Thank you for your answer. I think I get the idea. In practice, I found that decreasing the tolerance for the complementarity constraint helps disambiguate these cases.
Kind regards, Camille From: [email protected] [mailto:[email protected]] On Behalf Of Ray Zimmerman Sent: 26. oktober 2015 21:14 To: MATPOWER discussion forum <[email protected]> Subject: Re: Nonzero shadow prices for non-binding constraints Good question. Because the interior point solver does not explicitly define binding vs. non-binding constraints in the solution, any such classification must be done based on numerical tolerances. I’m pretty sure the intent of that code was to only zero out the shadow prices (i.e. mark the constraint as non-binding) if both the primal and dual solutions indicate that the constraint is not “active”. Not a very rigorous answer, I know, but it was an attempt at being conservative in marking a constraint as non-binding. If the shadow price is significant, then the constraint is still having an effect on the solution, even if the constraint value is not at the bound. Hope this helps, Ray On Oct 26, 2015, at 6:45 AM, Camille Hamon <[email protected]<mailto:[email protected]>> wrote: Dear all, I am implementing a search for the maximum loadability limit using an OPF with dispatchable loads in Matpower. To model PV buses, I set generator voltage set points as upper limits for the voltage magnitude at generator buses and set the lower voltage limits at a low number (0.1 for example). Also, I keep the reactive power upper limit. When the OPF has run, I check the shadow prices of the upper bound constraints on voltage magnitudes to identify the generators which are still PV buses and the generators which have become PQ buses. During my testing, I ran into cases where the upper bounds voltage constraints are non-binding, but the corresponding shadow prices are nonzero. I believe it is due to the different thresholds used to check binding constraints. In particular, as far as I understood, the shadow prices of all non-binding constraints are set to zero by the following lines (mips solver): %% zero out multipliers on non-binding constraints mu(h < -opt.feastol & mu < mu_threshold) = 0; My understanding is that the first condition h < -opt.feastol checks for non-bining constraints (deviations from the limit h is larger than the tolerance opt.feastol). I do not understand the second condition, however (mu < mu_threshold), which checks for the magnitude of the Lagrange multipliers. In my results, I have the case where h for a constraint is indeed larger than the opt.feastol threshold (h=7 e-4) but the corresponding Lagrange multiplier is larger than the mu_threshold (mu=2 e-4) so that the shadow price is not set to zero. Note also that the product of h and mu for the corresponding constraint is small enough for the last MIPS iteration to conclude on convergence (i.e. the relative complementarity constraint is satisfied). My question would then be: - What is the meaning of having the second check (mu < mu_threshold) in identifying the non-binding constraints? Kind regards, Camille Hamon Post doc Department of Electrical Power Engeneering Norwegian University of Science and Technology (NTNU) --_000_e8607519f2e44b44b83fcf43cc9ad18bitex10winntnuno_ Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: base64 <html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml"; xmlns="http://www.w3.org/TR/REC-html40";> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <meta name="Generator" content="Microsoft Word 15 (filtered medium)"> <style><!-- /* Font Definitions */ @font-face {font-family:Helvetica; panose-1:2 11 6 4 2 2 2 2 2 4;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {margin:0cm; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Times New Roman",serif;} a:link, span.MsoHyperlink {mso-style-priority:99; color:blue; text-decoration:underline;} a:visited, span.MsoHyperlinkFollowed {mso-style-priority:99; color:purple; text-decoration:underline;} span.apple-converted-space {mso-style-name:apple-converted-space;} span.EmailStyle18 {mso-style-type:personal-reply; font-family:"Calibri",sans-serif; color:#1F497D;} .MsoChpDefault {mso-style-type:export-only; font-size:10.0pt;} @page WordSection1 {size:612.0pt 792.0pt; margin:70.85pt 70.85pt 70.85pt 70.85pt;} div.WordSection1 {page:WordSection1;} --></style><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="1026" /> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1" /> </o:shapelayout></xml><![endif]--> </head> <body lang="NO-BOK" link="blue" vlink="purple"> <div class="WordSection1"> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">Thank you for your answer. I think I get the idea. In practice, I found that decreasing the tolerance for the complementarity constraint helps disambiguate these cases.<o:p></o:p></span></p> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></p> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">Kind regards,<o:p></o:p></span></p> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US">Camille<o:p></o:p></span></p> <p class="MsoNormal"><a name="_MailEndCompose"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US"><o:p> </o:p></span></a></p> <div> <div style="border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0cm 0cm 0cm"> <p class="MsoNormal"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> [email protected] [mailto:[email protected]] <b>On Behalf Of </b>Ray Zimmerman<br> <b>Sent:</b> 26. oktober 2015 21:14<br> <b>To:</b> MATPOWER discussion forum <[email protected]><br> <b>Subject:</b> Re: Nonzero shadow prices for non-binding constraints<o:p></o:p></span></p> </div> </div> <p class="MsoNormal"><o:p> </o:p></p> <div> <p class="MsoNormal">Good question. Because the interior point solver does not explicitly define binding vs. non-binding constraints in the solution, any such classification must be done based on numerical tolerances. I’m pretty sure the intent of that code was to only zero out the shadow prices (i.e. mark the constraint as non-binding) if both the primal and dual solutions indicate that the constraint is not “active”. Not a very rigorous answer, I know, but it was an attempt at being conservative in marking a constraint as non-binding. If the shadow price is significant, then the constraint is still having an effect on the solution, even if the constraint value is not at the bound.<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">Hope this helps,<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal"> Ray<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <p class="MsoNormal"><o:p> </o:p></p> <div> <p class="MsoNormal"><o:p> </o:p></p> <div> <blockquote style="margin-top:5.0pt;margin-bottom:5.0pt"> <div> <p class="MsoNormal">On Oct 26, 2015, at 6:45 AM, Camille Hamon <<a href="mailto:[email protected]";>[email protected]</a>> wrote:<o:p></o:p></p> </div> <p class="MsoNormal"><o:p> </o:p></p> <div> <div> <p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif">Dear all,<o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"> <o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">I am implementing a search for the maximum loadability limit using an OPF with dispatchable loads in Matpower. To model PV buses, I set generator voltage set points as upper limits for the voltage magnitude at generator buses and set the lower voltage limits at a low number (0.1 for example). Also, I keep the reactive power upper limit. When the OPF has run, I check the shadow prices of the upper bound constraints on voltage magnitudes to identify the generators which are still PV buses and the generators which have become PQ buses. During my testing, I ran into cases where the upper bounds voltage constraints are non-binding, but the corresponding shadow prices are nonzero. I believe it is due to the different thresholds used to check binding constraints. In particular, as far as I understood, the shadow prices of all non-binding constraints are set to zero by the following lines (mips solver):</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">%% zero out multipliers on non-binding constraints</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">mu(h < -opt.feastol & mu < mu_threshold) = 0;</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">My understanding is that the first condition h < -opt.feastol checks for non-bining constraints (deviations from the limit h is larger than the tolerance opt.feastol). I do not understand the second condition, however (mu < mu_threshold), which checks for the magnitude of the Lagrange multipliers. In my results, I have the case where h for a constraint is indeed larger than the opt.feastol threshold (h=7 e-4) but the corresponding Lagrange multiplier is larger than the mu_threshold (mu=2 e-4) so that the shadow price is not set to zero. Note also that the product of h and mu for the corresponding constraint is small enough for the last MIPS iteration to conclude on convergence (i.e. the relative complementarity constraint is satisfied).<span class="apple-converted-space"> </span></span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">My question would then be:</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div style="margin-left:36.0pt"> <p class="MsoNormal" style="text-indent:-18.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">-</span><span lang="EN-US" style="font-size:7.0pt"> <span class="apple-converted-space"> </span></span><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">What is the meaning of having the second check (mu < mu_threshold) in identifying the non-binding constraints?</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Kind regards,</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Camille Hamon</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Post doc</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Department of Electrical Power Engeneering</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> <div> <p class="MsoNormal"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Norwegian University of Science and Technology (NTNU)</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><o:p></o:p></span></p> </div> </div> </blockquote> </div> <p class="MsoNormal"><o:p> </o:p></p> </div> </div> </body> </html> --
