.2784 MW (8.2614 MW - 5.983 MW) and 1157.4 $/h(3790.4 $/h - 2633 $/h) respectively. This cost is allocated to the relevant loads in modified PQC method and shared among the generators and the loads in other methods as shown in Table 7. 4.2. IEEE 30 bus system IEEE 30 bus system with 6 generators, 18 loads and 41 transmission lines is considered for demonstration. The transaction data and quoted cost for the generators (assumed to be same as the fuel cost) is computed from the fuel cost equation of the generator as shown in Table 8. Additional available capacity of all the generators is assumed to be 1 MW. Case 1: Comparison of PQC and modified PQC method with other methods The total active power loss is 2.5183 MW and its associated cost is 521.86 $/h. The real power loss without reactive power loads is 1.8833 MW and its associated cost is 388 $/h. This real loss component cost is allocated to the transactions and later shared among the generators and the loads equally as given in Table 9. The loss due to reactive power load is 0.635 MW (2.5183 MW - 1.8833 MW) and its cost is 133.86 $/h (521.86 $/h - 388 $/h). This reactive loss component cost is allocated to the respective loads as shown in Table 10.

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379

Fig. 3. Comparison of PQC method with other methods.

Fig. 4. Comparison of modified PQC with other methods (after apportioning mathematical loss).

Fig. 5. Comparison of Modified PQC method with other methods (including reactive loads in other methods).

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Table 9 Comparison of PQC method with other methods. Bus No.

Cost allocation ($/h) PQC

PGPL

Zbus

Modified Zbus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

9.41 15.41 1.10 4.43 0 0 9.41 9.88 0 0.54 0 16.78 105.26 5.90 50.07 4.71 27.77 0.67 0.07 6.67 14.98 15.52 7.43 0.0174 0 1.30 40.95 0 9.048 30.587

26.92 47.23 2.83 8.97 0 0 26.92 35.42 0 6.85 0 13.22 44.99 7.32 9.68 4.13 10.63 3.78 11.22 2.60 20.66 27.51 17.59 10.27 0 4.13 29.76 0 2.83 12.52

27.33 37.75 0.83 5.48 0 0 42.26 59.36 0 13.30 0 2.70 −6.98 10.49 12.11 6.38 23.43 10.62 37.93 8.05 38.53 −40.42 12.97 15.56 0 11.01 −0.77 0 7.28 52.79

26.39 44.44 2.77 8.79 0 0 26.38 34.76 0 6.75 0 13.14 46.93 7.30 9.64 4.10 10.49 3.76 11.13 2.57 20.38 27.21 18.85 10.18 0 4.11 32.60 0 2.82 12.52

Total

388

388

388

388

The cost allocation to the buses without reactive power demand of the loads is given in Table 9. As discussed earlier, the incremental

cost for Zbus method is 206 $/MWh (388 $/h/1.8833 MW) and the cost of each line for modified Zbus method is 9.46 $/h (388 $/h/41). The dummy bus cost allocations are zero as shown in Tables 9 and 10 (bus 5, bus 6, bus 9, bus 11, bus 25 and bus 28). In this test system, generators are located at bus 1, bus 2, bus 13, bus 22, bus 23 and bus 27. From Table 5, the cost allocated to the generators are 1385.64 $/h (439.65 + 945.99) and to the loads are 1247.37 $/h (906.03 + 341.34). The total cost for provision of real power loss is 2633 $/h. Therefore, cost allocated to the generators is 52.62% and to the loads is 47.38%. In Zbus method [27], for IEEE 14 bus system the total real power loss and its associated cost is 13.5 MW and 678 $/h respectively. Cost allocated to the generators and the loads are 390 $/h and 288 $/h which are 57.52% and 42.48% respectively. As discussed in the introduction, the cost allocation in Zbus method varies with system size. For modified IEEE 14 bus system, the cost allocated to the generators and the loads are 36% and 64% respectively. Similarly, from Table 9, for Zbus method allocates 7.7% and 92.3% of cost to the generators and the loads and for modified Zbus method, it is 50.62% and 49.38% respectively. Case 2: Comparison of PQC method with other methods (considering mathematical loss) The real loss component is 1.8833 MW and its cost is 388 $/h. The mathematical loss is 0.1282 MW and its cost is 25.64 $/h which contributes to 6.8% of real loss component. This cost has to be allocated to the transmission owners and the remaining cost should be allocated to the respective bilateral contracts. Therefore, the loss and cost allocated to the bilateral contracts is 1.7551 MW and 362.36 $/h respectively. The cost allocated to the individual buses is shown in Fig. 3. From Fig. 3 it is clear that the cost allocation in PGPL and modified Zbus method goes hand in hand with overall difference of 1% i.e., PGPL method allocates cost to the generators and the loads equally but in modified Zbus method 51% of cost is

Table 10 Comparison of modified PQC method with other existing methods. Bus No.

LSF

Reactive loss component cost ($/h)

Cost allocation ($/h) Modified PQC

PGPL

Zbus

Modified Zbus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

– – 0.0120 0.0214 0 0 0.1327 0.4171 0 0.0222 0 0.0242 – 0.0117 0.0203 0.0208 0.0727 0.0152 0.0564 0.0128 0.0461 – – 0.0494 0 0.0372 – 0 0.0082 0.0197

1.6063 2.8646 0 0 17.7632 55.8330 0 2.9717 0 3.2394 1.5662 2.7174 2.7843 9.7316 2.0347 7.5497 1.7134 6.1709 6.6127 0 4.9796 0 1.0977 2.6370

9.41 15.41 2.70 7.29 0 0 27.17 65.71 0 3.51 0 20.02 105.26 7.46 52.78 7.50 37.50 2.70 7.62 8.38 21.15 15.52 7.43 6.63 0 6.28 40.95 0 10.14 33.22

26.92 47.23 4.43 11.83 0 0 44.68 91.25 0 9.82 0 16.46 45.00 8.88 12.40 6.91 20.36 5.81 18.77 4.31 26.83 27.51 17.59 16.88 0 9.10 29.76 0 3.92 15.15

27.33 37.75 2.43 8.34 0 0 60.02 115.19 0 16.27 0 5.93 −6.98 12.05 14.82 9.16 33.16 12.65 45.48 9.76 44.70 −40.42 12.97 22.17 0 16.00 −0.77 0 8.37 55.42

26.39 44.44 4.37 11.65 0 0 44.14 90.60 0 9.72 0 16.38 46.93 8.86 12.35 6.88 20.22 5.79 18.68 4.28 26.55 27.21 18.85 16.79 0 9.09 32.60 0 3.91 15.15

Total

1.0

133.86

521.86

521.86

521.86

521.86

K. Shafeeque Ahmed and S. Prabhakar Karthikeyan / Sustainable Energy, Grids and Networks 16 (2018) 370–379

allocated to the source and 49% to the loads. PQC and Zbus method shows wide variation in loss allocation. Case 3: Comparison of modified PQC with other methods (considering Mathematical Loss) The total real power loss is 2.5183 MW and its cost is 521.86 $/h. In this case the mathematical loss contributes to 5.09% of total active power loss. The loss and cost allocated to the individual buses is 2.39 MW and 496.22 $/h respectively. The real power loss cost allocation to the individual nodes is shown in Fig. 4. Case 4: Comparison of Modified PQC method with other approaches (without considering mathematical loss) In PGPL, Z bus and modified Zbus methods discussed in [25,28] and [29] respectively, reactive loads are not considered. In this case modified PQC method is compared with other methods like PGPL, Z bus and modified Zbus methods including reactive loads (without considering mathematical loss). The total loss and its cost is 2.5183 MW and 521.86 $/h respectively. The allocation of cost to the buses is shown in Fig. 5. The cost allocated to bus 3 and bus 23 is 0.97 $/h and 0.23 $/h respectively. 5. Conclusions In this paper, the proposed modified PQC method allocates real power loss cost to two components i.e. real and reactive loss component. Real loss component is allocated to transactions and reactive loss components to the loads. This method uses load flow results to compute load sensitivity factor. Application of load sensitivity factor to the reactive loss component pricing is fair and reasonable. This method can be claimed as a fairest among the existing approaches as it equally shares the transmission loss among the generators and the loads. This paper investigates on the reactive power demand and its impact on active power loss allocation. Additionally, the impact of mathematical loss on active power loss allocation is also discussed. This approach can be combined with any other existing active power loss allocation methods for pricing the reactive loss component. This approach is applicable for open access electricity market, where a set of load sensitivity factors can be used for every time block. References [1] Francisco D. Galiana, Mark Phelan, Allocation of transmission losses to bilateral contracts in a competitive environment, IEEE Trans. Power Syst. 15 (1) (2000) 143–150. [2] Garng M. Huang, H. Zhang, Transmission loss allocations and pricing via bilateral energy transactions, in: IEEE Power Engineering Society Summer Meeting, 1999, pp. 720–725. [3] Kyung-Il Min, Sang-Hyeon Ha, Su-Won Lee, Young-Hyun Moon, Transmission loss allocation algorithm using path-integral based on transaction strategy, IEEE Trans. Power Syst. 25 (1) (2010) 195–205. [4] Dibya Bharti, Mala De, A new graph theory based loss allocation framework for bilateral power market using diakoptics, Int. J. Electr. Power Energy Syst. 77 (2016) 395–403. [5] E.A. Belati, G.R.M. da Costa, Transmission loss allocation based on optimal power flow and sensitivity analysis, Int. J. Electr. Power Energy Syst. 30 (2008) 291–295. [6] M. Ilic, Francisco D. Galiana, L. Fink, Power Systems Restructuring: Engineering and Economics, Kluwer, Norwell, MA, 1998. [7] F.D. Galiana, A.J. Conejo, I. Kockar, Incremental transmission loss allocation under pool dispatch, IEEE Trans. Power Syst. 17 (1) (2002) 26–33. [8] J.W. Bialek, Tracing the flow of electricity, IEE Proc. Gener. Transm. Distrib. 143 (4) (1996) 313–320. [9] D. Kirschen, G. Strbac, Tracing active and reactive power between generators and loads using real and imaginary currents, IEEE Trans. Power Syst. 14 (4) (1999) 1312–1319.

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