© Strojni{ki vestnik 47(2001)5,199-209                   © Journal of Mechanical Engineering 47(2001)5,199-209
ISSN 0039-2480                                                                                                                                         ISSN 0039-2480
UDK 697.9(569.1):536.2:551.506:004.94                    UDC 697.9(569.1):536.2:551.506:004.94
Strokovni ~lanek (1.04)                                                                                                                        Speciality paper (1.04)
Identifikacija  sezonskih  modelov  temperatur zraka v podro~ju glavnega mesta Sirije Damask
An Identification of Seasonal Models for Air Temperature in the Capital Zone Damascus in Syria
Kamal Skeiker
Ta prispevek predstavlja matematično predstavitev vremenskih parametrov v glavnem mestu Sirije Damask Sezonski modeli, kot alternativa za uporabo urnih vremenskih vrednosti, so bili predlagani in uporabljeni za generiranje vremenskih podatkov za naslednje parametre:
-  temperaturo suhega zraka,
-  temperaturo vlažnega zraka,
-  temperaturo rosisča.
Matematični modeli so bili izdelani za ogrevalno sezono (od novembra do aprila) in za sezono klimatizacije
(od junija do septembra).
© 2001 Strojniški vestnik. Vse pravice pridržane.
(Ključne besede: projektiranje stavb, analize toplotne, modeli matematični, temperature zraka)
This paper presents a mathematical representation of weather parameters in the city of Damascus in Syria. As an alternative to using hourly historical weather data, seasonal models were suggested and used to generate synthetic weather data for the following parameters:
- air dry-bulb temperature,
-  air wet-bulb temperature,
-  air dew-point temperature.
These mathematical models were derived for the heating season (November to April) and for the air-conditioning season (June to September). © 2001 Journal of Mechanical Engineering. All rights reserved. (Keywords: building design, thermal analysis, mathematical models, air temperatures)
0 UVOD
Raba energije različnih panog v Siriji se deli na 42% za zgradbe in kmetijstvo, 41% za transport in 17% za industrijo [1]. Ker je največji del energije uporabljen v zgradbah, je smotrno nadaljevati raziskave na tem področju. Zato so raziskave usmerjene v: razvoj matematičnih modelov za preračune in simuliranje toplotnih sistemov, organizacijo in avtomatizacijo notranjih klimatizacijskih sistemov, oblikovanje pasivnih in aktivnih sončnih toplotnih sistemov, uporabo zunanjih površin za toplotno izolacijo in dvoslojnih oken.
V okviru razvoja matematičnih modelov za izračun in simuliranje toplotnih sistemov za stavbe je bil izdelan in preoblikovan računalniški program
0 INTRODUCTION
The energy consumption for the various sectors in Syria is as follows: 42% for buildings and agriculture, 41% for transportation and 17% for industry [1]. Since the greatest share of the energy consumption is accounted for by buildings, research in this field could prove to be useful, and so research has focused on the following areas: the development of mathematical models for thermal system calculations and simulations; the organization and automation of internal air-conditioning systems; the design of passive and active solar thermal systems; and the use of outer surfaces thermal insulation and double glazed windows.
As part of the development of mathematical models for the calculation and simulation of building thermal systems, the computer program LOS-A0 [2]
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LOS-A0 [2]. Spremenjena verzija CLIMA lahko računa poljubna obdobja v letu. V računalniškem programu CLIMA je izračun neustaljenega prevoda toplote v zgradbi izdelan glede na matematični model z enournim časovnim korakom. Potrebuje urne meteorološke vrednosti za lokalno področje kot del vhodnih podatkov. Tako je računalniški program CLIMA izdelan z urnim testnim referenčnim letom (TRL - RMY Reference Meteorological Year) za glavno mesto Sirije. Podatki TRL bazirajo na dostopnih urnih meteoroloških vrednostih , ki so bili dobljeni na Oddelku za meteorologijo. Posneti so bili na posebno datoteko z vrednostjo 347000 bytov Ti podatki so bili zbrani tudi v prejšnjih fazah dela [3].
Da bi zmanjšali celotni obseg računalniškega programa CLIMA in potrebne programske opreme, smo se odločili predstaviti TRL podatke v obliki matematične predstavitve vremenskih parametrov. V tem okviru so bili izdelani sezonski modeli za naslednje parametre:
- temperaturo suhega zraka;
- temperaturo vlažnega zraka;
- temperaturo rosišča.
Takšni matematični modeli so bili izdelani za ogrevalno sezono (od novembra do aprila) in za sezono klimatizacije (od junija do septembra) v Damasku. Potrebno je omeniti, da so številni avtorji uporabljali ta pristop za njihova specifična področja ([4] in [5]).
1 POSTOPEK IDENTIFIKACIJE MODELA
Ker je oblika grafičnega prikaza podatkovnih točk M(xi,yi) v tej študiji več-polinomska, se bomo osredotočili na modele nelinearne regresije. Splošna oblika polinoma, ki ustreza podatkom je podana v naslednji obliki ([6] do [10]):
was modified and re-organized at an earlier stage of this work. The modified version of CLIMA can calculate optional period during the year. In the CLIMA computer program the calculation of non-stationary heat transfer in a building is conducted according to the adopted mathematical model by using a one-hour time increment, and it requires hourly meteorological data for the locality as a part of its input. Therefore, the CLIMA computer program was provided with an hourly Reference Meteorological Year RMY database for the capital zone in Syria. The RMY database was based on the available hourly meteorological data, which was measured by the Department of Meteorology. It was recorded in a separate file with a size of 347000 byte. This database was also organized in the previous stage of this work [3].
To reduce the size of the CLIMA computer program and its relevant peripheral software we decided to represent the RMY database with a mathematical model of the weather. Therefore, a decision was made to identify seasonal models as an alternative to the use of hourly historical weather data and to generate synthetic weather data instead. Consequently, for this paper seasonal models were suggested for the following weather parameters:
- air dry-bulb temperature;
- air wet-bulb temperature;
- air dew-point temperature.
Such mathematical models were derived for the heating season (November to April) and for the air-conditioning season (June to September) in the Damascus zone. Several other authors have followed this approach for their specific localities ([4] and [5]).
1 METHOD OF MODEL IDENTIFICATION
Since the shape of the graphical outlay of the data points M(xi,yi) for a particular parameter under consideration in the present study suggested a multi-polynomial representation as a strong candidate, it was considered to be worth focusing on the nonlinear regression models. The general form of the polynomial used to fit the data is given by the following relation ([6] to [10]):
Y
b0 + b1 x i + b2 x i2 + ...+ bm x im + ei
2i
mi
(1).
Y je vrednost spremenljivke v i-tem poskusu in x je vrednost neodvisne spremenljivke v i-tem poskusu. Parametri modela so bk (k=0,1,2,...m), in napaka je ei. Yi predstavlja različne vremenske parametre opisane v tem prispevku kot so temperatura suhega zraka J, temperatura vlažnega zraka J in temperatura rosišča
Tehnika, ki jo uporabljamo za določevanje krivulj z najboljšim prilagajanjem, se imenuje metoda najmanjših kvadratov. Naj bo ak cenilec parametra bk. V metodi najmanjših kvadratov so a-ji izbrani tako, da je vsota kvadratov ostankov najmanjša. Z drugimi besedami parameter ak minimizira vrednost ([6] in [7]):
Y denotes the value of the response variable in the ith trial, and xi is the value of the explanatory variable in the ith trial. The parameters of the model are bk (k=0,1,2,.. ..m), and the error term is ei . Yi represents the various weather parameters predicted in this study such as the air dry-bulb temperature (J), air wet-bulb temperature (J ) and air dew-point temperature (J d). The technique used to determine the best-fitting curve was the least-squares method. If a denotes the estimators of the parameters bk, in the least-squares method the values of the a’s that make the sum of the squares of the residuals as small as possible are chosen. In other words, the parameter estimates ak to minimize the quantity ([6] and [7]):
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i=1
Obrazec za najmanjše kvadrate je precej zapleten. Zato se bomo osredotočili na razumevanje načela in pustili programski opremi, da opravi izračune.
Da bi raziskali zmožnot regresijskega modela moramo vpeljati naslednje parametre: - Standardna deviacija: Na splošno je standardna deviacija definirana kot ([6] do [8]):
Yi )2
(2).
The formula for the least-squares estimates is complicated. Therefore, we will be content to un-derstand the principle on which they are based and to let the software do the computations.
To examine the aptness of the regression model for the data at hand, the following statistical parameters need to be determined: - Standard Deviation: In general, the standard deviation is defined as ([6] to [8]):
s (Y)
1
\ n - p i= Vrednost n-p je prostostna stopnja povezana z s2 (Y). Stopnje prostosti so enake podatkom “n” minus “p=m+1”, število b pa moramo oceniti, da ustreza modelu. Standardna deviacija je vedno pozitivna in ima enako enoto kot vrednosti, ki jih obravnavamo. Srednja standardna deviacija sm (Y) je definirana kot ([6] do [8]):

-Yi)2
(3).
The quantity n-p is the degrees of freedom associated with s2(Y). The degrees of freedom equal the size of the data set “n” minus “p=m+1”, the number of b’s we must estimate to fit the model. The standard deviation is always positive and has the same units as the values under consideration. The standard deviation of the mean sm (Y)is defined as ([6] to [8]):
Ker so cenilke ak znane funkcije Y, in ker so napake v Y znane, so lahko napake ak določene z množenjem napak. Matrična algebra uporablja pravila varianc in ocenjuje, da je standardna deviacija cenilk ak podana z naslednjo relacijo [8]:
—    (Y)
n s
(4).
Since the estimators ak are known functions of Y, and the errors in Y are known, the errors in ak may be determined by error propagation. Some matrix algebra using the rules of variances establishes that the standard deviations of the estimators ak are given by the following relation [8]:
[s(a)] = sm(Y)[[X]T[X]\
(5).
[s(a)] predstavlja vektor standardnih deviacij cenilk z “m+1” elementi. [X] pa je matrika spremenljivk z “n” vrsticami in “m+1” kolonami. [X]Tin [X]1 sta transponirana in inverzna matrika matrike [X].
- t preizkus: Da bi pokazali ali je ničta hipoteza pravilna ali napačna in če potrebujemo polinom višjega reda, uporabimo test za bk. Da testiramo ali drži trditev bk =0 lahko uporabimo statistični test ([6] in [8]):
in pravilo odločitve:
[s(a)] denotes the column vector of the standard deviations of the estimators with “m+1” elements, and [X] is the matrix of the explanatory variable with “n” rows and “m+1” columns. [X]T and [X] 1 are the transpose and inverse of the matrix [X], respectively. - t Test: In order to show whether the null hypothesis is true or false, and whether higher power is important, tests for bk are set up in the usual fashion. To test whether or not b = 0 we may use the test statistic ([6] and [8]):
(6)
s (ak)
and the decision rule: p )             ^ bk = 0
(7),
It |  < t (1 - a/2 ; n
|t*|   > t (1 - a/2 ; n - p )             => b ^ 0
kjerčlen (1-a/2) predstavlja koeficient zaupanja               where the term (1 - a/2) represents the confidence
- Koeficient večkratne določenosti in koeficient        coefficient.
večkratne korelacije:  Koeficient večkratne        - Coefficient of multiple determination and coefficient of
določenosti, ki ga označimo z R2, je definiran na sledeč        multiple correlation: The coefficient of multiple determi-
način ([6] do [10]):                                                   n     nation, denoted by R2, is defined as follows ([6] to [10]):
T(yi-Y)2 R2 = 1- i=1                                                                                 (8).

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Izraz y = -!yi predstavlja aritmetično povprečje podatkov R 2 meri proporcionalno zmanjševanje variacije spremenljivke Y v odvisnosti od spremenljivk x. Zavzema vrednosti:
The term y = - Z yi denotes the arithmetic mean of the data values y. R 2 measures the propor-tionate reduction of total variation in Y associated with the use of the set of x variables. It takes the values as:
0<R2<1
Večji je R2 bolj je zmanjšano variiranje spremenljivke Y v odvisnosti od spremenljivke x. R2 zavzema vrednost 1 ko točke padejo direktno na odvisnostno črto, to je, ko je Yi=yi za vse i-je. R2 zavzema vrednost 0, ko so vsi ak =0 (k=1,2,...m), to je, ko je Y= y za vse i-je. V praksi R2 ni 0 in ne 1 ampak nekje med tema mejama.
Koeficient večkratne korelacije, označen z R je pozitivni kvadratni koren iz R2:
The larger R2 is, the more the total reduction of variation in Y caused by introducing the explanatory variable x. R2 takes on the value 1 when all data points fall directly on the fitted response curve, that is, when Yi=y for all i. R2 assumes the value 0 when all a =0 (k=1,2,.. .m), i that is, when Y= y for all i. In practice, R2 is not likely to be 0 or 1, but rather somewhere in between these limits.
The coefficient of multiple correlation, denoted by R, is the positive square root of R2:
R = yR2                                                                     (9).
Vredno je omeniti, da je za vsak R2, ki ni 0 ali 1, velja R2 <|R|, tako da R daje vtis”večje” odvisnosti med x in Y kot kaže vrednost R2. - Relativni odklon: i-ti odklon je razlika med opazovano vrednostjo in napovedano vrednostjo Y. Relativni odkloni od napovedanimi vrednostmi so podani z naslednjo relacijo ([6] do [10]):
Since for any R2 other than 0 or 1, R2 <|R|, R may give the impression of a “closer” relationship between x and Y than does the corresponding R2. - Relative Error: The ith error is the difference be-tween the observed value of the response variable yi and the corresponding predicted value Yi. The relative errors of the predicted values are given by the following relation ([6] to [10]):
yi -Yi
ii
(10).
Srednja relativna napaka je tudi merilo                     The mean relative error is also a fundamental
natančnosti in se lahko izračuna kot:                              measure of accuracy, and can be calculated as follows:
2 PREDLAGANI SEZONSKI MODELI
Za postavitev sezonskih modelov je bilo potrebno identificirati vremenske parametre, ki so bili omenjeni v uvodu. Opaziti je, da so značilnosti vremenskih parametrov spreminjajo od sezone do sezone in jih ne moremo predstaviti z enotni modelom skozi celo leto. Zato smo analizirali samo zimski (ogrevanje) in letni (klimatizacija) letni čas. Čas od 1 novembra do 30 aprila obravnavamo kot ogrevalno sezono, čas od 1 junija do 30 septembra pa kot sezono klimatizacije.
Podatki v tej analizi so vzeti iz urne TRL podatkovne baze za Damask. Podatki za merilno obdobje so podani v [3]. Čeprav so podatki v TRL podatkovni bazi dobljeni iz omejene okolice (mednarodno letališče v Damasku, zemljepisna širina 26°33' N in zemljepisna dolžina 32°36' E, nadmorska višina 608 m), predstavlja to velik del Damaska.
V začetku je bilo preizkušanih veliko modelov. Na koncu je prišel v poštev krivuljno-
n
(11).
2 SUGGESTED SEASONAL MODELS
The aim to set up seasonal models lead to the identification of the weather parameters mentioned in the introduction. It was noticed that the characteristics of the considered weather parameters differed from one season to the next, and can not be represented by single model throughout a particular year. Thus, only winter (heating) and summer (air-conditioning) seasons are analyzed in this paper. The period time from November 1 to April 30 is considered as the heating season, and the period from June 1 to September 30 is considered as the air-conditioning season for this analysis.
The data used in this analysis is extracted from the hourly RMY database for the Damascus zone. The recorded time of this data is given in [3]. Although the data considered in the RMY database had been derived from data collected over a limited area (Damascus International Airport, a latitude of 26o 33’ N and a longitude of 32o 36’ E, with an elevation of 608m above sea level), they represent a large area of Damascus.
Initially, several models were attempted. Finally, one important type of curvilinear response model, the multi-
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linearni model z imenom več polinomski reprezentančni model. Uporabljeni so bili polinomi z dvema spremenljivkama, parametri rezultata pa so definirali stopnjo polinoma, ki je najbolj ustrezal danim pogojem. Najvišji izbrani red polinoma je bil 7, ker nad tem redom nismo opazili pomembnih izboljšav.
Statistični način obdelave je bil predstavljen v prejšnjem poglavju. Z uporabo te metodologije je bil vsak set podatkov nekajkrat analiziran. To pomeni polinome od prvega do sedmega reda. V vsaki analizi smo izdelali naslednje izračune:
- cenilk a;
- standardne deviacije s(Y);
- srednje standardne deviacije sm (Y);
- standardne deviacije cenilk s(a);
- t-test t*;
- koeficient večkratne določenosti R2;
- srednje relativne napake e .
Ker delamo z velikim številom podatkov, je vrednost R2 0,75 ali več primerna [11]. Še več, velika vrednost R2 ni dovolj dobra razen, če je e dovolj majhen. Moderna programska oprema je bila uporabljena za te zahtevne izračune [12]. Ta oprema nam omogoča izračune ne da bi poznali vse podrobnosti. S korelacijo podatkov smo ugotovili tisti model, ki najbolje predstavlja podatke. Ta model je tudi izbran za objavo v tem prispevku.
Matematični modeli, ki predstavljajo v uvodu opisane vremenske parametre, so podani v naslednjih poglavjih. Dve spremenljivki sta t1 in t. t1 je čas v urah in zavzema vrednosti 1 (prva ura prvega dne v novembru) do 4344 (24 ura 30 dne aprila) za ogrevalno sezono. t1 zavzema vrednost 1 (prva ura prvega dne v juniju) do 2928 (24 ura 30 dne v septembru) za sezono klimatizacije. Nadalje t predstavlja uro v dnevu in zavzema vrednosti od 1 do 24.
2.1 Temperatura suhega zraka
Dobljeni so modeli za napoved temperature suhega zraka za poljubni čas med ogrevalno sezono in sezono klimatizacije. Ti modeli so izraženi z naslednjo glavno zvezo:
polynomial representation model, was considered. Thus, polynomials with two explanatory variables were attempted and the statistical parameters of the results dictated the degree of the multi-polynomial that was best for the particular parameter under consideration The highest degree of the multi-polynomial is taken as 7 since no appreciable improvements were noticed in the values computed by the resulting models when it was increased beyond this value. The statistical approach used is outlined in the previous section. In applying the outlined methodology, each set of data was analyzed several times. This covers multi-polynomials in the order of one to seven. In each analysis the following computations were performed:
- the estimators a
- the standard deviation s (Y);
- the mean standard deviation sm (Y);
- the standard deviation of the estimators s (a);
- the t-test t*;
- the coefficient of multiple determination R2;
- the mean relative error e.
Since we deal with large sets of data, a value of 0.75 or more for R2 is desirable [11]. Moreover, a high R2 value is not good enough unless e is also small enough. Modern software was used for th m se rather complex calculations [12]. Such software allows us to perform these calculations without an intimate knowledge of all of the computational details. In correlating the data by following the procedure outlined above, one of the models stands out as the best representative of the data. Such a model was selected for inclusion in this paper.
The mathematical models representing the data of the weather parameters mentioned in the introduction are given in the following subsections, where the two explanatory variables are t and t. t1 denotes the time series in hours and takes the values 1 (the first hour of the first day in November) through 4344 (the 24th hour of the 30th day in April) for the heating season. It takes the values 1 (the first hour of the first day in June) through 2928 (the 24th hour of the 30th day in September) for the air-conditioning season. t denotes the time in hours on the day under consideration, and takes the values 1 through 24.
2.1 Air Dry-bulbTemperature
The adequate models for predicting air dry-bulb temperature for an optional time during the heating and air-conditioning seasons were obtained. These models are expressed by the following general relation:
J = a0 + a1t1 + a2t12 + a3sin
12
(t2-10)
(12).
Cenilke a, koeficienti določenosti R2, standardna deviacija s(J) in srednje vrednosti relativne napake e so predstavljene v preglednici 1.
Vzemimo čas ob 12 uri 21 decembra (t1=1213, t=12) kot primer za primerjavo merjene in izračunane vrednosti. Izmerjena vrednost je 11,1°C, ki jo
The estimators a, the coefficient of determination R2, the standard deviation s(J) and the mean of the relative error e of these models are listed in Table 1.
As an example, we took 12:00 on 21 December (t = 1213, t = 12) to compare the recorded value and the computed value. The recorded value was
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primerjamo z 10,8 °C izračunane z gornjo zvezo. Ugotovimo malenkostno napako 0,3°C. Vzemimo še čas ob 12 uri 21 julija (t=1213, t=12) kot naslednji primer. Izmerjena vrednost je 33.6°C v primerjavi z 32.6°C izračunana z zgornjo zvezo. Ocenjena napaka v tem primeru znaša FC.
11.1 oC compared to 10.8 oC computed from the above relation. Thus a negligible error of 0.3 oC is observed for the predicted value. For 12:00 on 21 July (t1= 1213, t2=12) the recorded value was 33.6 oC compared to 32.6 oC computed from the above relation, an error of about 1 oC is observed in the prediction of this value.
Preglednica 1. Sezonski modeli cenilk in statističnih parametrov za temperaturo suhega zraka Table 1. Estimators and statistical parameters for air dry-bulb temperature seasonal models
Sezona Season
Ogrevanje Heating Klimatizacija Air-conditioning
a0
13,6280 22,2633
a1
-8,1386 10-3 6,1894 10-3
a2
2,1844 10-
-2,2553 10-6
2.2 Temperatura vlažnega zraka
Podobno kot sezonske modele za temperaturo suhega zraka imamo tudi modele za napoved temperature vlažnega zraka za poljubni čas. Ti modeli so podani z naslednjo glavno zvezo:
a3
5,3841 8,7402
R2
0,91 0,96
s(J)[oC]
em
1,65 1,30
0,120 0,048
2.2 Air Wet-bulb Temperature
As with the seasonal models of the air dry-bulb temperature, the models for predicting air wet-bulb temperature for an optional time during the heating and air-conditioning seasons were obtained. These models are expressed by the following general relation:
9w= a0 + a1t1 + a2t1
12
(t2-10)
(13).
Cenilke a in statistični parametri R2, s(J) in e so podani v preglednici 2. Ker se oblika grafičnega izhoda podatkov razlikuje v posameznih korakih, samo en model vseh podatkov ne more popisati. Zato imamo dve skupini cenilk za ogrevalno sezono in tri skupine za sezono klimatizacije.
Izmerjena vrednost za čas ob 12 uri 21 decembra je 7,6°C v primerjavi z 6,9°C izračunano iz zgornje relacije. Odstopanje v tem primeru je 0,7°C. Nadalje je izmerjena vrednost za čas ob 12 uri 21 julija 18,9°C v primerjavi z 18,7°C izračunane z gornjo relacijo. Ocenjena napaka v tem primeru znaša 0,2°C.
2.3 Temperatura rosišča
Statistični pristop, ki je bil uporabljen v prejšnjih podpoglavjih je bil uporabljen za napoved temperature rosišča zraka za poljubni čas med ogrevalno sezono in sezono klimatizacije. Ti modeli so podani v naslednji splošni odvisnosti:
The estimators ak and the statistical parameters R2, s(J) and e of these models are listed in Table 2. Since the shape of the graphical outlay of the data differed from one time interval to another, a single model throughout the season could not be used to represent them. Thus, two groups of estimators for the heating season and three groups for the air-conditioning season are listed.
The recorded value for 12:00 on 21 December is 7.6 oC compared to 6.9 oC computed from the above relation. Thus a negligible error of 0.7 oC is observed in the predicted value. The recorded value for 12:00 on 21 July is 18.9 oC compared to 18.7 oC computed from the above relation. Thus, the error in the prediction of this value is about 0.2 oC.
2.3 Air Dew-point Temperature
The general statistical approach used in the previous two subsections was applied to obtain models for predicting air dew-point temperature for an optional time during the heating and air-conditioning seasons. These models are expressed by the following general relation:
9d=A(9d1 + 9d2)
(14),
J   in J   so podani v naslednji odvisnosti:
d2
J   and J   are given by the following relations:
&d1 = a0
l7d 2      a3t2   +a 4t2    +a5 t2    +a 6t 2    +a7t2    +a8 t2    +a9 t2
(15), (16).
Cenilke ak teh modelov so podane v preglednici 3. Faktor množenja in statistični parametri
The estimators ak of these models are listed in Table 3. The multiplication factor A and the sta-
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K. Skeiker: Identifikacija sezonskih modelov - An Identification of Seasonal Models
R2, o(S) in e so podani v preglednici 4. Pomembno je upoštevati, da je imamo za ogrevalno sezono tri skupine cenilk.
Izmerjena vrednost za čas ob 12 uri 21 decembra je 4,9°C v primerjavi z izračunano vrednostjo 4,6°C. Standardni odstopek je v tem primeru 0,3°C. Izmerjena vrednost za čas ob 12 uri 21 julija je 13,5°C v primerjavi z izračunano vrednostjo 13,3°C. Pričakovano odstopanje pri napovedi teh vrednosti znaša 0,2°C.
tistical parameters R2, s(J) and em of these models are listed in Table 4. Three groups of estimators are listed for the heating season.
The recorded value for 12:00 on 21 December is 4.9 oC compared to 4.6 oC computed from the above relation. Thus a negligible error of 0.3 oC is observed in the predicted value. The recorded value for 12:00 on 21 July is 13.5 oC compared to 13.3 oC computed from the above relation. Thus, the error in the prediction of this value is about 0.2 oC.
Preglednica 2. Cenilke in statistični parametri za sezonske modele temperature vlažnega zraka Table 2. Estimators and statistical parameters for air wet-bulb temperature seasonal models
Sezona Season
Ogrevanje Heating
t1 =1-3912
t1 =3913-4344 Klimatizacija Air-conditioning
t1 =1-1008
t1 =1009-2136 t1 =2137-2928


a0
9,7880 469,373
13,5879 15,3478 17,1440

a1
-6,1693 10-3 -225,941 10-3

5,7522 10-3 3,2647 10-3 2,6868 10-3

a2
1,6346 10-6         3,1158
27,7682 10-6         2,9816
-2,5753 10-6          2,1476
-1,0701 10-6          1,3868
-1,2859 10-6          2,4088

a3
R2
s(Jw ) [oC]

0,93            0,83            0,109
0,89            0,66            0,033
em
Preglednica 3. Cenilke za sezonske modele temperature rosišča zraka Table 3. Estimators for air dew-point temperature seasonal models
Časovno obdobje t1 Period of time t1
t1=1-768                 7,3399
t1=769-3912           5,0994
t1=3913-4346       651,433
t1=1-2928

a0
a1 103
-6,8723 -3,6960 -311,23
a2 106
2,0226 1,0883 37,4567
Ogrevalna sezona Heating season

a3
2,7926 1,9023 0,0909
a4
-1,8822 -1,2328 -0,5833
a5
0,3698 0,2315 0,1614
a6 102
-3,0136 -1,8167 -1,5657
a7 103
1,0925 0,6369 0,6387
a8 105
-1,4633 0,8263 -0,9369
a9 106
Klimatizacijska sezona Air-conditioning season

9,7798 14,9859 -3,7733  6,2585  -5,1452  1,3678  -16,697 10,2364 -30,726  3,6015
Preglednica 4. Faktor množenja in statistični parametri za sezonske modele temperature rosišča zraka Table 4. Multiplication factor and statistical parameters for air dew-point temperature seasonal models
Sezona Season
Ogrevanje Heating Klimatizacija Air-conditioning
A
R2
0,87 0,94
s(Jd ) [oC]
0,89 0,98
em
0,144
0,080
3 VIZUALNA PRIMERJAVA REZULTATOV IN NAPOVEDANIH VREDNOSTI
V nadaljevanju obravnave rezultatov so bili izračunani in izmerjeni rezultati izrisani na istem diagramu, da smo lahko obravnavali vizualno primerjavo.
Napovedane in izmerjene vrednosti so bile izrisane za 21 december in 21 julij. Slika 1 prikazuje
3 VISUAL COMPARISON OF THE DATA AND THE PREDICTED VALUES
In a further study of the results, the predicted values along with the recorded data were plotted on the same scale for a visual comparison between them.
The predicted values along with the re-corded data for 21st December and 21st July were plot-
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K. Skeiker: Identifikacija sezonskih modelov - An Identification of Seasonal Models
izris napovedanih in izmerjenih vrednosti za temperaturo suhega zraka. Podobni izrisi so dobljeni za temperaturo vlažnega zraka in temperaturo rosišča.
Na slikah 2 in 3 so predstavljene vrednosti za temperaturo suhega zraka za december in julij. Podobni izrisi so dobljeni za ostala dva parametra na slikah 2 in 3.
40
izmerjeni podatki za 21. december observed data for 21st December izmerjeni podatki za 21. julij observed data for 21st July
ted for a visual comparison of the data and their predicted values. Figure 1 shows a plot of the pre-dicted and recorded data values for the air dry-bulb temperature. Similar plots for air wet-bulb temperature and air dew-point temperature were also obtained. In figures 2 and 3, the predicted values along with recorded data are presented for the air dry-bulb temperature of December and July, respectively. Plots for the remaining two weather parameters are similar to those in figures 2 and 3.
napovedane vrednosti za 21. december predicted values for 21st December napovedane vrednosti za 21. julij predicted values for 21st July
1                  3                  5                  7                  91
21       h     23
čas / time
Sl. 1. Potek temperature suhega zraka za 21. december in 21. julij Fig. 1. Behaviour of air dry-bulb temperature for 21st December and 21st July
»C
18
izmerjeni podatki za december observed data for December
napovedane vrednosti za december predicted values for December
101
201
301
401
501
601
701
čas / time
Sl. 2. Potek temperature suhega zraka za december Fig. 2. Behaviour of air dry-bulb temperature for December
grin^SfcflMISDSD
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K. Skeiker: Identifikacija sezonskih modelov - An Identification of Seasonal Models
40
izmerjeni podatki za julij observed data for July
napovedane vrednosti za julij predicted values for July
10
101
201
301
401
501
601
701
čas / time
Sl. 3. Potek temperature suhega zraka za julij Fig. 3. Behaviour of air dry-bulb temperature for July
4 SKLEP
Razvoj zelo zmogljivih osebnih računalnikov je eden izmed razlogov za razvoj simulacije sistemov. Numerične simulacije toplotnih sistemov zgradb potrebujejo kot vhodni podatek vremenske parametre.
Tehnika, ki je bila uporabljena za določitev najbolj odgovarjajoče krivulje, je metoda najmanjših kvadratov. Uporabljen pristop je nadrobno opisan.
Grafična oblika podatkov je narekovala uporabo večpolinomskega modela. Z uporabo polinomov so bili izdelani sezonski modli za naslednje vremenske parametre:
- temperaturo suhega zraka;
- temperaturo vlažnega zraka;
- temperaturo rosišča.
Ti matematični modeli so bili izdelani za ogrevalno sezono in za sezono klimatizacije v Damasku. Razviti so bili na podatkih dolgega časovnega obdobja.
Izbira najboljšega modela je temeljila na treh statističnih parametrih: koeficientu določenosti R2, srednji relativni napaki e in t testu. Ker delamo z velikim številom podatkov (4344/ali 2928 točk za ogrevalno/ali klimatizacijsko sezono), lahko predlagani modeli ustrezajo zgornjim parametrom z zadovoljivim R2 in e . Vizualno je bilo ugotovljeno, da obstaja dobra skladnost med napovedanimi in izmerjenimi vrednostmi.
Predlagane matematične modele lahko uporabimo v CLIMA programu kot alternativo urnim vremenskim podatkom.
4 CONCLUSION
The proliferation of high-power personal computers has helped in the advancement of the science of system simulation. The numerical simulation of building thermal systems calls for the input of local weather parameters.
The technique used in determining the best-fitting curve is the least-squares method. The ap-proach used in the analysis is outlined.
The shape of the graphical outlay of the data for the particular parameter under consideration suggested that a multi-polynomial representation was a strong candidate. By making use of a polynomial representation, mathematical seasonal models were suggested for the following weather parameters:
- air dry-bulb temperature;
- air wet-bulb temperature;
- air dew-point temperature.
Mathematical models were derived for the heating season and for the air-conditioning season in the Damascus zone. They were developed based on long-term data records.
The choice of the best model was based on three statistical parameters: the coefficient of determination R2, the mean relative error em and the t-test. Since we deal with large sets of data (4344 or 2928 data points for the heating or air-conditioning sea-son), the suggested models can estimate the above-mentioned parameters with acceptable R2 and em. It is clear that good agreements between the predicted and the observed data values were obtained.
The suggested mathematical models can be used in the CLIMA computer program as an alternative to using hourly historical weather data.
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K. Skeiker: Identifikacija sezonskih modelov - An Identification of Seasonal Models
Ta pristop moremo uporabiti za druge kraje in z uporabo dolgočasovnih nizov podatkov tudi za ostale vremenske parametre.
Zahvala
Avtor bi rad izrazil zahvalo direktorju prof. Dr. Ibrahim Othmanu za zanimanje in razumevanje pri delu. Hvaležen je tudi prof. dr. Tawfik Kassamu in dr. Mohamed Tlassu za dragocene diskusije in predloge. Nenazadnje pa še Oddelku za meteorologijo za posredovane urne vremenske podatke za Damask v preteklih letih.
Such an approach needs to be followed for the other sites, and to be applied using long-term data sets of the remaining relevant weather parameters.
Acknowledgments
The author would like to express gratitude to Prof. Dr. Ibrahim Othman, the Director General, for his continued interest and encouragement, and also to Prof. Dr. Tawfik Kassam and Dr. Mohamed Tlass, for valuable discussions and suggestions. Finally, thanks are due to the Department of Meteorology, for providing the hourly weather data of Damascus zone for the available previous years.
5 OZNAČBE 5 NOMENCLATURE
faktor množenja
cenilka parametra b
relativna napaka
niz podatkov
stopnja modela
obseg podatkov
število parametrov v modelu
koeficient večkratne korelacije
koeficient večkratne določenosti
t test
matrika spremenljivk
transponirana matrika [X]
inverzna matrika [X]
neodvisna spremenljivka
spremenljivka za napoved odgovora
matematični model krivulje podatkov
opazovana spremenljivka
aritmetično povprečje podatkov spremenljivke yi
funkcijska predstavitev točk s podatki
testno referenčno leto
stopnja zaupanja
modelski parameter
izraz za napako
temperatura
Ludolfovo število 3,141593
standardna deviacija
vektor standardnih deviacij cenilk
Indeksi
točka rosišča i-ti poskus k-ti
srednji vlažni
A            multiplication factor
a             estimator of parameter b
e             relative error
M           set of data points
m           model degree
n            size of the data set
p            number of model parameters
R            coefficient of multiple correlation
R2           coefficient of multiple determination
t             t-test
[X]        matrix of explanatory variable
[X]T       transpose of matrix [X]
[X]-1       inverse of matrix [X]
x           explanatory variable
Y          predicted response variable
Y=F(x)    mathematical model of the curve fitted to the data
y           observed response variable
y           arithmetic mean of data value yi
y=f(x)     functional representation of data points
TRL/RMY reference meteorological year
a           level of significance
b           model parameter
e           error term
J           temperature
p           Ludolphos number 3.141593
s           standard deviation
[s(a)]     vector of the standard deviations of estimators
Subscripts
d           dew-point
i            ith trial
k           kth
m          mean
w          wet-bulb
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K. Skeiker: Identifikacija sezonskih modelov - An Identification of Seasonal Models
[1]
[2]
[3]
[4]
[5]
[6]
[7] [8] [9] [10]
[11] [12]
6 LITERATURA 6 REFERENCES
Saman, H. (1995) Energy balance in the Syria Arab Republic, Internal statistical Report, Ministry of Petroleum, Syria.
Skeiker, K. (1990) LOS-A0, The computer program for dynamic analysis of heat transfer in building, Fac-ulty of Mechanical Engineering in Ljubljana, Slovenia.
Skeiker, K. (2000) Systematization organization development and utilization of an hourly reference meteo-rological year database for Damascus zone, Journal of Mechanical Engineering, 6/2000, Slovenia. Mahmoud, A. (1993) Empirical correlation of solar and other weather parameters, J. King Saud University, Vol.5.
Shuichi, H. (1991) statistical time series models of solar radiation and outdoor temperature - identification of seasonal models by Kalman filter, J. Energy and Building, Vol. 15-16.
David, S. (1999) Introduction to the practice of statistics, 3rd Edition, W.H. Freeman and Company, New York.
Preston, D.W. (1991) The art of experimental physics, John Wiley and Sons, New York. Neter, J. (1974) Applied linear statistical models, Richard D. Irwin INC, USA. Goon, A.M. (1981) Basic statistics, The World Press Private Limited, Calcutta.
Green, J.R. (1979) Statistical treatment of experimental data, Elsevier Scientific Publishing Company, New York.
Herbert, A. (1972) Statistical methods, Barnes & Noble Books, New York.
nd Wolfram, S. (1998) Mathematica, a system for doing mathematics by computer, 2     Edition, Addison-Wesley Publishing Company, New York.
Avtorjev naslov:
Kamal Skeiker
Department of Scientific Services
Atomic Energy Commission
P.O. Box 6091
Damascus, Syria
Author’s Address: Kamal Skeiker
Department of Scientific Services Atomic Energy Commission P.O. Box 6091 Damascus, Syria
Prejeto: Received:
23.8.2000
Sprejeto: Accepted:
27.6.2001