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Sensitivity of LMP Using an Iterative DCOPF Model Rui Bo, Student Member, IEEE, and Fangxing Li, Senior Member, IEEE FND-based Iterative DCOPF algorithm. The LMP sensitivity formulation can be expressed with a direct, explicit equation, which clearly shows that the LMP sensitivity is related to the loss component and linearly related to the sensitivity of Delivery Factors. Without loss component, the LMP sensitivity is zero if load is varied in a small range. The explicit equation in LMP sensitivity makes this work different from the previous ACOPF-based work , in which LMP sensitivity is implicitly available by solving a matrix equation, because the concept of Delivery Factor is neither applicable nor necessary to the ACOPF model. In addition, this paper shows that the LMP sensitivity may suddenly become infinite when the load grows to a critical level to cause a new binding constraint. This infinite, step change may be interesting to future research works. This paper is organized as follows. Section II briefly reviews the FND-based DCOPF model and the corresponding LMP calculation formulation. Section III presents the explicit equation for LMP sensitivity with respect to different load levels and validates the equation with the test systems, including a 5-bus system and the IEEE 30-bus system. Section IV discusses a special case of LMP sensitivity when there is a LMP step change due to the change of marginal units. Concluding remarks are presented in Section V.
Abstract--This paper firstly presents a brief review of the FND (Fictitious Nodal Demand)-based iterative DCOPF algorithm to calculate Locational Marginal Price (LMP), which is proposed in a previous work. The FND-based DCOPF algorithm is particularly suitable for simulation and planning purposes. Then, this paper employs the FND algorithm to analyze the sensitivity of LMP with respect to loads. Tests are performed on the PJM 5bus system and the IEEE 30-bus test system. A simple, explicit equation of LMP sensitivity is presented and validated. Also, a special case of infinite sensitivity under the step change of LMP is discussed. It is shown that if the operating point is close to the critical load level of LMP step change, the sensitivity is less reliable and may not be applied to a large variation of load. Finally, future works and concluding remarks are presented. Index Terms-- DCOPF, energy markets, Fictitious Nodal Demand, locational marginal pricing (LMP), optimal power flow (OPF), power markets, power system planning, sensitivity analysis.
C model of power system dispatch bears the feature of robustness and fast calculation speed, and therefore gains practical application in a number of industrial tools for market simulation and planning [1, 8]. The drawback of this model lies in the fact that power loss is typically ignored. However, loss plays an important part in practice, especially in deregulated environment. In fact, one of the challenges in LMP methodology is to address marginal loss price [3-6]. Therefore, a new algorithm is proposed in , which incorporates power loss in DCOPF model and obtains the solution by solving a sequence of Linear Programming (LP) problems. Besides the introduction of power loss into the power balance equation, Fictitious Nodal Demand (FND) is proposed to distribute the power loss into each individual line. In the discussion below, this algorithm is referred to as the FND-based Iterative DCOPF algorithm, or simply the FND algorithm. The accuracy and effectiveness of the FND algorithm for LMP simulation are demonstrated in . A subsequent question is raised naturally. If the proposed FND algorithm is a trustable approximation of ACOPF algorithm, what is the sensitivity of the LMP with respect to load variation? Besides the calculation of sensitivity at a specific load level, a more interesting question would be: what does the sensitivity curve look like? Or, what is the overall picture of LMP sensitivity with respect to load change? This paper presents a sensitivity analysis of LMP with the
II. FND-BASED DCOPF MODEL A. FND-Based DCOPF Model A new DCOPF model is proposed in  to address the power loss issue. In this model, Marginal Delivery Factor (DF) and Fictitious Nodal Demand (FND) are introduced for the purpose of power loss balance and distribution, respectively. The model is shown as follows. Min
est × Gi − ∑ DFi est × Di + Ploss =0
th × Gi − Di − Eiest ≤ Limit k for the k line
Gimin ≤ Gi ≤ Gimax where i = 1, 2, …, N; N = number of buses; k =1, 2,…, M; M= number of lines; ci = generation cost at Bus i ($/MWh); Gi = generation dispatch at Bus i (MWh); 1
Rui Bo and Fangxing Li are with Department of Electrical and Computer Engineering, The University of Tennessee, Knoxville, TN 37996, USA. Contact: [email protected]
, +1-865-974-8401 (F. Li).
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III. SENSITIVITY ANALYSIS OF LMP WITH RESPECT TO LOAD
DFi = delivery factor at Bus i; Di = demand at Bus i (MWh); Ploss = total power loss in the system (MWh); GSFk-i = generation shift factor to line k from bus i; Ei = Fictitious Nodal Demand at Bus i (MWh); Limitk = transmission limit of line k.
This section will examine the sensitivity of the LMP with respect to load changes based on the FND-based iterative DCOPF. A. LMP Sensitivity without Loss Based on DCOPF formulation, the sensitivity is strongly related to the loss model. In other words, if no loss is considered, the sensitivity of LMP should be zero if there is a very small change of demand (actually, as long as there is no change of marginal units). This is due to the linear characteristics in the DCOPF model. It can be shown as follows:
Marginal Delivery Factor (DF) at the ith bus, denoted by DFi, and Fictitious Nodal Demand (FND) at the ith bus, denoted by Ei , are defined as follows, respectively,
DFi = 1 − LFi = 1 −
Ei = ∑ 12 × Fk2 × Rk
LMPi no _ loss = LMP energy + LMPi cong = λ + ∑ µ × GSFk −i
where LFi = loss factor at Bus i; Pi = net power injection at Bus i (MWh); Mi = the number of lines connected to Bus i; Fk = line flow through line k; Rk = line resistance of line k.
no _ loss
∆LMPi ∆D j
From the above DCOPF model, LMP can be easily decomposed to three components: marginal energy price, marginal congestion price and marginal loss price. (7)
LMP energy = λ
LMPi cong = ∑ GSFk −i × µ k
LMPi loss = λ × ( DFi − 1)
∆µ ∆λ + ∑ k × GSFk −i = 0 ∆D j k =1 ∆D j
In the above equations, λ is independent of demand because it represents the change of dispatch cost with respect to the change of demand. If there is a small increase of demand, the same marginal unit(s) shall provide a matching amount of power to cover the demand increase. The reason is that the DCOPF model is based on a Linear Programming model. Hence, the change of generation of each marginal unit with respect to a load change at a specific bus should also be linear. This can be written as
Delivery Factor at the ith bus represents the amount of effectively delivered power to the customers with 1MW power injection at Bus i. The Fictitious Nodal Demand at Bus i is half of the total losses on all the lines connected with Bus i.
LMPi = LMP energy + LMPi cong + LMPi loss
∆Gl = α lj for all marginal unit l. ∆D j
With the assumption of a small load change without new binding constraints, the energy price can be written as Ml
(∆Gl ⋅ cl ) M l ∆Cost ∑ l =1 = λ= = ∑ (α lj ⋅ cl ) ∆D j ∆D j l =1
where Ml = number of marginal units.
where LMPi= LMP at Bus i; λ = Lagrangian multiplier of Eq. (2) = energy price of the system = price at the reference bus; µk = Lagrangian multiplier of Eq. (3) = sensitivity of the kth transmission constraint.
Hence, λ is independent of demand and
∆λ is equal to ∆D j
zero. Similarly, due to the linear formulation, µk represents the change of cost when there is a 1MW relax of the kth transmission constraint. As long as there is no new marginal unit, the reduced cost will remain constant or independent of
B. Comments on the FND-based DCOPF Model
It is shown in  that a deduction of system loss in energy balance equality constraint is necessary since the net injection multiplied by marginal delivery factor creates doubled system loss. Also, iteration is needed for the proposed DC model with losses considered, because the loss factor is dependent on generation dispatch, while generation dispatch is affected by loss factor as well. Then, a mismatch issue at the reference bus is identified and addressed with Fictitious Nodal Demand (FND) to model losses at each individual branch. Hence, the mismatch problem is eliminated as verified in a test in .
∆µ k is equal to zero. ∆D j
The LMP sensitivity is firstly tested on the PJM 5-bus system . The base case of the system is shown in Figure 1. Figure 2 shows the nodal LMP at each bus without considering losses, with respect to different load levels at Bus B from 300 MWh to 330 MWh. The LMPs remain constant in this small range.
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normalized LMP at each bus with respect to Bus B Load. The normalized values are used such that it is easier to observe the linear growth of DF or LMP at all buses versus Bus B Load. So, the LMP sensitivities with respect to load are the slopes of the straight lines in Fig. 5. Also, Fig. 6 plots the actual LMP sensitivity with respect to load at Bus B. Normalized Delivery Factor w.r.t. Load at Bus B (MWh) 1.0012 1.0010 1.0008 1.0006
Fig. 1. The Base Case of the PJM Five-Bus System.
LM P ($/M Wh) w.r.t. Load at Bus B (MWh) 40
310 Bus A
320 Bus C
Fig. 3. Delivery Factors normalized to base case at each bus with respect to Load at Bus B. The DFs at Base Case for the 5 buses are 0.98992, 1.01130, 1.01304, 1.00000, and 0.98561, respectively.
25 20 15 10
µ ($/MWh) w.r.t. Load at Bus B (MWh) 300
310 Bus A
320 Bus C
Bus E 50.9865
Fig. 2. LMP from Lossless DCOPF at each bus with respect to Load at Bus B.
B. LMP Sensitivity considering Loss As shown in the above analysis and test, the possible nonzero sensitivity of LMP in the paradigm of DCOPF must be attributed to the loss model. When load grows, loss grows quadratically with demand. Here, a misleading intuition is that LMP sensitivity to load shall only be related to delivery factor (or loss factor) since LMP sensitivity is zero when there is no loss, as shown in (12) and Fig. 2. However, a careful analysis shows that the change of load will lead to a change of not only DF but λ and µ. This is because the change of Delivery Factor in the DCOPF model shall lead to a new λ and µ, when load is varied. In summary, the sensitivity can be written as M ∆ DFi ⋅ λ + ∑ µ k ⋅ GSFk −i ∆LMPi k =1 = ∆D j ∆D j Hence, we have
∆LMPi ∆DFi ∆µ ∆λ = ⋅λ + ⋅ DFi + ∑ k ⋅ GSFk −i ∆D j ∆D j ∆D j k =1 ∆D j M
Fig. 4. µ of the Constraint of Line ED with respect to Bus B Load.
Normalized LMP w.r.t. Load at Bus B (MWh) 1.0020 1.0015 1.0010 1.0005
∆λ ∆µ k ≠ 0 and ≠ 0 in general. This makes the case ∆D j ∆D j
310 Bus A
320 Bus C
330 Bus D
Fig. 5. LMP normalized to base case with marginal loss at each bus with respect Load at Bus B. The LMPs of base case for the 5 buses are 15.86, 24.30, 27.32, 35.0, and 10.0 $/MWh, respectively.
with loss very different from lossless case. Figure 3 shows the normalized DF at each bus for the PJM 5-bus system with respect to Bus B Load in the range between 300 MWh and 330 MWh; Figure 4 shows the actual µ of Line ED with respect to Bus B Load; and Figure 5 shows the 3
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general. In fact, when the load level is around 360-390MW, the third part will play a major role in LMP sensitivity than the first two parts. Also, the next subsection provides test results from the IEEE 30-bus system, in which all three parts in LMP sensitivity are comparable to each other numerically. As shown in Figure 5, the LMP at marginal unit buses (e.g. Bus E) is constant, which is equal to the bidding price of the local generator Brighton, because this generator is always a marginal unit when Bus B Load varies between 300MW and 330MW. So, the local load increase at Bus E will be solely picked up by the local marginal generator Brighton. Thus the sensitivity of LMP at Bus E is zero, as shown in Fig. 6. This is also the case for Bus D because the local generator Sundance is also a marginal unit. For non-marginal-unit buses (A, B or C in this study), the LMPs increase linearly as the load increases. Since the loss is a quadratic function of the load, the generation is a quadratic function of the load as well. If there is no change of marginal units (i.e., due to very small change of load), the dispatch cost is quadratically related to the load. The LMP, defined as incremental cost over incremental load, should be a linear function of load, as shown in Fig. 5. Therefore, the LMP sensitivity at a bus without any marginal unit should be a nonzero constant, as shown in Fig. 6. C. LMP Sensitivity results from the IEEE 30-bus system The LMP sensitivity with loss considered is also tested on the IEEE 30-bus system . Network topology is shown in Figure 7. The system data is slighted modified for illustration purpose: 1) The bidding prices of the 6 generators are assumed as 10, 15, 30, 35, 40 and 45, respectively, all in $/MWh and 2) the transmission limit of line 6-8 is increased by 10%. Figure 8 shows the LMP sensitivity with respect to Load at Bus 8 between 27 and 35MWh in the system. Only the LMP sensitivities at a few buses are shown in the figure for better illustration. Again, LMP sensitivities have constant values at all these buses. Further tests show there will be a step change of LMP sensitivity because of a new binding constraint when load reaches around 36 MW. The diagram beyond 36MW is not shown simply because it is difficult to scale everything into one figure.
LMP Sensitivity ($/MWh ) w.r.t. Load at Bus B (MWh) 0.0015
310 Bus A
330 Bus E
Fig. 6. LMP Sensitivity ($/MWh ) with respect to Load at Bus B (MWh)
In addition to the results shown in Figs. 3-6, the energy component of LMP, or λ, is $35/MWh, constantly. However, this is a special case because the marginal unit happens to be the reference bus, so λ is constant. As previously stated, λ is usually not a constant in DCOPF model with loss, i.e.,
∆λ ≠ 0 . Also, the GSF of line DE to Bus B is -0.2176. ∆D j
Eq. (15) can be validated with Figs. 3 to 6. Taking LMP at Bus B as an example (i.e., i=j=Bus B), we have Normalized ∆DFi = 0.001028 (See Fig. 3) ∆D j 30 Actual ∆DFi = 1.01130 × 0.001082 = 3.647 × 10 -5 30 ∆D j
Actual ∆DFi ⋅ λ = 3.647 × 10 -5 × 35 = 1.277 × 10 -3 ($/MWh 2 ) ∆D j
Also, we have ∆λ ⋅ DFi = 0 ($/MWh 2 ) ∆D j ∆µ DE (50.9857 - 50.9863) ∆D j
⋅ GSFDE −i =
× (-0.2176) 30 =0.004×10-3 ($/MWh2) (See Fig. 4)
Therefore, we have M ∆DFi ∆λ ∆µ k ⋅λ + ⋅ DFi + ∑ ⋅ GSFk −i ∆D j
∆D j k =1 ∆ D j = (1.277+0+0.004) × 10-3 =1.281 × 10-3 ($/MWh2)
From Fig. 5, we have Normalized ∆LMPi = 0.001581 ∆D j 30 Actual ∆LMPi = 24.30 × ( 0.001581 ) = 1.281 × 10 -3 ($/MWh 2 ) 30 ∆D j ∆ Hence, the LMP sensitivity LMPi is very close to the value ∆D j M computed with ∆DFi ⋅ λ + ∆λ ⋅ DF + ∆µ k ⋅ GSF . This ∑ i k −i
validates Eq. (15) and also matches the data in Fig. 6 numerically. Similar validation can be made for other buses. Although the second part and the third part in Eq. (15) for this case is either zero or very small compared with the first part in Eq. (15), this does not mean they can be ignored in
Fig. 7. The Network Topology of the IEEE 30-Bus System.
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D. Notes on Delivery Factor and LMP sensitivity Note on Delivery Factor - Figure 3 and its caption show that the delivery factor may be greater than 1, which implies a negative penalty factor. Taking Bus C as an example, if there is a hypothetical injection increase at Bus C and it is absorbed by the reference bus D, this will reduce most of the line flows such as Line EA, AB, BC, and DC, and therefore reduce the system loss. So, the marginal loss factor is negative and delivery factor is greater than 1. Similar observation can be obtained at some other buses. Note on LMP Sensitivity - Reference  presents a generalized, ACOPF-based model for LMP sensitivity with respect to load and other variables. A matrix formulation needs to be solved to calculate LMP sensitivity eventually, therefore, there is no direct, explicit formulation available from  about LMP sensitivity to load. This paper does not intend to override the work in , instead, this research work does present an explicit formulation, Eq. (15), about LMP sensitivity to load, based on DCOPF with Delivery Factor, which is neither applicable nor necessary in ACOPF model. Hence, with the concept of Delivery Factor, the LMP sensitivity to load in DC model is more straightforward and simple such that it is more helpful to easily obtain a big picture about LMP sensitivity. This is very reasonable considering the simplifications from AC model to DC model. The observed results match the analytical equation (15) and clearly show that the LMP sensitivity is related to the loss component, linear to the sensitivity of delivery factors, and a constant numerically. Without loss component, the LMP sensitivity is zero if load is varied in a small range.
LMP Sensitivity ($/MWh ) w.r.t. Bus 8 Load (MWh) 0.025
0.020 bus 1 bus 5 0.015
bus 10 bus 15 bus 20
bus 25 bus 30 0.005
Fig. 8. LMP Sensitivity at a few buses with respect to Load at Bus 8 ranging from 27 MWh to 35 MWh (base case load=30MWh) in the IEEE 30-bus system.
When the new (and the only) binding transmission constraint appears, a non-zero µ occurs and its sensitivity is considerable. Equation (15) can be briefly verified using results at Bus 30 with respect to Bus 8 Load varied from 37.500 to 37.575 MW. This can be shown as follows (here i=30 and j=8): 2 ∆DFi ⋅ λ = (-0.0007781/0.075) × 26.96535 = -0.27976($/MWh ) ∆D j 2 ∆λ ⋅ DFi = (-0.012238/0.075) × 1.109955 = -0.18112 ($/MWh ) ∆D j ∆µ Line 6,8 ⋅ GSFLine 6,8− 30 = (-0. 26436/0.075) × (-0.12866) ∆D j = 0.45350($/MWh2) M 2 ∆µ k ∆λ ∆ DFi ⋅ GSF k −i = -0.007383($/MWh ) ⋅ DFi + ∑ ⋅λ + ∆D j ∆D j k =1 ∆ D j
IV. SENSITIVITY WHEN THERE IS A CHANGE OF MARGINAL UNITS Figures 10 and 11 show the normalized delivery factor and the LMP sensitivity, respectively, when load at Bus B is varied from 300 MWh to 390 MWh in the PJM 5-bus system. Again, other loads remain unchanged for simplicity. These two figures show there is a turn of delivery factors and a sharp change of LMP sensitivity, when Bus B Load increases from 346.50 to 347.25 MWh. This is due to a change of marginal units from Brighton and Sundance to Solitude and Sundance. This will change the LMP prices at each bus significantly. In addition, this will change the Delivery Factor (DF) growth pattern because DF is related to the generation locations. Therefore, the delivery factor sensitivity has a sharp change and so does the LMP sensitivity. As shown in Fig. 10, the delivery factor at Bus C decreases after the critical load level of the step change of LMP sensitivity. At the critical load level and above, the new marginal unit Solitude will generate more power to supply its local load. So, the power flow through Line DC and Line BC will be considerably reduced while the power flows through other lines are almost unchanged. This will reduce the line losses and the fictitious demand at Bus C. Thus, the delivery factor decreases. Hence, the delivery factor sensitivity changes sharply and so does the LMP sensitivity. This representative case well illustrates that delivery factors may be affected by generation scheduling. Hence, this also
Also, we have ∆LMPi = -0.00054264/0.075 = -0.007235 ($/MWh2) ∆D j Hence, the error of Eq. (15) is less than 2.0%. Figure 9 shows the LMP at Bus 30 with respect to Load at Bus 8 between 37.5 and 39 MWh. It can be easily verified that the slope of the LMP curve is roughly -0.007 $/MWh2. It remains this value since there is no new binding constraint when Load at Bus 8 is increased from 37.5 to 39 MWh. LMP ($/MWh) at Bus 30 w.r.t. Load at Bus 8 (MWh) 46.200
Fig. 9. LMP at Bus 30 with respect to Bus 8 Load from 37.5 to 39 MWh in the IEEE 30-bus system.
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shows the necessity to adopt the iterative DCOPF approach rather than using DF from a pre-defined typical scenario. The step change pattern implies the applicability of LMP sensitivities. When the present operating point is far from a change of marginal units, the LMP sensitivities (due to losses) can indicate how the LMP will change under load variation. On the other hand, when some of the current marginal units are near their generation limits or some transmission lines are very likely to be congested when there is a small load growth, the LMP sensitivities calculated in the present operating point are less reliable because a step change of LMP as well as LMP sensitivity may occur even with a small load growth.
step change when the load grows to a critical level to cause a new marginal unit. This infinite, step change may be an interesting topic to future research works. VI. REFERENCES
Normalized Delivery Factor w.r.t. Load at Bus B (MWh) 1.0018 1.0016 1.0014 1.0012 1.0010 1.0008
Fangxing Li and Rui Bo, “DCOPF-Based LMP Simulation: Algorithm, Comparison with ACOPF, and Sensitivity,” IEEE Trans. on Power Systems, vol. 22, no. 4, pp. 1475-1485, November 2007.
PJM Training Materials: LMP 101, PJM.
S. Stoft, Power System Economics – Designing Markets for Electricity, IEEE and John Willey Publication, 2002.
M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems, Wiley-Interscience, 2002.
E. Litvinov, T. Zheng, G. Rosenwald, and P. Shamsollahi, “Marginal Loss Modeling in LMP Calculation,” IEEE Trans. on Power Systems, vol. 19, no. 2, pp. 880-888, May 2004.
Fangxing Li, Jiuping Pan and Henry Chao, “Marginal Loss Calculation in Competitive Electrical Energy Markets,” Proceedings of the 2004 IEEE International Conference on Electric Utility Deregulation, Restructuring and Power Technologies, 2004 (DRPT 2004), vol. 1, pp. 205-209.
A. Conejo, E. Castillo, R. Minguez, and F. Milano, “Locational Marginal Price Sensitivities,” IEEE Trans. on Power Systems, vol. 20, no. 4, pp. 2026-2033, November 2005.
T. Overbye, X. Cheng, and Y. Sun, “A Comparison of the AC and DC Power Flow Models for LMP Calculations,” Proceedings of the 37th Hawaii International Conference on System Sciences, 2004.
R. D. Zimmerman, C. E. Murillo-Sánchez, D. Gan, MatPower – A Matlab Power System Simulation Package, School of Electrical Engineering, Cornell University, http://www.pserc.cornell.edu/matpower/matpower.html.
1.0006 1.0004 1.0002 1.0000 0.9998 300
350 Bus C
Fig. 10. Normalized Delivery Factor at each bus with respect to Load at Bus B ranging from 300 MWh to 390 MWh in the PJM 5-bus system.
LMP Sensitivity ($/MWh ) w.r.t. Load at Bus B (MWh) 0.0025 0.0020
Rui Bo (S’02) received the B.S. and M.S. degrees in electric power engineering from Southeast University of China in 2000 and 2003, respectively. He had worked at ZTE Corporation and Shenzhen Cermate Inc. from 2003 to 2005. He started his Ph.D. program at The University of Tennessee in January 2006. His interests include power system operation and planning, power system economics and market simulation.
0.0010 0.0005 0.0000 300
330 Bus B
350 Bus C
Fangxing (Fran) Li (M’01, SM’05) received the Ph.D. degree from Virginia Tech in 2001. He has been an Assistant Professor at The University of Tennessee (UT), Knoxville, TN, USA, since August 2005. Prior to joining UT, he worked at ABB, Raleigh, NC, as a senior and then a principal R&D engineer for 4 and a half years. During his employment at ABB, he had been the lead developer of GridViewTM, ABB’s market simulation tool. His current interests include energy market, reactive power, distributed energy resources, distribution systems, reliability, and computer applications. Dr. Li is the recipient of the 2006 Eta Kappa Nu Outstanding Teacher Award at UT.
Fig. 11. LMP Sensitivity with respect to Load at Bus B ranging from 300 MWh to 390 MWh in the PJM 5-bus system.
V. CONCLUSIONS This paper firstly reviews the FND-based DCOPF model and the associated LMP calculation formulation. This model is demonstrated to be a good approximation of the benchmark ACOPF model in a previous work. Then, based on the FND algorithm, a simple and explicit formulation of LMP sensitivity with respect to load is presented. It shows that LMP sensitivity is related to the loss component and linearly related to the sensitivity of delivery factors. Simulations are performed on the PJM 5-bus system and the IEEE 30-bus system to verify the formulation. Without loss component, the LMP sensitivity is zero if load is varied in a small range. With loss taken into account, the LMP sensitivities are constant values numerically. Furthermore, the LMP sensitivity may be subject to a sudden 6