OPTIMAL POINT TARIFFS IN TRANSMISSION AND DISTRIBUTION USING - PowerPoint PPT Presentation

The Ninth IASTED European Conference on Power and Energy Systems ~EuroPES 2009~ September 7 – 9, 2009, Palma de Mallorca, Spain OPTIMAL POINT TARIFFS IN TRANSMISSION AND DISTRIBUTION USING NONNEGATIVE LEAST SQUARES ESTIMATES by Egill Benedikt Hreinsson Department of Electrical and Computer Engineering, University of Iceland, Hjardarhagi 6, Reykjavik, (Iceland) Email: egill@hi.is 1

Presentation overview • Introduction • Model for optimal point tariffs – Least squares deviations from nodal prices • Simple computational examples • Discussion and conclusion

Introduction • Assume a power system with many nodes • Assume at each node we have Lagrange multipliers reflecting the nodal energy price at each note ? – These prices are the multipliers to real power constraints for each node in an OPF formulation. • The transmission cost between nodes is given by the difference in nodal prices ? • How should we set up point tariffs to reflect as accurately as possible these transmission prices?

Transmission costs The marginal cost difference between an injection node i and extraction node j is λ = λ − λ ij i j u • Assume the injection and extraction involves quantity ij • Marginal transmission cost (congestion rent) is skew λ = − λ symmetric, or ij ji − λ − λ � � � 0 N 12 1 � � λ − λ 0 � � • The cost matrix: N = � � 12 2 � � � � � � λ λ � � � 0 N N 1 2 N N = �� u λ C • total transmission charges ij ij TXMCOST = = i j 1 1 4

An over-determined set of linear equations The problem involves an over-determined set of linear equations with nonnegative constraints: + = λ ∀ u r s u i j ( ) , ij i j ij ij ≥ ≥ ∀ r s i j 0 ; 0 , i i This set can be rewritten: Ax b x 0 = ≥ 5

The least squares problem Ax b x 0 − ≥ Minimize subject to where: A × ≥ m n m n is a known coefficient matrix, , b m is the element vector with known constants, and x n is the element unknown solution vector.

A simple (trivial) 2 node system r 1 3 r 2 2 + = r s 0 1 1 + = r s λ 1 2 2 4 λ 2 1 2 1 2 + = − r s 2 2 1 2 1 s 1 + = s 2 r s 0 2 2 1 1 = = = = r s r s 0.8889 , 0 , 0 , 0.8889 1 1 2 2

A simple 3 node system λ = λ = λ = 1 ; 2 ; 3 Assume that 1 2 3 and all contracts are for simplicity sake = u unit from one node i to node j . 1 equal or ij Then we have the following set of equations: + = r s 0 1 1 λ + = − λ r s 1 1 2 1 2 1 2 + = − r s 2 1 3 + = r s 0 λ 3 2 2 3 + = s r 1 1 2 + = − r s 1 2 3 + = s r 2 1 3 + = s r 1 2 3 + = r s 0 3 3

Matlab formulation and results A=[1 1 0 0 0 0;1 0 0 1 0 0;1 0 0 0 0 1; 0 0 1 1 0 0;0 1 1 0 0 0;0 0 1 0 0 1; 0 1 0 0 1 0;0 0 0 1 1 0;0 0 0 0 1 1] b=[0;-1;-2;0;1;-1;2; 1; 0] x = x= lsqnonneg(A,b) b = 0 0 A = -1 1 1 0 0 0 0 -2 0.7500 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 -1 0 1 1 0 0 0 2 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0.7500 0 0 0 1 1 0 0 0 0 0 1 1 0

Non zero diagonal line of A If we for instance relax the requirement of the diagonal line λ by setting a price for the input/output at the same bus in ij in the first, 4 th and the last equation of the previous eq., for instance 0.2, 0.4 and 0.6 units respectively, we get (rearranged for simplicity and compactness): b = x = 0.2000 0 -1.0000 0.7500 -2.0000 0 0.4000 0 1.0000 0.9500 -1.0000 0 2.0000 1.0000 0.6000 r = s = This means that: while and other 0.75 0.95 3 1 charges are zero so it does not alter basically the structure of the least squares solution.

Discussions and conclusions • Optimal instantaneous marginal costs . • Snapshot of operating conditions . • Economic efficiency . • Fixed cost recovery • Wider range spatial equalization of charges .

Suggestions for further research Define a system of realistic size with tens, hundreds or thousands of buses • and define an appropriate time period with time steps. Define the basic operating conditions such as a basic load and basic • bilateral contracts on top of that basic load as injections and extractions of specified MWh’s in each time step (on top of that basic load). Run an OPF for all time steps and obtain Lagrange • multipliers for each node and each time step as nodal marginal prices. Consider other requirements such as recovering fixed cost for • each transmission link or for the system as a whole and how this condition is merged with the problem definition and model formulation. Run the least squares calculation considering the zonal partitioning in the • system, if any, and adaptation to covering the fixed costs recovery during the time frame selected. Interpret the results by calculating the total charges and how these total • charges compare for instance with marginal prices and total transmission costs accumulated for all transaction.

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