Using Load Research Data to Model Weather Response

Similar documents
The Link Between Load Research and Forecasting

Scientific research on the biological value of olive oil

ENERGY CONTENT OF BARLEY

WSU Tree Fruit Research and Extension Center, Wenatchee (509) ext. 265;

2. Hubs and authorities, a more detailed evaluation of the importance of Web pages using a variant of

Optimizing Metam Sodium Fumigation in Fine-Textured Soils

Review TEACHING FOR GENERALIZATION & MAINTENANCE

Reducing the Risk. Logic Model

Analytic hierarchy process-based recreational sports events development strategy research

Not for Citation or Publication Without Consent of the Author

Invasive Pneumococcal Disease Quarterly Report. July September 2017

The Measurement of Interviewer Variance

P AND K IN POTATOES. Donald A Horneck Oregon State University Extension Service

Check your understanding 3

Chapter II. THE PREVALENCE METHOD John Bongaarts*

Clinical Study Report Synopsis Drug Substance Naloxegol Study Code D3820C00018 Edition Number 1 Date 01 February 2013 EudraCT Number

2 nd Properties of the Exponential Functions

Chapter. Lesson. arrays. as skip-counting. multiplication

HEMOGLOBIN STANDARDS*

Integrated Biological Control of Woolly Apple Aphid

CheckMate 153: Randomized Results of Continuous vs 1-Year Fixed-Duration Nivolumab in Patients With Advanced Non-Small Cell Lung Cancer

Article from: Risks & Rewards. February 2010 Issue 55

Time trends in repeated spirometry in children

Using Paclobutrazol to Suppress Inflorescence Height of Potted Phalaenopsis Orchids

Math 254 Calculus Exam 1 Review Three-Dimensional Coordinate System Vectors The Dot Product

Chapter 02 Crime-Scene Investigation and Evidence Collection

Single-Molecule Studies of Unlabelled Full-Length p53 Protein Binding to DNA

XII. HIV/AIDS. Knowledge about HIV Transmission and Misconceptions about HIV

Teacher motivational strategies and student self-determination in physical education

Introduction. Lance Baumgard. Introduction con t. Research Emphasis at AZ. Teaching and Advising. Research Emphasis at ISU 4/29/2010

Input from external experts and manufacturer on the 2 nd draft project plan Stool DNA testing for early detection of colorectal cancer

JOB DESCRIPTION. Volunteer Student Teacher. Warwick in Africa Programme. Warwick in Africa Programme Director

EVALUATION OF DIFFERENT COPPER SOURCES AS A GROWTH PROMOTER IN SWINE FINISHING DIETS 1

Optimisation of diets for Atlantic cod (Gadus morhua) broodstock: effect of arachidonic acid on egg & larval quality

STATISTICAL DATA ANALYSIS IN EXCEL

Cattle Producer s Library

Comparison of three simple methods for the

Interrelations of Age, Visual Acurty, and Cognitive Functioning

CLPNA Pressure Ulcers ecourse: Module 5.3 Quiz I page 1

3. DRINKING WATER INTAKE BACKGROUND KEY GENERAL POPULATION STUDIES ON DRINKING WATER INTAKE RELEVANT GENERAL POPULATION

THE EVALUATION OF DEHULLED CANOLA MEAL IN THE DIETS OF GROWING AND FINISHING PIGS

Invasive Pneumococcal Disease Quarterly Report July September 2018

The step method: A new adaptive psychophysical procedure

SUPPLEMENTARY INFORMATION

There has been little systematic

Chapter 5: The peripheral nervous system Learning activity suggested answers

Finite-Dimensional Linear Algebra Errata for the first printing

Diabetes affects 29 million Americans, imposing a substantial

Summary. Effect evaluation of the Rehabilitation of Drug-Addicted Offenders Act (SOV)

BIOSTATISTICS. Lecture 1 Data Presentation and Descriptive Statistics. dr. Petr Nazarov

Extraction and Some Functional Properties of Protein Extract from Rice Bran

Wireless Hearing Products

Maximize Your Genetic Return. Find your Genetic Solution with Boviteq West

3/10/ Energy metabolism o How to best supply energy to the pig o How the pig uses energy for growth

EFFECT OF DIETARY ENZYME ON PERFORMANCE OF WEANLING PIGS

BMI and Mortality: Results From a National Longitudinal Study of Canadian Adults

Relationship between food availability, glycerol and glycogen levels in lowtemperature challenged rainbow smelt Osmerus mordax

Dissociable effects of the implicit and explicit memory systems on learning control of reaching

SEIZURES AND EPILEPSY

Addendum to the Evidence Review Group Report on Aripiprazole for the treatment of schizophrenia in adolescents (aged years)

Mecadox. Improves pig performance in a wide range of health and growing conditions. (Carbadox) Talk With a Phibro Expert:

Effect of Somatic Cell Count on Milk Yield in Different Parities and Stages of Lactation in Holstein Cows of Iran

Assessment of Depression in Multiple Sclerosis. Validity of Including Somatic Items on the Beck Depression Inventory II

MANAGING ANTHRACNOSE BLIGHT AND BOTRYOSPHAERIA AND PHOMOPSIS CANKERS OF WALNUT PART 1: BOTRYOSPHAERIACEAE AND PHOMOPSIS CANKERS OF WALNUT

which, in case the fertility rates are specific for five age groups, becomes f =

AR Rice Performance Trials (ARPT) Color as a Quality Indicator. Functional Property Analyses. Cause of Chalkiness in Rice Kernels

Paper-based skin patch for the diagnostic screening of cystic fibrosis

Copy Number ID2 MYCN ID2 MYCN. Copy Number MYCN DDX1 ID2 KIDINS220 MBOAT2 ID2

Analysis of Regulatory of Interrelated Activity of Hepatocyte and Hepatitis B Viruses

Effects of age, density, and seasonality on molt pattern in the mammal genus (Peromyscus)

Outline. AMH as a diagnostic marker for cryptorchidism 2/4/2018. Use of. in diagnosing ovarian tumors and cryptorchidism. Anti-Müllerian hormone (AMH)

EE247 Lecture 4. EECS 247 Lecture 4: Filters 2005 H.K. Page 1. This Lecture

DIFFERENTIAL REINFORCEMENT OF VOCAL DURATION1

PNEUMOVAX 23 is recommended by the CDC for all your appropriate adult patients at increased risk for pneumococcal disease 1,2 :

WORKSHOP FOR SYRIA. A SHORT TERM PROJECT A Collaborative Map proposal Al Moadamyeh, Syria

Will All Americans Become Overweight or Obese? Estimating the Progression and Cost of the US Obesity Epidemic

Geographical influence on digit ratio (2D:4D): a case study of Andoni and Ikwerre ethnic groups in Niger delta, Nigeria.

Table 1. Sequence and rates of insecticide sprays in experimental plots of apples, Columbus, Ohio, Treatment

1980, 133, NUMBER 4 (WINTER 1980) UNIVERSITY OF NOTRE DAME. be viewed as a response and defined a priori by

The Effects of Diet Particle Size on Animal Performance

SOME MECHANISTIC CONCEPTS IN ELECTROPHILIC ADDITION REACTIONS TO C=C BONDS

Appendix J Environmental Justice Populations

The diagnosis of autism and Asperger syndrome: findings from a survey of 770 families

SAMPLE YEAR 6 MASTERING THE MATHEMATICS CURRICULUM. Written by Laura Sumner. Including CD-ROM for whiteboard use or printing

URINARY incontinence is an important and common

A LAYOUT-AWARE APPROACH FOR IMPROVING LOCALIZED SWITCHING TO DETECT HARDWARE TROJANS IN INTEGRATED CIRCUITS

Feeding state and age dependent changes in melaninconcentrating hormone expression in the hypothalamus of broiler chickens

CHOICE BETWEEN CONCURRENT SCHEDULES' RONALD L. MENLOVE2, MARILYNNE MOFFITT, AND CHARLES P. SHIMP

Goal: Evaluate plant health effects while suppressing dollar spot and brown patch

5 WAYS VCARE? What is GLAUCOMA. has earned the highest MEDICAID? Are you on. Are you at Risk? Viva Medicare Plus. to Avoid Hospital Readmissions

CONCEPT IDENTIFICATION BY SCHIZOPHRENIC AND NORMAL SUBJECTS AS A FUNCTION OF PROBLEM COMPLEXITY AND RELEVANCE OF SOCIAL CUES 1

FHWA/TX lF IMPACT OF AGGREGATE GRADATION AND TYPE ON ASPHALT MIXTURE CHARACTERISTICS. Research Report 1158-lF. Final

SROC Curve. S. D. Walter McMaster University, Hamilton, Ontario, Canada. Petra Macaskill University of Sydney, NSW, Australia INTRODUCTION

Fertility in Norwegian testicular cancer patients

Simulating the Effect of Exercise on Urea Clearance in

What s Going On? The Question of Time Trends in Autism

Factors affecting orthodontists management of the retention phase

RADIATION RESEARCH 158, (2002) /02 $ by Radiation Research Society. All rights of reproduction in any form reserved.

Quantifying perceived impact of scientific publications

Thebiotutor.com A2 Biology OCR Unit F215: Control, genomes and environment Module 1.2 Meiosis and variation Answers

Transcription:

30 25 20 15 10-5 1,950 2,025 2,100 2,175 2,250 2,325 2,400 2,475 2,550 2,625 2,700 2,775 2,850 2,925 3,000 3,075 3,150 3,225 3,300 3,375 3,450 3,525 3,600 MW 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Frequency Cumultive Frequency Usg Lod Reserch Dt Model Wer Response Sturt McMenm EFG Meetg My 3-4, 2007 2005, Itron Inc. Frequency Cumultive Freq

Overview 1. Fncil ecstg models generte ecsts monthly sles. y estimted usg billg dt. Sles = F m ( HDD,CDD, EconVrs, Pr ices ) m m 2. HDD CDD vribles ten defed sgle brekpot, exmple CDD65. 3. System lod dt tells us tht reltionship between wer lod non-ler tht not ll degrees eqully powerful. 4. Lod reserch dt cn be used see th t clss level construct multi-prt DD vribles tht improve models. 5. Proly constructed models cn n be used :. Estimte models tht expl hricl sles vritions b. Clendrize: Estimte clendr month sles by clss c. Normlize: Estimte cycle clendr month sles norml wer d. Estimte unbilled sles e. Construct vrince nlys m m 2006, Itron Inc. 2

Wht Model A model model n n eqution eqution tht tht summrizes summrizes reltionships reltionships dt. dt. coefficients coefficients eqution eqution summrize summrize level level dt dt verge verge impct impct chnge chnge drivg drivg fcrs fcrs ( ( X s) X s) on on vrible vrible beg beg expled. expled. Sttticl Sttticl methods methods fd fd prmeters prmeters ( ( b) b) tht tht mke mke errors errors (e) (e) smll. smll. When When ler ler regression regression used, used, it it importnt importnt construct construct X vribles vribles so so tht tht reltionship reltionship between between Y X ler. ler. Y Y = + b X + t t e t X 2006, Itron Inc. 3

Wht We Lern from Net System Lod Gs Dy Sendout (MCF) Ech Ech Pot Pot One One Dy Dy Dt Dt from from 2000-2004 2000-2004 Weekdys Weekdys Blue Blue Sturdys Sturdys Ornge Ornge Sundys Sundys Red Red Holidys Holidys -- -- Green Green Dily Zone Output (GWh) Dily Averge Temture Averge Gs Dy Temture re re stble stble predictble predictble reltionship reltionship between between dily dily system system lods lods wer. wer. Clendr Clendr impcts impcts lso lso cler. cler. reltionship reltionship between between wer wer system system lod lod nonler. nonler. 2006, Itron Inc. 4

Focus on Electricity Dily Zone Output (GWh) reltionship reltionship between between temture temture lod lod not not sgle sgle stright stright le. le. It It cn cn be be pproximted pproximted by by series series connected connected slopes. slopes. Models Models cn cn estimte estimte reltive reltive slopes slopes ech ech segment. segment. Dily Energy 2002-2004 Dily Averge Temture 2006, Itron Inc. 5

Wht We Lern from Lod Reserch Dt Lod Lod Reserch Reserch Dt. Dt. Ech Ech pot pot one one dy. dy. Dt Dt 2003 2003 Res SCI LCI Lod Lod Reserch Reserch dt dt cn cn help help clrify clrify how how wer wer effects effects work work t t revenue revenue clss clss level. level. se se dt dt suggest suggest different different HDD HDD CDD CDD triggers triggers different different clsses. clsses. reltive reltive power power degrees degrees ech ech rnge rnge cn cn be be estimted estimted from from se se dt dt used used models models monthly monthly sles. sles. 2006, Itron Inc. 6

Lod Reserch Power Dily Dt Lod Lod reserch reserch dt dt very very powerful powerful understg understg wer wer reltionships. reltionships. re re 30 30 pots pots month, month, strong strong dvntge dvntge over over monthly monthly dt. dt. A dy dy dy dy cn cn be be mtched mtched clenly clenly wer wer tht tht dy. dy. Monthly Monthly sles sles dt dt cn cn be be mtched mtched pproximtely pproximtely dys dys wer wer bsed bsed on on cycle cycle dtes. dtes. re re more more wer wer vrition vrition t t dily dily level. level. Billg Billg month month wer wer dt dt ggregted ggregted weighted weighted over over roughly roughly 60 60 clendr clendr dys dys tht tht impct impct sles sles billg billg cycle. cycle. extremes extremes lost lost both both X Y directions. directions. 2003 2003 Lod Lod Reserch Reserch Dt. Dt. Ech Ech pot pot one one dy. dy. 2003 2003 Billg Billg Dt. Dt. Ech Ech pot pot one one month. month. 2006, Itron Inc. 7

Inmtion from Neurl Network Dily Energy (GWh) Dily Dily energy energy from from lod lod reserch reserch hourly hourly priles priles cn cn be be used used estimte estimte neurl neurl network network models. models. Dily Averge Temture Neurl Neurl network network models models flexible flexible nonler nonler models models tht tht work work well well th th type type dily dily dt. dt. denergy/dtemp (Gwh/deg) Estimted Estimted neurl neurl network network models models hve hve derivtives derivtives respect respect X vribles. vribles. A simple simple model model Averge Averge Temtures Temtures gives gives derivtives derivtives shown shown here. here. An extr CDD dds 5 GWh An extr HDD Blnce dds.7 Pot t GWh 60 Dily Averge Temture 2006, Itron Inc. 8

Cpped vs Uncpped DD Vribles Sles slope = 10 slope = 25 slope = 15 Uncpped Exmple CDD60 = Mx(T-60,0) CDD70 = Mx(T-70, 0) Sles = + 10 CDD60 + 15 CDD70 60 70 Avg Temp Sles slope = 10 slope = 25 Cpped Exmple CDD60Cp10 = M(Mx(T-60,0),10) CDD70 = Mx(T-70, 0) Sles = + 10 CDD60Cp10 + 25 CDD70 60 70 Avg Temp 2006, Itron Inc. 9

Estimtion Multiple CDD & HDD Vribles Uncpped Dily Dily energy energy from from lod lod reserch reserch hourly hourly priles priles cn cn be be used used estimte estimte reltive reltive strength strength CDD CDD HDD HDD vribles. vribles. Models Models cpped cpped uncpped uncpped vribles vribles sme. sme. Predicted Predicted vlues vlues ll ll residul residul stttics stttics identicl. identicl. However However coefficient coefficient strd strd errors errors t-stts t-stts stronger stronger cpped. cpped. Cpped Accumulte Tke Differences 2006, Itron Inc. 10

Constructg Multi Prt (Sple) Vribles lod lod reserch reserch slope slope estimtes estimtes cn cn be be used used construct construct multi-prt multi-prt hetg hetg coolg coolg vribles. vribles. We We cll cll se se HDDSple HDDSple CDDSple. CDDSple. HDD Sple.107*Wthr.HDD60 +.718*Wthr.HDD55 +.175*Wthr.HDD50 -.213*Wthr.HDD30 CDD Sple.061*Wthr.CDD60 +.675*Wthr.CDD65 +.058*Wthr.CDD70 +.206*Wthr.CDD75 2006, Itron Inc. 11

How Regression Sees Dily Dt Residentil Residentil Lod Lod Reserch Reserch Dt. Dt. Ech Ech pot pot one one dy. dy. Dt Dt 2003 2003 Dily Energy (GWh) Dily Energy (GWh) Dily Energy (GWh) Multi Multi prt prt sples sples turn turn nonler nonler reltionship reltionship two two ler ler reltionships. reltionships. Th Th very very cler cler dily dily dt dt becuse becuse ech ech dy dy hs hs CDD CDD or or HDD HDD but but not not both. both. HDD Sple CDD Sple 2006, Itron Inc. 12

How Defe Averge Temture July 8 Avg24 = 88.0 Hi/Low = 88.5 July 10 Avg24 = 82.4 Hi/Low = 87 Averge Mx & M hs been dustry strd decdes. Averge hourly vlues will work better on dys where re sudden temture chnges, such s cused by fternoon thundersrms summer. 2006, Itron Inc. 13

Defition Averge Temture Actul Dily Energy (GWh) Act Act vs. vs. Pred Pred usg usg Avg Avg 24 24 hourly hourly vlues vlues Dily Dily energy energy from from lod lod reserch reserch hourly hourly priles priles more more strongly strongly relted relted verge verge temture temture thn thn (Mx+M)/2. (Mx+M)/2. Actul Dily Energy (GWh) Act Act vs. vs. Pred Pred usg usg (Mx+M)/2 (Mx+M)/2 Predicted Dily Energy (GWh) Neurl Neurl Network Network Actul Actul Predicted Predicted vlues vlues usg usg dily dily verge verge temture temture (verge (verge 24 24 hourly hourly vlues) vlues) MAPE MAPE = = 3.45% 3.45% Predicted Dily Energy (GWh) Neurl Neurl Network Network Actul Actul Predicted Predicted vlues vlues usg usg verge verge Mx Mx M M temtures. temtures. MAPE MAPE = = 3.87% 3.87% 2006, Itron Inc. 14

Sesonl Wer Response System Lod Dily Zone Output (GWh) Dily Zone Output (GWh) Dt Dily Energy 2002-2004 Jn July Hot wer hs bigger impcts lter summer months thn sprg months. Dily Zone Output Dily Zone Output My Th suggests models slopes tht vry by month or seson. Oct Averge Temture Averge Temture 2006, Itron Inc. 15

Sesonl Response from Lod Reserch Res Dily Energy (GWh) Res Dily Energy (GWh) July & Aug Lod Lod reserch reserch dt dt supports supports sesonl sesonl differences differences system system lod lod dt. dt. Jn & Feb Res Dily Energy (GWh) Res Dily Energy (GWh) Apr & My Oct & Nov Averge Temture Averge Temture 2006, Itron Inc. 16

Trnsition Monthly Dt 1. lessons lerned nlys from dily lod reserch dt cn be used strengn models tht use monthly dt. 2. followg models monthly billg dt. monthly dt billg cycles, so it necessry construct pproprite wer vribles. 3. Averge temture (verge hourly vlues) computed ech dy n HDD CDD vlues ccumulted cross dys ech cycle. cycle vlues n weighted cross cycles get monthly HDD CDD vlues. 4. gol estimte models tht:. Use multi-prt HDD CDD vribles (sples) constructed usg prmeters from lod reserch dt nlys. b. Allow HDD CDD slopes vry by month/seson. c. Cn be used wer normliztion, clendr month, unbilled clcs. 2006, Itron Inc. 17

Detils on Monthly Wer Vribles 1. Compute verge temture by dy 2. Compute CDD60, CDD65,... billg dys ech cycle. 3. Compute weighted verge CDD60, CDD65,..., billg dys usg pproprite weights ech cycle. 4. Compute CDD Sple HDD Sple vribles from CDD60, CDD65,... 5. Divide CDDSple HDDSple by number billg dys. Th will look like dily dt. For exmple, vlue 10 dictes tht verge dy month hd 10 degree dys. 6. If pproprite do th ll THI DD coolg where THI cludes dily temture humidity. 2006, Itron Inc. 18

How Regression Sees Monthly Dt UPC Per Dy (KWh) Residentil Residentil Billg Billg Dt. Dt. Ech Ech pot pot one one month. month. Dt Dt 1991 1991 2006. 2006. Cycle Averge Temture UPC Per Dy (KWh) CDD Sple CDD Sple Reltionships Reltionships less less cler cler monthly monthly dt. dt. Mny Mny months months hve hve both both HDD HDD CDD CDD over over cycle cycle dys. dys. UPC Per Dy (KWh) Ech Ech pot pot billg billg month. month. Color Color codg codg by by seson seson Blue Blue = = Wter, Wter, Green Green = = Sprg Sprg Red Red = = Summer, Summer, Ornge Ornge = = Fll Fll HDD Sple CDD Sple HDD Sple 2006, Itron Inc. 19

Models Monthly Dt Dependent Dependent vrible vrible monthly monthly use use cusmer cusmer billg billg dy. dy. Addg Addg Monthly Monthly bries bries ccounts ccounts lightg lightg or or sesonl sesonl effects. effects. 2006, Itron Inc. 20

Models Sesonl CDD Slopes Dependent Dependent vrible vrible monthly monthly use use cusmer cusmer billg billg dy. dy. Sesonl Sesonl CDD CDD Sple Sple slopes slopes constent constent expecttions. expecttions. CDD CDD effects effects lower lower Sprg Sprg months. months. 2006, Itron Inc. 21

Wer Effects Chnge Over Time Wer slopes chnge over time Cusmer growth implies more hetg/coolg equipment Chnges hetg/coolg sturtion levels chnge CDD HDD slopes cusmer Chnges hetg/coolg efficiency chnge CDD HDD slopes cusmer Dt Dt from from 2004 2004 2005 2005 Dt Dt from from erly erly 1990 s 1990 s For For residentil residentil cusmers, cusmers, response response hot hot wer wer hs hs cresed cresed significntly significntly between between erly erly 1990 s 1990 s 2005 2005 on on cusmer cusmer bs. bs. Th Th constent constent ntionwide ntionwide trend trend wrd wrd lrger, lrger, more more completely completely cooled cooled homes. homes. Models Models tht tht use use hricl hricl dt dt must must ccount ccount th th growth growth wer wer sensitivity, sensitivity, project project contued contued chnges chnges future. future. 2006, Itron Inc. 22

Models Sesonl Slope Trends Dependent Dependent vrible vrible monthly monthly use use cusmer cusmer billg billg dy. dy. Trend Trend vrible vrible creses creses by by 1 1 ech ech yer yer 0 0 2005. 2005. So So estimted estimted slopes slopes 2005 2005 slopes. slopes. Slopes Slopes erlier erlier yers yers reduced reduced by by.066.066 KWh/degree KWh/degree yer. yer. For For exmple, exmple, July July slope slope 1.64 1.64 1993 1993 2.43 2.43 1995. 1995. 2006, Itron Inc. 23

Models AR1 Residul Addg Addg n n AR1 AR1 term term improves improves overll overll fit. fit. coefficients coefficients not not ltered ltered significntly. significntly. Coefficients Coefficients out out AR1 AR1 2006, Itron Inc. 24

Actul & Predicted Vlues Fl Model Monthly Use Per Cusmer Per Billg Dy (KWh) Actul Actul vlues vlues Red Red Predicted Predicted vlues vlues Blue Blue Vlues Vlues Monthly Monthly Use Use Per Per Cusmer Cusmer Per Per Billg Billg Dy Dy 2006, Itron Inc. 25

Usg Structured HDD, CDD SAE Approch With th pproch, structured vribles constructed hetg coolg uses. CDDSple HDDSple vribles cn be used usge equtions. For exmple: XCool = CoolIndex CoolUse CoolIndex t y t = rmleff y Weight Type t Type St St Type y Type by Eff Eff Type y Type by CoolUse t = P P t 98 Inc Inc t 98 b HHSize HHSize t 98 c CDDSple CDDSple t 98 2006, Itron Inc. 26

Buildg Models on Per Dy Bs 1. By buildg models on dy bs, y cn esily be used : -- Budget ecstg on clendr month or cycle bs -- Wer normliztion booked billed sles -- Clendriztion billed sles -- Estimtion unbilled energy 2. Estimte models use billg dy s Y vrible (or use cusmer billg dy). 3. Estimte models CDD billg dy HDD billg dy s X vribles. 4. Evlute models pproprite wer -- Clendr month CDD/HDD clendr dy -- Unbilled corner CDD or HDD unbilled dy n multiply by number dys 5. Usg dy models mkes billg dt look like dily lod reserch dt... except it still just one number month. 2006, Itron Inc. 27

Conclusions Lod reserch dt cn be used refe improve models wer response. Improved models will do better job > Forecstg > Wer normliztion > Estimtion clendr month energy > Estimtion unbilled energy > Vrince nlys 2006, Itron Inc. 28