Question

Formulate the optimal hydrothermal scheduling problem considering the inequality constraints on the thermal generation and water storage employing penalty functions. Find out the necessary equations and gradient vector to solve the problem.

Explain clearly about above mentioned topic.

Answer 1

ptimal stheduling_o _hydso- Thermal Spitem : operahion of and themal planty j sptein having both hydro complex Mare cost hydro plant have hegligble pera onstra perate under available for yro generabion in Tequired aut are to fCuater thuy belons Eo name re problem cast of holro- thermal sptem ofPLAmising plant unolcr te C storage the real mode F Optimi salion. the operating rimising mint vteweal Can he pilel cost termal cvater auaiabills the one Conjoaint and inlo fos hyolo gven pentod operation generation Over a JCcuater unyloe c short and long term) : ReyanvoLr Cstoragex laad demand contrauized ba d. t어 Hydro pla nt themal Plant cischarge) LL uater ake of simplicity and the FOT undergtanding solution techrue the Problem omlation and yated trough a are Simipced hydrD hermal syster g

hydro andhemal Thus stem coniy f one plantpplyng and repered to optcmization poer generalton ac caunted for by the loss fomula pocuer to a cenbralized laad ay fundamental cof be conbol vanable, with tarymsa stem Camed out wi eal CU KK CP + 2 - BT P BetH ( GT Mathemalical fmulation: period operaon Certain FoT a Cone ear day/depending assiemed that menith one ohe / the requirement), it stora ge f hydso- reyervotr at the begining Specipied and water inoco to 7ervoir Capter accounting tor irnqabion demand pon cnd the perod and are the ye and load knocon are on colth Compiete certainiky PToblem diseharge cost of ermal hime functhons Cdetorminijia cae). The to Cater ctthe Minimize detcTnine te a to generation abjecbive funebion c'C PaTCdt conybrainty min CT fol locng loaddemand under the ) Meebing Por CPaC - RC-c the

This is called pocser lbalanee eauation avaiabig xCTxCoj ct uhere JCt j the coater nplow Coate) water ytorage and uote end f xCoxC at the pecipied beginning and at are storage the ophimizahion interval the funetion cate Storage Corhead), hyro diseharge and can be handled convenently optinizaticn hterval T MSub intervels each subiterval it problem disretization. The Supdivided into each time assimed length that all The OVer AT pixed in valie. vaniables remain as. the posed noco proelem min AT CC min M21 follocoing Constraintg the unden euation balance pocver + where thernal generation in the far M terval hydlro-generation in tke hterval aymiSsion loss h ihe M interval

load demand in the Mth intcnoal. Continuity euabien CM АT- 0 where cvater sturage at the end of mth interva cater inflocCTate) in the the ICM M. interval discharge Crabe) in te interval water above The Can be Cntten CM-1o wbt wr rate stora ge cohere xIM AT hage uncs Xo and x are the pecipied Can storages at the optimization ihteva andcrd p the anysubinterval begi Gi HdDenerahion etpresred ay ho Ct o5e Cx+x GH where ho baic cwater head Chead correpond ho to dead Jtorage). pactor to coitk Corechion vanation aater head for head auccount storge Ef fective diharge C cwater non to n hyelro-generat digeharge needed load)

above problem formulation, it j e Ph the convienient to subihtervoy cicept while hydro genieratons, hermal generationy and breated that aater Subinterva i hoon anter dicha vanable independ eht choase one af tora g in all dependent vaniabley. The fat dijcharge in one aгe Subihtervad water fthe vaniale depencknt a belocs Alding tke folloaing aabon xMx (+. cwater auailabilidg ean khoon Becauye the remaining be specipicd independenty anad can be determined from thii euahion one dependknt chesen For Vanable. Cand therepore depend ent Corite a Convencence can cohuch Varicble fos Techniaue:- Solution Scved here sing problem programmingtechniaue i'n coniineta Airyt order Lagragian Lis fomulated he The hon-inear coith gradient method the he augmenting csit eaualiby conjtvaun f ean's thraugh Lagrange mulbipuery Caial vanabl cost punchion of ean D ano and nTRuj.

M -CcC)-Ct ) + 2 C- ho Cit oseCxtx Mel daual obtaihed beuobing The Vantabe are the partial derivatives the lagralanc to zaro coith, repert to yelding the folbadpndent variabl dc C p the equahony m C1 he 번일e 5hoe ce-e- Cntl Mtl Mt WEWwe ho dual vanable for any ub thteval ma tollocs froM The obtained a be U) ootaun cbtain J2 ean egnis trom vate li cbtain from and 애her een ean gradent Lagngton ciryp vector i veh yhe The pagtal dervative to the indeperdent vanaHe h L fi+ose Cax" +- 13