Dear Jovan and Ray, Thanks a lot for the very interesting discussion. I would like to include an analysis accomplished to determine the performance of the both methodologies. I realize that this idea was no new, so I would like to find out references or studies to understand the state of the art.
I would like to keep on working and researching in this very interesting field. I will wait your comments and observations with respect to the results and the proposal. Regards, Vh DC Optimal power flow (DC-OPF) The DC-OPF problem is concerned with the optimization of steady-state power generation, subject to various equality and inequality constraints. Mathematically, the convex programming problem may be represented as: Subject to the following linear constraints: Where: represents a scalar objective function to be minimized. In DC-OPF problem, this function is the fuel cost of thermal units that can be expressed in terms of real power generation, and it is modeled by a quadratic cost function. denotes the linear equality constraint called nodal power balance constraint, which includes the network DC power flow equations. is the vector of the inequality constraints with lower limit and upper limit. These constraints are active power flow in the transmission lines and active generation capabilities. Using DC network modeling, it is quite known that real power flows are linear functions of voltage angles. is the set of control variables (called decision variables in the optimization theory). In the DC-OPF problem, these variables are the real power output for each generator. In addition, there are dependent variables in the problem, which depend on control variables. Considering the DC power flow problem, the voltage phase angles variables are considered as dependent variable, where the angle in the slack bus is known. Matpower includes a primal-dual interior point solver called MIPS. It is implemented in Matlab code [12, 19]. In DC-OPF problem, the decision variables are the following: In this proposed methodology, the study considers that the optimization problem can be solved using the power generation as decision variables (control variables), therefore the DC-OPF problem can be formulated as: Subject to: Where: The nodal power balance constraint is not needed, just the global supply demand balance constraint, and is the total load of the customers. is the vector of active power flow in the transmission lines and active generation capabilities. It is very important to notice that the real power flows can be computed as [PLj-k]=[b]*[S]*[]. Where, the [b] matrix is the susceptance of line j () and non-diagonal elements are zero, and [S] is the bus-branch incidence matrix. Besides, the power flow equations in the matrix form and the corresponding matrix relation for flows through branches are represented by []=*[P], where [B] is the admittance matrix with [R]=0, and [P] is bus active power injections for buses j. In addition, we need to define the slack bus, so that the row reference in [P], and the reference row and column of [B] are disregarded. When you consider that the power flows are computed by the following matrix as [PLj-k]=[b]*[S]**[P], it is very easy to figure out the linear sensitivity factors called PTDF in the literature, where [PTDF]=[b]*[S]*. These factors show the approximate change in the line flow for changes in generation on the network configuration. Finally, for a given value of , the Lagrangian for the equality constrained problem can be formulated as follow: Where: is the number of generators and is the number of power lines. variable is a real value, and it gives information about the marginal cost in the global node; i.e., the nodal price of the slack bus. [] is the Lagrange value associated to both constraints real power flow in the transmission lines and active power capacity. The main advantage of this study is the reduction of the dimension. This is very important when the algorithm needs to compute and . We have programmed the same algorithm (MIPS) to solve the proposed problem considering the same tolerance criterions (feasibility, gradient, complementary, cost condition, and maximum number of iterations). Results Five test power systems are considered to inspect and verify the proposed algorithm, and the simulations results have been compared with the solution obtained through Matpower (power and angles as decision variables). The data of the IEEE systems (14-bus, 30-bus, 39-bus, 57-bus and 118-bus) data were obtained from Matpower. Several experiments were conducted to determine the performance of the proposed methodology in different power systems (small and medium dimension). In the next Table is shown the results obtained by Matpower and the proposal accomplished. The results show that both methodologies obtained the same solution in the DC-OPF problem. As mentioned Jovan Ilic, it is possible to compute the bus angle in each bus to draw some conclusions about the system. Moreover, when there is congestion in the transmission lines, it is possible to compute the nodal prices using the incremental cost for each generator. De: [email protected] [mailto:[email protected]] En nombre de Ray Zimmerman Enviado el: lunes, 05 de agosto de 2013 13:08 Para: MATPOWER discussion forum Asunto: Re: DC-OPF on matpower Thanks, Jovan, for this interesting discussion. You've convinced me. After a bit more thought, I believe I agree with you after all. The formulations should be equivalent. -- Ray Zimmerman Senior Research Associate B30 Warren Hall, Cornell University, Ithaca, NY 14853 phone: (607) 255-9645 On Aug 5, 2013, at 11:34 AM, Jovan Ilic <[email protected]> wrote: Ray, No and yes. No, I am not sure what you mean by "line constraints will be affected by the choice of slack bus". A line constraint is either binding or not and slack bus choice will not affect it. It can be shown mathematically and is clear intuitively, once you show it mathematically, I suppose. :-) Yes, this formulation will give you a single LMP (at the slack bus) and line Lagrange multipliers (mus) and then you use the found LMP, DF and mus to calculate the rest of LMPs: LMPs = ones(Nb-1,1)*LMP + transpose(DF)*mu You can derive this from KKT conditions. You can also find a hint of it in Felix Wu's paper "Folk Theorems in Power Systems" or something like that. If I remember the paper, Felix uses relative LMPs or something like that, stopped half way if you ask me. Jovan Ilic On Mon, Aug 5, 2013 at 11:12 AM, Ray Zimmerman <[email protected]> wrote: For an unconstrained network, I agree that the slack is irrelevant and the problems are equivalent. However, I don't think the problems are equivalent when you have binding line constraints. Then I'm pretty sure the line constraints will still be affected by the choice of slack used when forming the PTDF. Another clue that the problems are not equivalent in this case is that it seems there is no way to recover the nodal prices from the PTDF-based approach. In a case with a single binding line limit you only have two non-zero constraint shadow prices in the problem (using PTDF), but you can have many different nodal prices from the traditional DC OPF. -- Ray Zimmerman Senior Research Associate B30 Warren Hall, Cornell University, Ithaca, NY 14853 phone: (607) 255-9645 <tel:%28607%29%20255-9645> On Aug 5, 2013, at 10:52 AM, Jovan Ilic <[email protected]> wrote: Ray, Yes, as I mentioned in my original post, dense PTDF is the drawback of this approach and you might want to know bus angle differences to draw some conclusions about the system but if you code it carefully you already have what you need to quickly calculate the angles. However, there is no approximation introduced by the slack bus, with or without the slack bus the result is the same. I coded it with both LP and QP and the results are the same, well if the cost functions are all linear of course. It takes about an hour to code it in any language and test it if you already have decent LP/QP functions. I actually expected people to question the method because I said that nodal equations are not needed, just the global supply demand balance constraint. It seems that I underestimated the audience. :-) Jovan Ilic On Mon, Aug 5, 2013 at 9:59 AM, Ray Zimmerman <[email protected]> wrote: MATPOWER does not use the dense distribution factor matrix (PTDF) to formulate the DC OPF. One other important distinction is that a PTDF matrix is an approximation that requires an assumption about the slack. The sparse formulation that includes the voltage angles does not require this assumption. So the two formulations are not quite equivalent. In MATPOWER, the DC OPF is formulated as an LP or QP problem and the algorithm used to solve it varies depending on the solver chosen via the OPF_DC_ALG option. -- Ray Zimmerman Senior Research Associate B30 Warren Hall, Cornell University, Ithaca, NY 14853 phone: (607) 255-9645 <tel:%28607%29%20255-9645> On Aug 5, 2013, at 9:17 AM, Victor Hugo Hinojosa M. <[email protected]> wrote: Dear Jovan, Thank you so much for your message. I agree with your comment. Considering that [Y_bus]*[theta_angles]=[Power_injections], the balance nodal constrained can be formulated as a global balance power equation. Hence, the balance nodal matrix is reduced to one single equation. In addition, the power transmission constraint depends on the angles, but the angles can be transformed using the equation [theta_angles]=[Y_bus]^-1*[Power_injections], so the transmission constraint depends on the power injections, and its easy to find out about the distribution factors. With this proposal, its possible to model the DC-OPF problem using only the power generation as decision variable. Therefore, there are many advantages of this proposal. Do you address the DC-OPF problem with this proposal? Id like to know which algorithm is applied by you to solve the OPF problem? Best Regards, Víctor De: <mailto:[email protected]> [email protected] [mailto:bounce-104920604-12657875@ <http://list.cornell.edu/> list.cornell.edu] En nombre de Jovan Ilic Enviado el: jueves, 01 de agosto de 2013 20:11 Para: MATPOWER discussion forum Asunto: Re: DC-OPF on matpower Victor, Yes, you can do it without bus angles but you'd end up with a formulation with a dense distribution factors matrix which could be a problem for large systems. One place where you can speed up such DCOPF is by using a global power balance equation instead of nodal equations. You do not need nodal balance equations if you have the distribution factors matrix. An added benefit of using the distribution matrix would be loss estimation. I rarely use Matpower so I am not sure which algorithm Ray uses. Maybe I should've let Ray answer the question since it was addressed to him. Jovan Ilic On Thu, Aug 1, 2013 at 7:06 PM, Victor Hugo Hinojosa M. < <mailto:[email protected]> [email protected]> wrote: Dear Dr Zimmerman, Id like to ask you a question about the DC optimal power flow (DC-OPF). The optimization problem in Matpower is modeled using as decision variables the active power generation and the bus voltage angles. These variables are solved using the primal-dual interior point solver (MIPS) considering that both variables are independent. When the AC transmission system is transformed using the DC approach, the voltage angles and the active power injections are related through the Y_bus matrix, so the decision variables are dependent. Its possible to model the DC-OPF problem using only the power generation as decision variable? Thank you so much for your comments in advance. Best Regards, Víctor
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