Transmission Loss Allocation Based on a New ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006

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Transmission Loss Allocation Based on a New Quadratic Loss Expression Qifeng Ding, Member, IEEE, and Ali Abur, Fellow, IEEE

Abstract—This paper presents a derivation of a new quadratic expression for the transmission loss in a power system. The main advantage of this expression is the improvement in accuracy over the other existing approximations. Furthermore, this improved quadratic form allows the transmission loss to be expressed in terms of nodal power injections. This paper builds on the derived loss expression and proposes a bus-loss matrix. This matrix is useful in quantifying the interactions among different bus power injections and establishing a loss allocation scheme for individual buses. Numerical examples are used to illustrate the accuracy of the derived quadratic form and its use in loss allocation. Index Terms—Loss allocation, power flow, Taylor series, transmission loss.

I. INTRODUCTION

E

LECTRIC power industry is experiencing important changes brought about by deregulation. Transmission losses, which used to be treated as an extra “load” by the vertically integrated utility companies, have become a complicated quantity to allocate among system buses after deregulation. This is due to the fact that losses are expressed as a nonlinear function of line flows, making it impossible to exactly calculate the amount of loss caused by each generator, load, or transaction in the system. Based on different assumptions and approximations, four classes of methods [1], [2] have been proposed for loss allocation so far. Some propose to allocate system losses to nodal injections in a pool market, while others allocate them to individual transactions in a bilateral contracts market. Examples can be found for pro rata methods [3], incremental transmission loss (ITL) methods [4]–[7], proportional sharing procedures [8]–[11], and loss formula methods [12]–[15]. A comparison among pro rata methods, ITL methods, and proportional sharing procedures is given in [1]. Pro rata methods are not considered sound since the network topology is not considered. Proportional sharing procedures are based on the neither provable nor disprovable principle that the power injections are proportionally shared among the outflows of each bus [2]. Test results obtained using the ITL methods in [1] also show that ITL methods are volatile and depend on the selection of the slack bus. Among all these methods, loss formula methods are the most appropriate in terms of expressing losses with individual nodal injections or transactions. However, because of the

complexity of the loss formula, it usually leads to significant errors after making strong assumptions or approximations. A number of loss approximation methods, including those based on B-coefficients [16], explicit polynomial expressions based on bus generations [17], quadratic approximations in terms of phase angles [18], and, more recently, first-order Taylor expansion of loss expressions [19], are proposed. A “physical-power-flow-based” approach [12], one of the loss formula methods, uses the quadratic loss approximation to distribute losses among the transactions in a multiple-transaction framework. The method provides loss allocations that are appropriate and behave in a physically reasonable manner, despite having up to 16% error in some cases due to the approximations, such as dc power flow assumption, used in its derivation. It is well known that the transmission losses are caused by generation and load. If an appropriate approximation can be developed to express losses in terms of nodal injections, then it can be utilized to derive a reasonable loss allocation method. This paper exploits the natural truncation of Taylor series expansion of power flow equations when expressed in rectangular coordinates, to expand the system loss function into a simpler and more accurate Taylor-series approximation in terms of nodal injections. Extensive tests show that the proposed quadratic loss expression, unlike fore-mentioned methods, is quite accurate, representing the loss value with less than 3% error. This paper uses the derived loss expression to propose a reasonable loss allocation scheme. A bus-loss matrix will be derived to show the interactions between different nodal injections. Different allocation methods among parties with a quadratic relationship are discussed in [14]. In this paper, the allocation between involved parties linked via a quadratic coupling term will be assumed to be equal. This is one of several possible allocation methods listed in [14] since the interactions between different injections are clearly calculated in the bus-loss matrix. All losses can then be allocated to each individual bus based on this bus-loss matrix. This paper is organized in such a way that the proposed formulation is presented first, followed by its implementation algorithm. Numerical examples are included at the end to illustrate the application of the proposed method to typical power systems. II. PROBLEM FORMULATION

Manuscript received January 10, 2006; revised February 13, 2006. Paper no. TPWRS-00858-2005. Q. Ding is with CenterPoint Energy, Houston, TX 77251-1700 USA (e-mail: [email protected]). A. Abur is with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2006.876652

A. Power Flow Formulation [20], [21] Expressing the bus voltages in rectangular coordinates as , the power flow problem can be formulated as the solution of the following power flow equations:

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P-Q buses: (1a)

Defining any given state as state 1 and an initial state 0, at which no transaction exists, the following power flow equations can be formed: and and

(1b) P-V buses: (1c)

Since the nodal injections are known values, the loss expression needs to be constructed in terms of instead of bus voltages . By using the Taylor series expansion, the loss expression at state 0 can be described as follows:

(1d) Slack bus: (1e) (1f) and

and , as well as and , are known values specified for PQ buses, PV is the buses, and the slack bus, respectively. element of the admittance matrix. to are Assuming that buses 1 to are PQ buses, PV buses, and is the slack bus, the following vectors can be defined: where

Each equation of (1a)–(1f) can then be expressed as

(5) where

,

and , and

derivation

of are

given in the Appendix. Theoretically, it is possible to calculate higher order derivatives. However, extensive experiments have proven that the “second-order” approximation is already sufficiently accurate to represent the loss value. Numerical results that show less than 3% errors in loss calculations for different systems will be presented in the next section. Neglecting the third and higher order terms, losses will be expressed by a quadratic form of bus injections

(2) is a sparse, constant, and symmetric matrix. where As a result, we will have .. .

.. .

.. .

which can also be rewritten as

(3) Hence, the solution to the above equation (3) is the result to of the original power flow problem (1a)–(1f). If we use represent the transmission losses, the following loss expression can be constructed: (4) where is the line index of the line is given as pression for

(6)

, and the detailed exin the Appendix.

B. Loss Expression For a given state of an operating power network, it is noted that the real and reactive power loads and generation are responsible for the power flows and losses on the transmission lines.

(7) where is a

matrix, and

is a matrix. Consequently, the transmission loss expression has been built in terms of nodal injections and . Compared to is very small and is the loss value even without transactions, only incurred by maintaining voltage values of PV buses. Theoretically, should not be allocated to individual nodal injections, and changes in specified voltage values of PV buses will affect its calculation. represents changes in losses, incurred by changes in generation and load , and related through (6) and (7). Clearly from (7), once matrices and are calculated, it is possible to discuss how to allocate losses to each individual injection in the system.

DING AND ABUR: TRANSMISSION LOSS ALLOCATION BASED ON A NEW QUADRATIC LOSS EXPRESSION

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III. PROPOSED ALLOCATION SCHEME Since bus voltages of PV buses stay the same from state 0 are approximately equal to zero, we will to state 1 and approximately as have (8) and

in (7) can be expressed as .. .

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.

.. .

(9) Fig. 1. Flowchart.

.. . The relationship between nodal injections and transmission losses is clearly stated in (9). However, (9) can be used to construct a bus-loss matrix BL in order to express the interactions in a concise form as below:

those that depend on pairs of generators. Hence, it facilitates derivation of creative loss allocation schemes among generators. Different possibilities of allocating the off-diagonal cross terms are discussed extensively in [14]. In this paper, cross terms will be evenly allocated between involved parties. As a result, for each single bus injection, the allocated loss can be expressed as (13)

.. . .. .

.. . .. .

.. . .. .

.. .. .

.

.. . .. .

.. . .. .

(10)

Thus, the expression for total system loss increase (8) will become (14)

where

(11)

(12) in (11) is solely due to the bus injection The diagonal term , including real power injection and reactive power injection . The cross term in (12) represents losses accrued due to the interactions between bus injections and . The so-called loss matrix BL is a compact representation of components of system losses that depend solely on individual generators and

In [2], the concept of “center of losses” (CL) is discussed. The CL is defined as a fictitious bus with zero loss allocation, where the generators deliver their produced energy and load get their consumed energy. In reality, it is very hard to know where the CL is. In loss allocation calculation, we want to select the reference bus to be close to the CL. In implementing this procedure, an initial power flow analysis is executed, and the phase angles of all buses are compared. A bus, which does not have any load or generation and has the phase angle value closest , is chosen as the best reference bus to available in the system. Different methods to choose the best reference bus are studied in [2]. The flowchart shown in Fig. 1 is used for loss allocation. After the initial power flow calculation, the best reference bus is selected, and another power flow analysis is executed by setting transactions to zero in order to obtain the base case bus voltage

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TABLE I APPROXIMATED LOSSES OF DIFFERENT TEST SYSTEMS (MW)

magnitudes and phase angles. Next, the bus-loss matrix and the losses allocated to bus injections are calculated. allocated to individual buses is not exactly the same as since is not considered. Usually, the true system losses is very small, and loss allocation is normalized as

(15) Fig. 2. Comparison of the true and approximate system losses for the IEEE 118-bus system with increased loading.

IV. NUMERICAL RESULTS The overall performance of the quadratic approximation to the exact loss formula is examined first. As in [12], various test systems are used to compare the loss approximations with the true loss values calculated from the power flow solutions. Table I shows the loss approximations, true values, and errors. The results show less than 3% discrepancy between the approximated and true loss values. Furthermore, the approximation remains accurate as the system loading is increased for the IEEE 118-bus system, as is evident from the results shown in Fig. 2. A simple four-bus system is used to illustrate the proposed expression. Bus-4 is chosen as the reference bus. The parameters and system diagram are shown in Fig. 3 and Table II. The initial power flow results and the power flow results without loads are listed in Table III Fig. 3. Four-bus test system. TABLE II LINE CHARACTERISTICS FOR A THREE-BUS SYSTEM

Following the formulas listed in the Appendix, matrices and in (9) are calculated as shown in the equation at the bottom of the page.


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TABLE III POWER FLOW RESULTS OF A FOUR-BUS TEST SYSTEM (P.U., BASE POWER: 100 MW)

Then, the bus-loss matrix BL in per unit can be constructed following (10) and (11) as

Using (13), the approximated loss value will be 2.104 MW, and the following allocation results can be obtained:

In this paper, the well-known IEEE-RTS 24-bus system whose diagram is shown in Fig. 4 is also tested. It has 24 buses, 38 branches, and 14 generators. The generation and load data are slightly modified to better illustrate the proposed method. The total system load is 2800 MW. Choosing bus 13 as the system slack bus, the initial power flow yields the total system losses as 46.98 MW. After the initial power flow calculation, the phase angle of , bus 24 is found to be very close to and it is set as the best reference bus. Hence, another power flow calculation is carried out using bus 24 as the reference bus and without considering any transactions to obtain the initial system state. The algorithm in Fig. 1 is then used to calculate the losses allocated to each bus. The results are shown in Table IV. In Table IV, generators at buses 21–23 output about 50% of the total generation to the distant loads, so they are allocated the higher percentages of losses. Also, it is noted that some buses, for example, bus 15, are allocated negative losses. As also described in [13], these buses are “well-positioned” and are rewarded with negative losses. Comparisons with other loss allocation methods described in [1], like the pro rata method, the ITL method, as well as the Z-bus method [13], are made as shown in Table V. Note that the pro rata allocates the most losses to bus 23 since it has the largest active power injection. The Z-bus method has similar allocation results as the pro rata method because it is based on the magnitudes of current injections. The ITL method and the proposed method in this paper have similar loss allocation patterns, even though the ITL method only uses the first-order derivatives in terms of real power injections. Bus 23 is assigned smaller losses in both the ITL and the proposed methods because it is closer to loads than buses 21 and 22.

Fig. 4. IEEE RTS 24-bus system. TABLE IV RESULTS OF LOSS ALLOCATION

A test similar to the one used in [2] is also used here to test the validity of the proposed method. Consider that two new buses, 25 and 26, are connected to bus 22 through two transmission

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APPENDIX

TABLE V COMPARISON OF DIFFERENT LOSS ALLOCATION METHODS

The loss expression of a single transmission line follows:

where

and

is as

(A1) are line resistance and reactance. Hence

(A2) where four elements can be easily calculated as

TABLE VI SENSITIVITY ANALYSIS

Then, can be derived as shown in (A3) at the top of the next page, where

lines. Three cases (detailed description can be found in [2]) are tested: Case 1: identical lines and identical generations; Case 2: different lines and identical generations; Case 3: identical lines and different generations. The results are shown in Table VI. As in [13], identical losses are found to be allocated to both bus 25 and bus 26 in case 1. In case 2, since the resistance of line 22-26 is larger than that of line 22–26, bus 26 is allocated more losses. Also, due to more generation output coming from bus 26 in case 3, bus 26 is allocated larger losses. V. CONCLUSIONS This paper approximates the transmission losses with a quadratic form of nodal injections by using the Taylor-series expression in rectangular coordinates. Tests show that the approximated loss values of different power systems are very accurate. Based on the proposed quadratic form, a bus-loss matrix is built. The bus-loss matrix facilitates the investigations of the interactions of nodal injections. Furthermore, it is shown that loss allocation can be easily done using the derived bus-loss matrix. Numerical examples are given to illustrate that the proposed method yields loss allocation results that are intuitively reasonable and consistent with expectations.


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only has 16 elements and is a constant Clearly, matrix. Also, since we have the power flow equations as .. .

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.. . (A4)

Consequently, the derivatives can be obtained as (A5) Then (A6) (A7) (A8) (A9) It is evident from (A5)–(A9) that matrices can be easily built.

and

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(A3)

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Qifeng Ding (M’03) received the Ph.D. degree from Texas A&M University, College Station, in 2004. Currently, he is a Senior Engineer in the Control Systems Division of CenterPoint Energy, Houston, TX.

Ali Abur (F’03) received the B.S. degree from Orta Dogu Teknik Universitesi, Ankara, Turkey, in 1979 and the M.S. and Ph.D. degrees from the Ohio State University, Columbus, OH, in 1981 and 1985, respectively. He joined the Department of Electrical Engineering, Texas A&M University, College Station, in late 1985 and worked as a Professor until 2005. Since November 2005, he has served as Professor and Chair of the Electrical and Computer Engineering Department, Northeastern University, Boston, MA.