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 > > > > 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 >> >> >> >> >> >> 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 it’s easy to find out about >>> the distribution factors. With this proposal, it’s 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? >>> I’d like to know which algorithm is applied by you to solve the OPF problem? >>> Best Regards, >>> Víctor >>> >>> De: [email protected] >>> [mailto:[email protected]] 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. >>> <[email protected]> wrote: >>> Dear Dr Zimmerman, >>> I’d 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. It’s 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 >>> >>> >> >> > >
