Some Observations on the Locational Marginal Prices and Loads

Transcription

1 on the Locational Marginal Prices and Loads Xinbo Geng Texas A&M University October 15, 2014 Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

2 Overview 1 Preliminary Economic Dispatch Locational Marginal Prices and Shadow Prices Some Theorems 2 Some Observations Prof.Leigh Tesfatsion s Observations 3 Summary Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

3 Preliminary Economic Dispatch Economic Dispatch Assume there are generators and load at every bus, therefore n g = n d = n b. Let bus#1 be the slack bus. minimize P G subject to C T P G n b P Gi = P Dj : λ 1 i=1 n b j=1 H (P G P D ) F : µ + H (P G P D ) F : µ P G PG max : η + P G PG min P G 0 : η Further reading: History of optimal power flow and formulations. FERC Staff Technical Paper (2012). Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17 (1)

4 Preliminary Locational Marginal Prices and Shadow Prices Some Important Theorems Definition of Locational Marginal Price (LMP) (C: total dispatch cost, P Di : load at bus i): λ i = C P Di (2) Definition of Shadow Prices µ ij If the transmission capacity of line ij were increased by ɛ MW, total system cost would be changed bu µ ij ɛ. µ ij = C F ij = µ + ij µ ij (3) Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

5 Preliminary Some Theorems Some Theorems Relationship between LMPs and Shadow Prices λ 2 λ 1 λ 3. = λ 1 HT µ +. λ n λ 1 We can use the definition of LMP, take partial derivatives on the lagrangian dual. If strong duality holds, then it can be proved. no congestion µ = 0 all the λ i are the same. Consider this process: no load in the system load increasing lines congested µ = 0 µ = 0 some elements of µ are non-zero. λ i = c 1 are the same and constant λ i = c 2 are the same and constant λ i are different from each other. (4) Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

6 Preliminary Some Theorems The features of LMP λ and shadow prices µ Continuous or Discrete? µ Shadow prices: there are some sudden changes in the process. λ LMPs: there are some sudden changes in the process. LMPs and shadow prices are very likely to be descrete. Will the λ and µ vary? Yes In the large-scale, e.g. the whole process in the previous slide. No In a small scale, e.g. no-congestion. Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

7 Step Changes of LMP The step changes of LMPs are first reported in System Load: the summation of all the load in the system, grows from 0 to the possible maximum value. Load Pattern: the load at each bus grows proportionally to a given pattern. Bus 1 Bus 2 Bus 3 System Table : Example of Load Pattern (MW) 1 Fangxing Li, Continuous Locational Marginal Pricing (CLMP), IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007 Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

8 An Illustrative Example - 5 Bus System Assumptions Fixed generators bids, system topology, load pattern, commitment results. Figure : Base case modified from the PJMfive-bus example Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

9 Step Changes Found Figure : Step Changes of LMP Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

10 Some Analysis (1) No congestion, marginal unit is Brighton (2) No congestion, marginal unit is Alta (3) No congestion, marginal unit is Brighton (4) Line E D is congested (5) Line A B is congested Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

11 Some Analysis Why the step changes happen The active constraints change. E.g. transmission capacity limits, generation upper/lower limits. Critical Load Level (CLL) Defined as: a load level at which a step change occurs, and the corresponding new binding limit, either a congested transmission line or a marginal unit reaching generation limit. LMP Forecast If we can indentify the CLLs, then we can perfectly forecast the LMPs. Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

12 More Work From Prof.Fangxing Li s Group Continuous LMP Step Changes are not good signals = smooth the step changes = Continuous LMP Fangxing Li, Continuous Locational Marginal Pricing (CLMP), IEEE Trans on Power Systems, Identificatoin of CLLs and LMP Forecast A small perturbation around current system condition, calculate the LMP sensitivity matrices = indentify Critical Load Levels (CLLs) = LMP forecast. Fangxing Li, Rui Bo, Congestion and Price Prediction Under Load Variation, IEEE Trans on Power Systems, Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

13 Possible Improvements Load Pattern = Load Space Current: The load at every bus grows proportionally to a given pattern. Reality: There is no such a pattern! One dimensional problem (system load) = N dimension problem (in the load space ) Piecewise Linear Cost Curve = Quadratic Cost Curve Current: piecewise linear cost curve of generators. What if: quadratic cost curve? Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

14 Possible Improvements DC OPF = AC OPF Rui Bo, Fangxing Li, Kevin Tomsovic, Prediction of critical load levels for AC optimal power flow dispatch model, Electrical Power and Energy Systems 42 (2012) Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

15 Prof.Leigh Tesfatsion s Observations In the Load Space Prof.Leigh Tesfatsion s Observations Zhou, Qun, Leigh Tesfatsion, and Chen-Ching Liu. Short-term congestion forecasting in wholesale power markets. Power Systems, IEEE Transactions on 26.4 (2011): Zhou, Qun, Leigh Tesfatsion, and Chen-Ching Liu. Global sensitivity analysis for the short-term prediction of system variables. Power and Energy Society General Meeting, 2010 IEEE. IEEE, Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

16 Summary Summary Report the step changes of LMPs. Give some intuitive explanations. Introduce the basic ideas behind related researches. Discuss the drawback/limitation/possible improvement of current researches. Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17

17 Summary Any Questions? Some Observations on the Locational Marginal Prices and Loads Xinbo Geng Texas A&M University October 15, 2014 Xinbo Geng (TAMU) Relationship between LMP and Load October 15, / 17