Re: Question about LMP

[Carlos E Murillo-Sanchez]

Dear Victor: Derivation of the lagrangian function with respect to Pg1 yields the first order optimality condition f'(Pg1) + mu_+ - mu_-    - \lambda_P1 = 0 or \lambda_P1 = f'(Pg1) + mu_+ - mu_- where \lambda_P1 is the multiplier on bus' 1 balance constraint; mu_+ is the multiplier on the upper limit constraint for Pg1, and mu_- is the corresponding multiplier for the lower limit constraint.    In your example, \lambda_P1 = 11.899 (in fact for all buses), Absent congestion, the balance constraint multiplier must be the same for all buses.  In the solution, the generator is against its lower limit. So mu_+ = 0,  mu_- = 0.303 and lambda_P1 = 11.899  .  You should convince yourself that at the solution, the marginal cost of the generator evaluates to 11.899+0.303 = f'(Pg1) . Carlos. Victor Hugo Hinojosa M. wrote: Thank you so much for the information Prof. Zimmerman. I’d like your explanation about the simulation for the 6-bus system (Wood & Wollemberg). When I run a DCOPF with the original case (rundcopf(case6ww)), the LMP shown for Matpower are the same (11.899 $/MWh) for all buses because there isn’t congestion in the transmission lines. Despite of the fact that the Lagrange multiplier for generator 1 is active (0.303 $/MWh), the LMP are the same. In my opinion, the LPM from bus 1 should be 12.202 $/MWh. I’ll wait for your comments. Regards, Vh     MATPOWER Version 5.1, 20-Mar-2015 -- DC Optimal Power Flow Gurobi Version 6.0.4 -- automatic QP solver   Converged in 0.16 seconds Objective Function Value = 3046.41 $/hr ================================================================================ |     System Summary                                                           | ================================================================================   How many?                How much?              P (MW)            Q (MVAr) ---------------------    -------------------  -------------  ----------------- Buses              6     Total Gen Capacity     530.0           0.0 to 0.0 Generators         3     On-line Capacity       530.0           0.0 to 0.0 Committed Gens     3     Generation (actual)    210.0               0.0 Loads              3     Load                   210.0               0.0   Fixed            3       Fixed                210.0               0.0   Dispatchable     0       Dispatchable          -0.0 of -0.0      -0.0 Shunts             0     Shunt (inj)             -0.0               0.0 Branches          11     Losses (I^2 * Z)         0.00              0.00 Transformers       0     Branch Charging (inj)     -                0.0 Inter-ties         0     Total Inter-tie Flow     0.0               0.0 Areas              1                             Minimum                      Maximum                  -------------------------  -------------------------------- Voltage Magnitude   1.000 p.u. @ bus 1          1.000 p.u. @ bus 1   Voltage Angle      -3.67 deg   @ bus 5          0.00 deg   @ bus 1   Lambda P           11.90 $/MWh @ bus 3         11.90 $/MWh @ bus 4   Lambda Q            0.00 $/MWh @ bus 1          0.00 $/MWh @ bus 1     ================================================================================ |     Bus Data                                                                 | ================================================================================ Bus      Voltage          Generation             Load          Lambda($/MVA-hr)   #   Mag(pu) Ang(deg)   P (MW)   Q (MVAr)   P (MW)   Q (MVAr)     P        Q   ----- ------- --------  --------  --------  --------  --------  -------  -------     1  1.000    0.000*    50.00      0.00       -         -      11.899     -     2  1.000   -0.299     88.07      0.00       -         -      11.899     -     3  1.000   -0.278     71.93      0.00       -         -      11.899     -     4  1.000   -2.986       -         -       70.00      0.00    11.899     -     5  1.000   -3.666       -         -       70.00      0.00    11.899     -     6  1.000   -3.087       -         -       70.00      0.00    11.899     -                         --------  --------  --------  --------                Total:    210.00      0.00    210.00      0.00   ================================================================================ |     Branch Data                                                              | ================================================================================ Brnch   From   To    From Bus Injection   To Bus Injection     Loss (I^2 * Z)    #     Bus    Bus    P (MW)   Q (MVAr)   P (MW)   Q (MVAr)   P (MW)   Q (MVAr) -----  -----  -----  --------  --------  --------  --------  --------  --------    1      1      2      2.61      0.00     -2.61      0.00     0.000      0.00    2      1      4     26.06      0.00    -26.06      0.00     0.000      0.00    3      1      5     21.33      0.00    -21.33      0.00     0.000      0.00    4      2      3     -0.15      0.00      0.15      0.00     0.000      0.00    5      2      4     46.91      0.00    -46.91      0.00     0.000      0.00    6      2      5     19.59      0.00    -19.59      0.00     0.000      0.00    7      2      6     24.33      0.00    -24.33      0.00     0.000      0.00    8      3      5     22.75      0.00    -22.75      0.00     0.000      0.00    9      3      6     49.03      0.00    -49.03      0.00     0.000      0.00   10      4      5      2.97      0.00     -2.97      0.00     0.000      0.00   11      5      6     -3.37      0.00      3.37      0.00     0.000      0.00                                                              --------  --------                                                     Total:     0.000      0.00   ================================================================================ |     Generation Constraints                                                   | ================================================================================ Gen   Bus                  Active Power Limits   #     #     Pmin mu     Pmin       Pg       Pmax    Pmax mu ----  -----   -------   --------  --------  --------  -------    1     1      0.303     50.00     50.00    200.00      -     De: bounce-119913926-12657...@list.cornell.edu [mailto:bounce-119913926-12657...@list.cornell.edu] En nombre de Ray Zimmerman Enviado el: jueves, 19 de noviembre de 2015 12:03 Para: MATPOWER discussion forum Asunto: Re: Question about LMP   The LMPs for a DC OPF problem do incorporate any generator limits as well as generation cost. Consider the case with no congestion, where the LMPs are uniform at all nodes. For nodes with generators that are dispatched between their lower and upper limits, the LMP equals their marginal cost of generation. For a node with a generator at a binding upper (lower) limit, the LMP will equal the marginal cost of generation plus (minus) the shadow price on the binding upper (lower) generation constraint.      Ray   On Nov 19, 2015, at 8:27 AM, Victor Hugo Hinojosa M. <victor.hinoj...@usm.cl> wrote:   Dear Jovan and Sarmad, I agree with your comments about LMP. In this analysis I’m not considered the congestion. If the generation inequality constraints aren’t active, Matpower prints this information correctly, and It’s possible to realize different prices when the lines is congested. Sarmad, I’ve verified your idea. Despite the fact that the shadow price for the minimum or maximum is active, the LMP shown are the same for all buses. My question is about why LMP doesn’t include the Lagrange multipliers related to generation inequality constraints. I did a model using the dual problem for the DCOPF, and I realized that dual constraints are the prices for each buses. It’s very clear in those constraints that those “prices” take into account the marginal cost, the congestion cost through the partial transmission distribution factors (PTDF) and the generation constraints. In the technical literature for the DCOPF (losses are neglected), the LPM are modeled considering energy cost and congestion cost. However, in the book “Spot pricing of electricity” from F. Schweppe et all, authors include these shadow prices in order to compute the spot prices. I’d like to know your feedback about these comments. Regards, Vh   De: bounce-119912654-12657...@list.cornell.edu [mailto:bounce-119912654-12657...@list.cornell.edu] En nombre de Jovan Ilic Enviado el: jueves, 19 de noviembre de 2015 1:13 Para: MATPOWER discussion forum Asunto: Re: Question about LMP     Dear Victor,   If there is no congestion in the network, there is the same LMP at all the nodes.  The LMP consists of loss, congestion, and energy costs. DCOPF has no  losses, and if there is no congestion only the energy cost is accounted for.  You can think of it as if since there is no congestion or loss cost the energy can be distributed to all nodes at the same price.    Regards,  Jovan Ilic   On Wed, Nov 18, 2015 at 4:37 PM, Victor Hugo Hinojosa M. <victor.hinoj...@usm.cl> wrote: Dear Prof. Zimmerman, I have a question about Local Marginal Prices (LMP) that are shown in Matpower. The definition of the LMP is the marginal cost of supplying, at least cost, the next increment of electric demand at a specific location (node) on the electric power network, taking into account both supply (generation/import) bids and demand (load/export) offers and the physical aspects of the transmission system including transmission and other operational constraints. When it is performed a DCOPF, Matpower shows LMP for each bus considering the marginal cost (energy cost) and the congestion cost so that I'd like to know why the generation constraints (maximum and minimum power) aren't considered in the LMP. Thank you so much for your ideas and comments. Regards, Vh