tutorial 4: transmission losses - simplexnodal.com · tutorial 4: transmission losses page 167...

ElectricityMarketTutorialusingtheSimplexNodalapp 20-May-2020

Tutorial4:TransmissionLosses Page163

Tutorial 4: Transmission Losses This tutorial explains how transmission losses areincluded in the electricity market model and theimpactthatincludingtransmissionlosseshasontheresults.

Modelling transmission losses

Transmissionlosses=PowerlostasheatWhen electrical power flows through a conductorthemovementofelectronscausestheconductortoheatup.Themorepowerthatflows,themoreheat.If a conductor gets too hot then its structure ispermanently altered… the maximum flow limit issettopreventthisfromhappening.Power that is lost as heat is power that is nottransmitted to the bus at the other end of thebranch. In themodel, this lostpower isreferredtoasbranchlosses.Ourmarket model uses the simplified power flowequation to calculate a branch’s power flow. Thebranch loss calculation corresponding to thesimplified power flow equation is shown inEquation9.



𝐿𝑜𝑠𝑠𝑒𝑠!"#$%& = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒!"#$%& × 𝑀𝑊𝐹𝑙𝑜𝑤!"#$%&' Equation 9: Branch loss calculation for simplified AC powerflow

ModellingbranchlossesBecause the branch loss calculation in Equation 9includes a quadratic term, it represents a curve(specificallyaparabola)and therefore it cannotbedirectly included in the LP (Linear Programming)model used by the simplex algorithm because thismodel requires its constraints to be linear, i.e.,straightlinesnotcurves.Tomeettherequirementforlinearconstraints,theloss curve ismodelled as a series of straight lines.Toseethis,buildatwo-bus,one-branchmodel(tapBus-Bus-Gen-Load-Branch) and solve with the“Include Losses” option set to ON as shown inFigure118.TheresultisshowninFigure119.

Figure118:SettingbranchlossestoON



Figure119:Resultthatdemonstratesbranchlosses

To view the branch loss parabola, double tap thebranch to see its Data Display then tap the “Loss”buttononthetoolbar,asindicatedinFigure120

Figure120:BranchDataDisplaywithLossbuttonindicated

The “Loss” button leads to the “Branch Segments”display in Figure 121, which shows the individuallosssegments in table formandalsoasaplot.Thedisplay also shows the settings that were used tocalculatethesegments.



Figure121:Branchflow-losssegments

In the plot the flow-loss result from the latestsolutionisindicatedbyadotwithacirclearoundit.Thedisplayshowstheindividuallinearsegmentsinblue, with the parabola that they are estimatingshowningrey.Using thepinchgesture tozoomin,you can see a close up of the differences betweenthepiecewisecurveandtheparabola,asshownbythedetailinFigure122.



Figure122:Zoomintoseedetailsofparabolavssegments

Toseeafullscreenversionoftheplot,asshownin

Figure123,tapthe button.

Figure123:Fullscreenflow-losscurve



HowthesegmentsarecalculatedThere are different ways of modelling the branchsegments. The app allows you to investigate theimpactofthefollowingmodellingdecisions:- Howmanybranchsegments- What algorithm is used to determine theendpointsofthesegments

NumberofsegmentsvstotalsegmentsThe number of segments refers to the number ofsegments in each direction, i.e., selecting threesegmentsresultsinthemodelusingthreesegmentsto model flow in the forward direction and threesegmentsinthereversedirection.

SegmentendpointsTo determine the endpoints of the segments, theNew Zealand electricity market uses an algorithmthatminimizestheerrorbetweentheparabolaandthesegments.Thealgorithmisderivedfromaleastsquareserrorfunction.Theappreferstothisasthe“LeastSq”(leastsquares)method.The Singapore electricity market calculates thesegmentsbydividingthemaximumflowintoequaldivisions.Thelosspointattheendofthesegmentsistakendirectlyfromtheparabola. Intheappthis



is referred to as the “Equal Div” (equal divisions)method.Note:Unlessotherwisestated,thetutorialexamplesalluse“LeastSq”andthreesegments.

BranchconstraintstomodellossesAs shown in Equation 10, the branch flow isconstrained to be equal to the sum of the flow ontheindividualflow-losssegments.

𝐹𝑙𝑜𝑤!"#$%& = 3 𝐹𝑙𝑜𝑤()*+)$,-)*+)$,(!"#$%&

Equation10:Branchflowissumofsegmentflows

The “branch flow is sum of segment flows”constraints can be viewed on the Constraintsdisplay,asshowninFigure124.



Figure124:Constraintsdisplay:Branchflowissumofsegmentflows

Becausethesegmentconstraintslinkbranchflowtosegment flows, in order to schedule flow on thebranch the solver must schedule flow on one ormore of the branch segments…. and in order toscheduleflowonabranchsegmentthesolvermustalso schedule the corresponding loss, due to the



constraint shown in Equation 11 which requiresthateveryMWofsegmentflowhasacorrespondingloss, determined by the segment’s loss-flow-ratio.Figure125showsthe“lossforflow”constraintsforoneofthethreesegments.

𝐿𝑜𝑠𝑠()*+)$, = 𝐿𝑜𝑠𝑠𝐹𝑙𝑜𝑤𝑅𝑎𝑡𝑖𝑜()*+)$,× 𝐹𝑙𝑜𝑤()*+)$,

Equation11:Segmentloss-for-flowconstraint

Each segment also has its own flow limit, asdescribedbyEquation12,andshowninFigure126.

𝐹𝑙𝑜𝑤()*+)$, ≤ 𝑀𝑎𝑥𝐹𝑙𝑜𝑤()*+)$,Equation12:Segmentmaximumflowconstraint

This segment flow limit is necessary in order toeffectively model the parabola… without it thesolverwouldschedulealloftheflowonthesegmentwiththelowestloss-flowratio.



Figure 125: Constraints display: Segment flow linked tosegmentloss(onesegmentshown)



Figure126:Constraintsdisplay:Eachbranchsegmenthas itsownlimit

OptionsforassigningbranchlossesThe constraints shown above describe how lossesarecalculated.Therealsoneedstobesomewayofsubtractingtheselossesfromthebranchflow.This



isachievedbyincludingthebranchlossesasanout-flowinthenodebalanceconstraint.Theappofferstwooptionsforapplyingthebranchlossestothenodebalanceconstraint:- Assign half the branch losses to the sendingbusandhalftothereceivingbus.

- Assignallbranchlossestothereceivingbus.The app provides the option of selecting eithermethod,viathesolversettingshowninFigure127.

Figure127:Solversettingforlossallocation

BusconstraintstomodelbranchlossesIfthe“RcvBus”optionforLossLocationisselectedthenabranch’s lossesareassigned to thebus thatreceives power from the branch, and the nodebalanceconstraintisasshowninEquation13.

3𝐵𝑢𝑠./012$ −3𝐵𝑢𝑠./0134,

− 3 𝐿𝑜𝑠𝑠()*+)$,()*+)$,./012$

= 0

Equation13:Nodebalance:Lossesassignedtoreceivingbus



If the “50/50” option is selected then the nodebalanceconstraintisasshowninEquation14.

3𝐵𝑢𝑠./012$ −3𝐵𝑢𝑠./0134,

−0.5 × 3 𝐿𝑜𝑠𝑠()*+)$,()*+)$,./012$./0134,

= 0

Equation14:Nodebalance:Lossesassigned50/50tosendingbusandreceivingbus

Viewed via the app these constraints appear asshowninFigure128andFigure129.Note:Unlessotherwisestatedtheexamplesusethe“Rcv Bus” setting, i.e., losses are assigned to thereceiving end bus. The New Zealand electricitymarket originally assigned dynamic losses 50-50butnowassignsthematthereceivingend.



Figure128:Busconstraintsforlossesatreceivingend



Figure129:Busconstraintsforlossesassigned50/50

Per-unitvaluesandlosscalculationThe calculation of power flow is potentiallycomplicated by the fact that different parts of thepowersystemrunatdifferentvoltages. Inordertobe able to ignore these voltage differences thesimplified power flow equation uses susceptancevalues that have been adjusted so that they take



into account the branch’s nominal voltage. This isreferred to as using per-unit values. As you mayhave noticed, the resistance, reactance andsusceptancevaluesusedbythemodelareallquotedinper-unit,abbreviatedtop.u.,e.g.,seeFigure130.

Figure130:Themodelusesper-unitvalues

Theseper-unitvalueswerecalculatedpriortobeingenteredintotheapp.Per-unitvaluesarecalculatedby dividing the actual values by a base value, e.g.,per-unitresistanceiscalculatedasfollows:Rper-unit=Ractual/RbaseThe base value depends on the voltage. The baseresistance is calculated from a base voltage valueandabasepowervalue…Rbase=Vbase2/Pbase



…where the base voltage is the nominal voltage ofthecomponent,e.g.,forabranchdesignedtorunat220kV,thebasevoltageis220kV.Tocalculatethebaseresistancetherealsoneedstobeabasepowervalue,whichiseffectivelyarbitrarybutthesamevaluemustbeusedconsistentlyinthecalculation of all the per-unit values that areprovidedtoamodel.For a power system the per-unit values arecommonly specified in terms of a 100MVA basepower value. When combined with the typicalvoltages of a power system, the 100MVA basepower value results in per-unit resistance,reactance and susceptance values that are not toobigandnottoosmall.For thepower flowcalculationwhereall flowsarerelative, the fact that the per-unit values arecalculated using a 100MVA base can be ignored.However, to calculate the branch losses the100MVA base will impact the calculations.Therefore,aspartofthebranchlosscalculationtheappdividestheper-unitresistancevalueby100inorder to account for the presence of the 100MVAbaseintheoriginalper-unitcalculation.Ifyouwanttoenterper-unitvaluesthatdidnotusea100MVAbase,andyouwanttomodellosses,then



youwillneedtoconverttheperunitvaluesbeforeenteringthemintotheapp.Forexample,ifyouhadper-unit resistance and reactance values thatwerecalculatedusinga10MVAbasethenyouwouldneedtomultiplytheirvaluesby10beforeusingthemintheapp.

Transmission rentals

CongestionchargesandtransmissionrentalsIn the transmission tutorialwe saw that a bindingbranchcanresultinthetotalpaymentsmadebytheload exceeding the total payments received by thegenerator.Thedifferencebetweentheseamountsisdisplayedasthe$GridvalueontheResultsdisplay.Branch lossescanalsogiverise to$Grid,asshownbytheresultsinFigure131,whichareforthesinglebranch model of Figure 119, where the branch isnotbinding.These$Gridaredueto line lossesandarereferredtoastransmissionrentals.



Figure131:Branchlossesgiverisetotransmissionrentals

Non-parabolic loss model has no transmissionrentalsThe cause of transmission rentals is the parabolicnatureoftherelationshipbetweenbranchflowandbranchlosses.Ifthebranchlossesarenotparabolicthen there are no transmission rentals. We candemonstratethisbyeditingthesinglebranchmodelso that the branch only uses the first flow-losssegment; if the solver only uses the first segment



then the relationship between flow and loss is astraightline.For the line loss example that we already have(Figure 119), the branch segments in Figure 121showthatthefirstsegmenthasamaximumflowof93.03MW. To only use this first segment, edit theload toreduce itsquantity from100MWto90MW.As shown by the result in Figure 132 thetransmissionrentalsarenow$0.

Figure132:Rentals$0whenonlyfirstbranchsegmentused



WhystraightlinelosseshavenolossrentalsWhen only the first branch segment is used, therelationship between flow and losses is a straightline. Although there is a price difference betweenthe load bus and the generation bus this pricedifference only represents the value of the powerthatwasdissipatedaslosses;theloadpaysahigher$/MWpricebecause it is effectivelypaying for thequantity of power that was generated, but onlyreceivingthepowerthatwasdelivered.Overalltheload pays the same amount as the generatorreceivesandthereisnoextrapayment.Becausethebuspricerepresentsthe$/MWcostofthenextMW,with a straight-line loss function the$/MWcostofthenextMWisthesameasthecostofthe scheduledMW, because the losses perMW forthenextMWarethesameasthelossesperMWforthescheduledMW.

WhyparaboliclosseshavelossrentalsThe actual physical losses are parabolic. Hence, ifwe couldmodel a parabola then the $/MW lossesassociatedwiththenextincrementalMWwouldnotbe the same as the perMW losses associatedwiththescheduledMW.



As a crude example… with parabolic losses if ascheduledflowof1MWincursalossthatis10%offlow,thenwecaninferthattheresistanceis0.1perunit, i.e., loss = 0.1 x flow2. An “incremental” 1MWwouldtaketheflowto2MWandincuralossof0.1x22 = 40%. The bus price is set by the cost of theincrementalMW therefore the bus price would besetbasedona40%loss,eventhoughthescheduledloadisonlyincurringa10%loss.Theloadpaysforlosses that are not actually incurred, but whichwouldbeincurredbythe“next”MW.Hencetheloadpaysmorethanthegeneratorreceives.

Howthepiece-wiseparabolacauseslossrentalsTheLPmodelcannot includeaparabola.Butwhenthepiece-wiselinearlossmodelschedulesflowthatuses more than the first flow-loss segment, it willbegin to approximate a parabola and this will bereflectedinthelossrentals.Toseehowthishappens,wewilladjust themodelso that it is using more than the first flow losssegment. We could achieve this by adjusting theload again, but it will be more interesting if weincreasethenumberofflowlosssegments…ontheBranch Segments display for br00, tap the “4”button to increase thenumberofbranchsegmentsfrom 3 to 4. As shown in Figure 133, the display



adjustsimmediatelytoshowthenewsegmentsandwecanseethatgoingtoafour-segmentmodelwillmovethescheduledflowintothesecondsegment.

Figure133:Changeto4segments(notyetre-solved)

The“Segments”buttonishighlightedredtoindicatethat this change has not yet been applied to themodel…thiswillhappenwhenyouleavethedisplayvia theBackbutton.Tap theBackbuttonand then



tap “OK” when you are prompted to apply thechange.Aftersolvingwith4segments,theflow-lossresultisshown in Figure 134 and the corresponding lossrentalsareshowninFigure135.

Figure134:Flow-lossresultaftersolvingwith4segments



Figure 135: Increasing to 4 segments has brought back thelossrentals

This result includes loss rentals because the loadreceives the final 25.25MW of its 90MW via thesecond flow-loss segment. The value of the nextMW,andhencethebusprice,reflects the flow-lossratioofthesecondsegment.Hence,theloadpaysasifallofthe90MWweredeliveredwiththisflow-lossratio.



If all of the 90MW had incurred losses at the rateassociated with the second segment then thepayments made by the load would match thepaymentsreceivedbythegeneration.However, the first 67.42MW is only subject to the0.0152 flow-loss ratio of the first segment,comparedtothe0.0652ofthesecondsegmentthatsettheprice.Thegenerationquantity,andhencetheamountpaid to thegenerator, reflects the fact thatsome of the losseswere incurred at a lower ratio,but the loadpaysataprice thatwassetas ifallofthe generation quantity was subject to the higherflow-loss ratio of the second segment. Hence theloadpaysmorethanthegenerationreceives…withthedifferencereferredtoastransmissionrentals.

Non-physical losses

Non-physicallossesduetonegativepricesWhen branch losses are included in the model,negative prices can lead to non-physical losses. Asexplained in the Spring Washer tutorial, negativepricesoccuratabuswhereadecrease inavailablepowerwouldbenefittheobjectivevalue.Onewaytodecreasetheavailablepoweratabusisviaload.Anotherwayisviabranchlosses.Hence,anegative price at a bus indicates that there is a



benefit to increasing the branch losses assigned tothatbus.

IncreasingbranchlossesWithintheconstraintsofthemodelitispossibletoincreasebranchlossesbecausewhilebranchflowisthesumofthebranchsegmentflows,theonlythingensuring that the “correct” segments are used isthat usually the solver is trying to minimize thebranch losses.Using the “correct” segmentsmeansthat as the scheduled flow increases, its flow-lossratioapproximatesaparabola.When there are negative prices the solver can usebranch segments that maximize the losses. Thescheduled flow-lossno longer follows theparabolaandtheresultinglossesareanover-estimateoftheactual physical losses that would occur for thescheduledflow.The losses that are over-estimated are referred toas non-physical losses. The associated resultschedulesmore generation thanwould actually berequired.

Demonstratingnon-physicallossesTo demonstrate non-physical losses, build thespringwashermodel shown in Figure 136. Leaveall components with their default values except



change the limitonbr01 tobe20MW.Ensure thatbranch losses are set to “Least Squares” and 3segments (via the Branch-Losses display), withlosses assigned at the receiving end of the branch(viaSolveSettings).

Figure136:Springwasherwithlosses

Figure136alsoshowstheresults.This isthesamemodel thatwebuilt in the springwasher example,andagainwehavenegativepricesatbus01.Whatisdifferent is that this time the model includes linelosses and the line losses on br00 are 20.187MW.



This level of losses is not physically realistic on abranchthatisonlytransmitting38.24MW.Thelosscurve in Figure 137 shows how the solver hasachievedthenon-physicallossesonbr00.

Figure137:Springwasherwithnon-physicallossesonbr00

Because the solver can use any of the segmentsincludingthoseinthereversedirections,ithasfullyscheduled segment 1 in the reverse direction,balancingthisflowbyfullyschedulingsegment3in



the forward direction… the solver is schedulingcirculating branch flows in order to increase thelossesassignedtobus01.Whilethescheduledflowof38.24istransferredviasegment2,thelossesarethesumoftheclearedlossesonallthreesegments.

BranchsegmentsasviewedbythesolverAlthough the appdisplays the flow loss curve as aparabola, a graphical view of the constraints is asshowninFigure138.

Figure138:Branchsegmentconstraints

Inthenormalsituation,withnonegativeprices,thesolver is minimising losses and will use thesegmentswith the lowest loss-flow ratio first. Thesolver will schedule flow on segment 1 to itsmaximum,thensegment2andthensegment3…in



thiswaythescheduled flowand losswill track theparabola.When negative prices incentivise the solver tomaximise losses it can utilize circular flows toschedulethesegmentflows“outoforder”,resultingin the total scheduled flow leaving theparabola aswesawinFigure137.

Removalofnon-physicallossesIn an actual electricity market, any non-physicallossesaredetectedinpost-processingandremovedbyre-solvingtheschedulewithsomeofthebranchsegmentsremoved.Theremovalofbranchsegmentsfromthemodelisachieved either via post-processing logic whichremovesthebranchsegmentsandthenre-runsthesolve, or via post-processing logic that re-runs thesolve as a mixed integer solution whereby thesolverisforcedtomakebinarydecisionsregardingwhichsegmentstoincludeinthesolution.

Summary In this section we demonstrated how parabolictransmission lossesaremodelled ina linearsolverbyusingapiece-wiselinearcurve.We explained why the modelling of transmissionlosses gives rise to transmission rentals, whereby



the amount paid by the load is greater than theamountpaidtothegenerator.We demonstrated that negative prices have thepotential tocausethesolvertoproducecirculatingbranchflowsthatresultinnon-physicallosses,andweexplainedhowandwhythishappens.

tutorial 4: transmission losses - simplexnodal.com · tutorial 4: transmission losses page 167...

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