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Improved multiple linear regression based models for solar collectors
Renewable Energy 91 (2016) 224e232
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Improved multiple linear regression based models for solar collectors rd Kicsiny Richa n University, Pa ter K. u. 1., 2100 Go € do €llo }, Hungary Department of Mathematics, Institute for Mathematics and Informatics, Szent Istva
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 November 2015 Received in revised form 9 January 2016 Accepted 19 January 2016 Available online xxx
Mathematical modelling is the theoretically established tool to investigate and develop solar thermal collectors as environmentally friendly technological heat producers. In the present paper, the recent and accurate multiple linear regression (MLR) based collector model in Ref. [1] is empirically improved to minimize the modelling error. Two new, improved models called IMLR model and MPR model (where MPR is the abbreviation of multiple polynomial regression) are validated and compared with the former model (MLR model) based on measured data of a real collector field. The IMLR and the MPR models are significantly more precise while retaining simple usability and low computational demand. Many attempts to decrease the modelling error further show that the gained precision of the IMLR model cannot be significantly improved any more if the regression functions are linear in terms of the input variables. In the MPR model, some of the regression functions are nonlinear (polynomial) in terms of the input variables. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Solar collectors Mathematical modelling Black-box model Multiple linear regression Polynomial regression
1. Introduction It is highly important nowadays to study and develop solar thermal collectors within the framework of environmental protection. The theoretical tool for this purpose is mathematical modelling. Two main types of mathematical models exist for collectors: physically-based (or white-box) models describe exact and known physical laws, while black-box models represent some experienced or measured correlations empirically. In the literature, there are numerous physically-based models. One of the most important and still often used ones is the HottelWhillier-Bliss model [2,3], which is also among the earliest models. The collector temperature is determined as a function of s et al. time and a space coordinate in this distributed model. Buza [4] proposed a simpler model based on the piston flow concept assuming that the collector temperature is homogeneous in space. This model is a linear ordinary differential equation (ODE) validated in Ref. [1] and is probably the simplest such (physically-based and ODE) model used in the practice (see e.g. Refs. [5e8]), which describes the transient processes of a collector with a proper accuracy. The greatest advantage of black-box type models is that it is not needed to know precisely the physical laws of a collector in order to create an usable model. Nevertheless, the model may be rather precise even if it is mathematically simple as in the case of Ref. [1].
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.renene.2016.01.056 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
In the scope of solar thermal systems, the most widely used blackbox model type may be the artificial neural network (ANN). In Ref. [9], the useful heat gained from a solar heating system as well as the temperature rise of the storage water were predicted with an ANN with an error of 7e10%, which is considered proper accuracy with respect to such systems [10]. The layer temperatures are modelled in a solar storage by means of an ANN elaborated in Ref. [11]. In particular, ANNs are frequently applied to model solar collectors separately as well. Generally speaking, ANNs are precise modelling approaches but quite troublesome to use because of the necessary training/learning process. A lot of measured data must be collected under various working conditions to train the ANN for gaining a satisfactory precision. E.g. in Ref. [12], the training requires the measured data of three months or in Ref. [13], the measurements of 17 days are needed to work out an ANN, which models a collector under similar circumstances as during the measurements. Only one so-called back-propagation algorithm (which is needed in the training) from the 11 available ones provides a proper approximation in the latter work, based on which, it can be easily concluded that the success of an ANN, and its computational demand, depend highly on the user's expertise. Also, the convergence of the algorithm indicating the end of the training can be time-consuming. Ref. [14] demonstrates the problem of uncertainty as well, where six separate ANNs are used to identify several collector parameters, so there is no general and exact instruction for designing appropriate ANNs. In fact, an
R. Kicsiny / Renewable Energy 91 (2016) 224e232
Nomenclature t Tout
I Ta Tin V v
tA tB t1 t2
time, s homogeneous temperature inside the collector (assumed to be the same as the outlet collector temperature), C global solar irradiance on the collector surface, W/ m2 ambient temperature of the collector, C inlet collector (fluid) temperature, C volume of the collector, m3 (constant) flow rate inside the collector, m3/s time delay before Case A or A3, s time delay before Case B, s time of flowing inside the collector from the inlet to the outlet when the pump is switched on permanently, s length of time between successive measurements on the collector, s
universal applicable algorithm, which ensures a reliable and fast design of a proper ANN for a collector is still missing based on Fischer at al [15]. Because of the above difficulties on complexity/uncertainty and time/computational demand, a precise and general but still simple black-box model has been proposed recently in Ref. [1], which can be applied fast and easily for many types of solar collectors. The model (called MLR model in short) is founded on standard methods from mathematical statistics, in particular, on multiple linear regression (MLR). Based on studies in the literature, the author found that MLR is a relatively rare black-box modelling tool in the field of solar collectors in spite of its simplicity. MLR can be applied to identify collector parameters in white-box models (see Refs. [16e18]) and to model collector efficiency in collector test methods [19,20]. In Ref. [21], the annual thermal performance of collectors is modelled as a function of the annual solar radiation by means of linear regression. In the mentioned MLR model, a typical day has been divided into sub-cases, for which, separate regression equations have been proposed to reach higher precision. The regression equations describe empirical relations immediately between input and output variables of collectors. Considering the high precision (with an error of 4.6%), simple usability and low computational demand of the MLR model, it is definitely worth improving further to maximize its precision while retaining simple usability and low computational demand. Accordingly, this improvement has been set as a future research task in the Conclusion of Ref. [1]. The following are the contributions of the present paper in details: the MLR model is empirically improved by means of inserting new operating sub-cases in a rather natural way. On the basis of measured data of a real collector, two new, improved models called IMLR model and MPR model (where MPR is the abbreviation of multiple polynomial regression) are identified (based on four days), validated (based on two months) and compared with the MLR model in view of accuracy. In the MPR model, some of the regression functions/equations are nonlinear (polynomial) in terms of the input variables. Polynomial regression is generally considered as a special linear regression, since the regression functions are linear in terms of the constant parameters (which are to be identified), although not linear in the input variables. If the regression functions are all linear, the regression can be
225
called simple linear regression. Matlab [22] has been used to carry out the needed calculations numerically. This software is widely used in the field of solar engineering to simulate different systems (see e.g. Ref. [23]). The paper is structured as follows: Section 2 recalls the details of the recent MLR model for the Reader's convenience. In Section 3, the new IMLR and MPR models are worked out. These models are validated in Section 4 by means of measured and simulated data. Finally, conclusions and future research suggestions can be found in Section 5.
2. MLR model For the Reader's convenience, the MLR model [1] is recalled in details in this section. Fig. 1 shows the studied solar collector. The inputs in the MLR model are from appropriate Tin, I, Ta and Tout values. The output is from appropriate Tout values. The flow rate v is a prefixed positive constant or 0 according to the differential control, which is not only the most frequent control method but also the optimal or nearly optimal one many times [24,25]. Since the flow rate is bounded, only Tin(t t1) can function as an input in the MLR model, where t1 is a (positive) time delay and the formed output is Tout(t). Similarly, only former I(t t2) and Ta(t t2) values can function as inputs corresponding to the output Tout(t) because these effects have bounded propagation speed. (For simplicity, the same delay (t2) is assumed for I and Ta.) Of course, an adequate former value of Tout affects Tout(t) itself and functions essentially as the initial value in the MLR model at the time (t t2). In the (black-box type) MLR model, distinct sub-models were identified for separate operating conditions. It is clear, for example, that the collector behaves very differently if the pump is off (v ¼ 0) or on (v > 0) permanently. Under the same conditions, including a high enough solar irradiance, the collector temperature and thus Tout increases much more fast when the pump is off. The effect of Tin was neglected in permanently switched off case, since there is no flowing from the inlet to the outlet in the collector. Assuming a typical day, when the increase of Tout is relatively high, three separate operating conditions were distinguished in accordance with Fig. 2. The exact specification of each case is the following: Case A: The pump is switched off permanently. Case A contains the term started at the beginning of the day and finished, when the pump is first switched on. All the terms, which begin at a time when the pump is permanently off for exactly tA time and finish at the next switch-on or at the end of the day, also belong to this case. (tA is the time, which is generally enough for Tout to become not fluctuating but permanently monotone, since, intentionally, frequent fluctuations are characteristics of Cases C1 and C2.)
Fig. 1. The studied solar collector.
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R. Kicsiny / Renewable Energy 91 (2016) 224e232
Case C1 :
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ þ cout;C1 Tout ðt t2 Þ þ cC1 (1c) Case C2 :
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ þ cout;C2 Tout ðt t2 Þ þ cC2 (1d) The constant parameters cI,A, ca,A, cout,A, cA, cin,B, cI,B, ca,B, cout,B, cB, cin,C1, cI,C1, ca,C1, cout,C1, cC1, cin,C2, cI,C2, ca,C2, cout,C2, cC2 are to be identified in the model. The measurements take place at times t ¼ 0, t2, 2t2, 3t2,... Accordingly, Tout,mod that is the modelled value of Tout is calculated at times t ¼ t2, 2t2, 3t2,… from corresponding values of I(t t2), Ta(t t2), Tout(t t2) and Tin(t t1) based on Eqs. (1a)e(1d). t1 occurs in Eqs. (1b)e(1d), where t > t1 naturally holds because of the relatively small collector volume and the time schedule of the pump operation (see Fig. 2).
3. IMLR and MPR models
Fig. 2. Outlet temperature, solar irradiance and pump operation on a typical day.
Case B: The pump is switched on permanently. Case B contains all the terms, which begin at a time when the pump is permanently on for exactly tB time and finish at the next switch-off. (tB is the time, which is generally enough for Tout to become not fluctuating but permanently monotone.) Case C relates to frequent switch-ons and -offs. It can be admitted that two further cases can be significantly distinguished in Case C: in general, Tout is increasing before and decreasing after the solar noon, respectively, so this case is divided into Cases C1 and C2. Case C1: Terms not within Case A and Case B before the solar noon (12:30 according to summer time). Case C2: Terms not within Case A and Case B after the solar noon. For making the modelling accuracy as well as possible, different sub-models were elaborated based on MLR for all operating cases. Tin is an input in the sub-models of Case B, Case C1 and Case C2, since some fluid is flowing from the inlet to the outlet of the collector in accordance with that the pump is permanently or intermittently on in these cases. For lack of flowing, Tin is neglected in the sub-model of Case A as the pump is permanently switched off. The below linear equations form the sub-models of the operating cases in the MLR model:
Case A :
Tout;mod ðtÞ
¼ cI;A Iðt t2 Þ þ ca;A Ta ðt t2 Þ þ cout;A Tout ðt t2 Þ þ cA (1a) Case B :
Tout;mod ðtÞ
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ þ cout;B Tout ðt t2 Þ þ cB
(1b)
The MLR model is empirically improved in this section. The resulted new models will be called IMLR and MPR models. The steps of the improvements are the following (see also Fig. 2): 1. The (largest) operating case Case A is divided into four sub-cases as follows: Case A1: This case consists of the time period from the beginning of the day to the time when the solar irradiance is first greater than 10 W/m2. This case practically belongs to the term of no irradiance in the first half of the day. Clearly, the effect of the irradiance can be neglected here. This makes the corresponding mathematical relation simpler and refines the model globally. Based on experiments, this step decreases the modelling error. Case A2: This case consists of the time period from the end of Case A1 to the time when the solar irradiance is first greater than 100 W/m2. Usually, this time is followed by a very intensive increase in the irradiance, so this is apparently the time of sunrise, when the irradiance changes from (mostly) diffuse to (mostly) direct. Based on experiments, it is worth distinguishing this case in the model to decrease the modelling error. Case A3: Time periods besides Cases A1, A2 and A4 (see below) within Case A. Case A4: The last three hours of the day. In essence, the term after Case C2 corresponds to the free cooling of the collector from a relatively high temperature. Based on experiments, this section cannot be modelled well with a single relation, so it should be divided into sub-sections. Empirically, the problem can be solved well with only two sub-parts if the last three hours are separated. This may be caused by that the temperature is rather high in the beginning of the cooling process, where the highly nonlinear effect of heat radiation (to the colder environment) is probably considerable. After 21:00, the temperature is always much lower (and decreases much more slowly), so the heat radiation is more negligible. In addition, the temperature range after Case C2 is still rather big, which makes for a single linear relation impossible to model the cooling process precisely. Clearly, the effect of the irradiance can be neglected here because of the night.
R. Kicsiny / Renewable Energy 91 (2016) 224e232
2. To avoid unneeded overestimates in Case A3, the minimum of Tout,mod(t) from Eq. (2c) below and the previously modelled Tout,mod(t t2) is considered as the modelled outlet temperature at time t in Case A3 if I(t 2t2) I(t t2)10 W/m2 and the pump stagnates for at least 20 min. This inequality means in essence that the solar irradiance has not increased since the last measured time (10 W/m2 compensates the accidental measuring error), so it is meaningful to assume that the collector temperature has neither increased considering also that the pump stagnates for at least 20 min. This time is assumed to be enough for the collector (with a rather small volume) to warm up during a stagnation (cf. Case A in the MLR model). This correction is more important in the IMLR and MPR models since Case A3 has much less measured data for identification than Case A in the MLR model because of its much shorter length. In addition, the temperature range occurring during Case A3 is usually rather wide, which makes the precise modelling difficult. It is also suggested to apply the maximum of the measured Tout temperature occurring in the identification as an upper limit for the modelled Tout temperature. Based on experiments, this upper limit does not improve significantly the overall modelling precision for a long time (for several months) but rarely occurring rough overestimates can be avoided in this way. 3. The coefficients of the zeroth-order members (cf. cA, cB, cC1 and cC2 in Eqs. (1a)e(1d) in the MLR model) are set zero in Eqs. (2a)e(2g) and (3a)e(3g) below. (The corresponding identification of each case can be carried out easily with a suitable statistical or spreadsheet program (SPSS, Excel, etc.).) This set is in line with the physical phenomenon that the collector (outlet) temperature must be zero if all the inputs Tin, I, Ta and the previous collector temperature are zero. Based on experiments, this natural constraint results in a bit lower precision in the identification but higher precision in the validation, that is the modelling error decreases.
Case C1 :
Tout;mod ðtÞ ¼ ca;A1 Ta ðt t2 Þ þ cout;A1 Tout ðt t2 Þ
þ cout;C1 Tout ðt t2 Þ (2f) Case C2 :
Case A2:
þ cout;C2 Tout ðt t2 Þ (2g) ca,A1, cout,A1, cI,A2, ca,A2, cout,A2, cI,A3, ca,A3, cout,A3, ca,A4, cout,A4, cin,B, cI,B, ca,B, cout,B, cin,C1, cI,C1, ca,C1, cout,C1, cin,C2, cI,C2, ca,C2, cout,C2 are constant parameters to be identified. The following second-order equations correspond to the above operating cases:
Case A1 :
Case A3 :
Tout;mod ðtÞ
¼ ca;A1 Ta ðt t2 Þ þ cout;A1 Tout ðt t2 Þ þ ca;A1;2 Ta2 ðt t2 Þ (3a) Case A2 :
Tout;mod ðtÞ
¼ cI;A2 Iðt t2 Þ þ ca;A2 Ta ðt t2 Þ þ cout;A2 Tout ðt t2 Þ þ cI;A2;2 I 2 ðt t2 Þ þ ca;A2;2 Ta2 ðt t2 Þ (3b) Case A3 : Tout;mod ðtÞ ¼ cI;A3 Iðt t2 Þ þ ca;A3 Ta ðt t2 Þ þ cout;A3 Tout ðt t2 Þ þ cI;A3;2 I 2 ðt t2 Þ þ ca;A3;2 Ta2 ðt t2 Þ (3c) Case A4 :
Tout;mod ðtÞ
¼ ca;A4 Ta ðt t2 Þ þ cout;A4 Tout ðt t2 Þ þ ca;A4;2 Ta2 ðt t2 Þ (3d) Tout;mod ðtÞ
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ 2 ðt t1 Þ þ cI;B;2 I 2 ðt t2 Þ þ cout;B Tout ðt t2 Þ þ cin;B;2 Tin
Tout;mod ðtÞ ¼ cI;A2 Iðt t2 Þ þ ca;A2 Ta ðt t2 Þ þ cout;A2 Tout ðt t2 Þ
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ
Case B : (2a)
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ
The following linear equations correspond to the above operating cases:
Case A1 :
227
þ ca;B;2 Ta2 ðt t2 Þ
(2b)
(3e) Case C1 :
Tout;mod ðtÞ
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ
¼ cI;A3 Iðt t2 Þ þ ca;A3 Ta ðt t2 Þ þ cout;A3 Tout ðt t2 Þ
2 ðt t1 Þ þ cout;C1 Tout ðt t2 Þ þ cin;C1;2 Tin
(2c)
þ cI;C1;2 I2 ðt t2 Þ þ ca;C1;2 Ta2 ðt t2 Þ Case A4 :
Tout;mod ðtÞ ¼ ca;A4 Ta ðt t2 Þ þ cout;A4 Tout ðt t2 Þ
(3f) (2d)
Case B :
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ
Tout;mod ðtÞ
2 ðt t1 Þ þ cout;C2 Tout ðt t2 Þ þ cin;C2;2 Tin
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ þ cout;B Tout ðt t2 Þ
Case C2 :
(2e)
þ cI;C2;2 I2 ðt t2 Þ þ ca;C2;2 Ta2 ðt t2 Þ (3g)
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R. Kicsiny / Renewable Energy 91 (2016) 224e232
ca,A1, cout,A1, ca,A1,2, cI,A2, ca,A2, cout,A2, cI,A2,2, ca,A2,2, cI,A3, ca,A3, cout,A3, cI,A3,2, ca,A3,2, ca,A4, cout,A4, ca,A4,2, cin,B, cI,B, ca,B, cout,B, cin,B,2, cI,B,2, ca,B,2, cin,C1, cI,C1, ca,C1, cout,C1, cin,C1,2, cI,C1,2, ca,C1,2, cin,C2, cI,C2, ca,C2, cout,C2, cin,C2,2, cI,C2,2, ca,C2,2 are constant parameters to be identified. In the IMLR model, all regression functions are linear with respect to the input variables, that is, the regression equations are Eqs. (2a)e(2g). To improve the precision of the IMLR model, the linear regression equations of Cases A4, C1 and C2 have been changed to the polynomial equations Eqs. (3d), (3f) and (3g), since polynomial regression generally provides more flexible curvefitting functionality and thus higher precision. The gained model is the MPR model. (See Note 2 of Remark 3.1 below for more details.) Remark 3.1 1. The above second-order equations (Eqs. (3a)e(3g)) may be the simplest higher order relations, where each measured input occurs at first and second orders. Tout(tt2) occurs at first order only, since it works as an initial function, which has a very direct and simple connection with Tout,mod(t) based on the following simple assumption: Tout,mod(t) would be equal to Tout(tt2) if the other inputs took no effect. 2. Many other attempts have been made in the present research to decrease the modelling error further. Because of limits in volume, only a few of them can be mentioned here. For example, Cases B, C1 and C2 have been also divided into sub-cases and/or more and more days have been used for identification. Based on experiments, these complexities have not significantly improved the precision. Similarly, the precision of the MPR model cannot be improved significantly if Eqs. (3a)e(3c) and (3e) are used instead of (2a)(2c) and (2e). That is why the simpler linear relations are remained for Cases A1eA3 and B in the MPR model in accordance with the high R2 values of the IMLR model in these Cases (see Table 2).
4. Validation and comparison In this section, the IMLR model (Eqs. (2a)e(2g)) and the MPR model (Eqs. (2a)e(2c), (3d), (2e), (3f), (3g)) are identified and validated based on simulation and measured data then they are compared with the MLR model in view of precision. The results and figures of the MLR model used in the comparison are from Ref. [1]. The identification and the validation of the IMLR and MPR models are based on the same days as in case of the MLR model in Ref. [1]. The used real flat plate collector field of 33.3 m2 [26] at the Szent n University (SZIU) in Go € do €llo }, Hungary (SZIU collector in Istva short) is also the same. Tout, Tin, I, Ta and v are measured once a minute at the SZIU collector. The measured value of Tout(0) is used as initial condition in the models. The later measured values of Tout is used only in the identification and for comparison purposes. The measured inputs are Tin, I, Ta in the IMLR model (see Eqs. (2a)e(2g)). 2 , I2 and T 2 can be considered as further In the MPR model, Tin a measured inputs (see Eqs. (3d) and (3f)e(3g)), in terms of which, the regression equations are linear, so standard MLR routines can be used in the identification below. tB is set 4 min in all of the three models, while tA has been changed to 10 min in the IMLR and the MPR models (from 20 min in the MLR model). This shorter time for considering the pump already permanently switched off removes naturally some terms from Cases C1 and/or C2. This change results in less measured data for Cases C1 and C2, which makes their identification less accurate but also refines the model on a day
globally. Empirically, the second effect is more dominating and the average daily modelling error decreases. The other technical details of the identification and validation of the IMLR and MPR models are very similar as in case of the MLR model [1], so they are not fully specified below.
4.1. Identification In the identification, four measured days are chosen from a selected season (summer) so that they cover a wide variety of possible operating conditions. Two of them (2nd July 2012, 24th June 2012) are with relatively few pump switches (smooth operation) and two others (28th June 2012, 8th June 2012) are with relatively many pump switches (intermittent operation) because the operating conditions can be characterized well with the pump operation (if the pump is switched on or off). According to many computer experiments, such four days are enough to gain a rather precise identified model. These four days are chosen from the first third of the summer. In this manner, the identified model can be already applied in the remained part of the summer (two months). Ten standard MLR routines are used independently to identify parameters ca,A1, cout,A1, …, ca,C2,2 in the IMLR and MPR models (see Eqs. (2a)e(2g), (3d) and (3f)e(3g)) based on the measured data of the operating cases (Cases A1, A2, A3, A4, B, C1, C2). According to the minute-based measuring, t2 is set 1 min v ¼ 0.98 m3/h if the pump is switched on and V ¼ 0.027 m3, from which t1 z 1.5 min. The measured value of Tin(tt1) should be used in the equations in the identification (and in the validation). t1 ¼ 1.5 min is not suitable for the minute-based measuring, so Tin(tt1) is replaced with (Tin(tt2) þ Tin(t 2t2))/2 in the identification. The well-known, standard MLR routine (using least squares method) is available and easy-to-apply in most spreadsheet or statistical programs (Excel, SPSS, etc.) with low computational demand, so it is not detailed here. The parameters of the IMLR model identified with the ten independent MLR routines can be seen in Table 1 along with the corresponding values of the MLR model (rounded to four decimals). Usually, the results of the MLR routine is evaluated with the square of the corresponding correlation coefficient R2. Table 2 contains the R2 values resulted from the whole identification for each case (rounded to three decimals). The IMLR and the MPR models can be called reasonable as R2 > 0.75 in most cases, even, it is close to 1 in Cases A1, A2, A3, B and C2 for both models and close to 1 in Case A4 for the MPR model (see Table 2). Nevertheless, the indices of Table 3 are more expressive and important in our case, especially, in light of the comparison with the MLR model accepted already in the literature. Table 3 presents the average of error (time average of the difference between the modelled and measured outlet temperatures) and the average of absolute error (time average of the absolute difference between the modelled and measured outlet temperatures) values for two days (2nd July 2012, 28th June 2012) of the identification of all models. The average of absolute error values are also shown in proportion to the difference between the daily maximal and minimal measured outlet temperature values (in %). Table 3 also contains the mean of these % values with respect to all of the four days of the identification (3.2% for the IMLR model). Figs. 3 and 4 compare the measured and modelled outlet temperatures in case of the identification of the MLR and the IMLR models for a day with smooth and intermittent pump operation, respectively. The figures also show the operating state of the pump.
R. Kicsiny / Renewable Energy 91 (2016) 224e232
229
Table 1 Parameter values of the MLR, IMLR and MPR models.
cI,A, m2K2/W ca,A, e cout,A, e c A, C ca,A1, e cout,A1, e cI,A2, m2K2/W ca,A2, e cout,A2, e cI,A3, m2K2/W ca,A3, e cout,A3, e ca,A4, e cout,A4, e ca,A4,2, 1/ C cin,B, e cI,B, m2K2/W ca,B, e cout,B, e cB, C cin,C1, e cI,C1, m2K2/W ca,C1, e cout,C1, e cin,C1,2, 1/ C cI,C1,2, m4K3/W2 ca,C1,2, 1/ C cC1, C cin,C2, e cI,C2, m2K2/W ca,C2, e cout,C2, e cC2, C cin,C2,2, 1/ C cI,C2,2, m4K3/W2 ca,C2,2, 1/ C
MLR model
IMLR model
MPR model
0.0048 0.0191 0.9815 0.0543 e e e e e e e e e e e 0.0354 0.0024 0.0086 0.9552 0.5324 0.0277 0.0056 0.2289 0.7736 e e e 2.9295 0.0655 0.0021 0.0823 0.8916 0.3457 e e e
e e e e 0.0048 0.9945 0.0008 0.0054 0.9939 0.0062 0.0298 0.9723 0.0004 0.9978 e 0.0436 0.0021 0.0036 0.9451 e 0.0493 0.0093 0.3361 0.7225 e e e e 0.1714 0.0026 0.0935 0.7981 e e e e
e e e e 0.0048 0.9945 0.0008 0.0054 0.9939 0.0062 0.0298 0.9723 0.0002 0.9978 0.00003 0.0436 0.0021 0.0036 0.9451 e 0.1818 0.0179 0.2666 0.6926 0.0013 0.00002 0.00004 e 0.3629 0.0163 0.1961 0.7593 e 0.002 0.00002 0.0054
Table 2 R2 values of the IMLR and MPR models from the identification.
IMLR model MPR model
A1
A2
A3
A4
B
C1
C2
1.000 1.000
1.000 1.000
0.999 0.999
0.631 1.000
0.994 0.994
0.793 0.794
0.961 0.962
Fig. 3. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 2nd July 2012 in case of the MLR and IMLR models.
temperature is to be modelled in the validation of course and not to be measured. The modelled days are from 3rd July 2012 to 31st August 2012. It means 56 days in the validation because of minor technical interruptions during the operation. The measured and modelled outlet temperatures are compared and evaluated in case of each model. Table 3 presents the average of error and average of absolute error values for the above two days for the validation in case of all models. The average of absolute error values are also shown in proportion to the difference between the daily maximal and minimal measured outlet temperature values as well (in %). Table 3 also contains the mean of these % values with respect to the whole time period for the validation 3rd July e 31st August in case of all models (4.6% for the MLR model, 4.1% for the IMLR model and 4.0% for the MPR model). Figs. 5 and 6 present the measured and modelled outlet temperatures in case of the MLR and the IMLR models for a day with smooth (3rd August 2012) and intermittent operation (5th August 2012), respectively. The figures also show the operating state of the pump.
Table 3 Average of error values and average of absolute error values with the models. MLR model Identification
2nd July (smooth operation) 28th June (intermittent operation)
Validation
Mean % value for the whole identification (four days) 3rd August (smooth operation) 5th August (intermittent operation) Mean % value for the whole validation (3rd July e 31st August)
Average Average Average Average Average Average Average Average Average Average
of of of of of of of of of of
error absolute error absolute absolute error absolute error absolute absolute
error error error error error error
0.47 C 2.79 C; 4.6% 0.23 C 3.01 C; 5.2% 4.7% 1.31 C 2.85 C; 4.5% 1.58 C 3.07 C; 5.2% 4.6%
IMLR model
0.53 C 1.64 C; 2.7% 0.80 C 1.84 C; 3.2% 3.2% 1.18 C 1.92 C; 3.0% 1.90 C 2.30 C; 3.9% 4.1%
MPR model 0.64 C 1.59 C; 2.6% 0.55 C 1.80 C; 3.1% 3.1% 1.41 C 1.89 C; 3.0% 1.86 C 2.22 C; 3.8% 4.0%
4.2. Validation
4.3. Comparison and assessment
All identified models are applied with the related measured inputs from the remaining part of the summer in the validation. One input is changed in comparison with the inputs in the identification that is the modelled value Tout,mod(tt2) is used as Tout(tt2) in the models (in Eqs. (1a)e(1d), (2a)e(2g), (3d) and (3f)e(3g) (not Tout,meas(tt2))) according to that the outlet
It has been shown that the collector field (outlet) temperature can be modelled precisely with the IMLR and the MPR models, in particular, more precisely than with the also accurate MLR model. On the basis of the validation, the mean modelling error is 4.1% with the IMLR model, 4.0% with the MPR model and 4.6% with the MLR model, so the accuracy of the IMLR and the MPR models can be
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Fig. 4. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 28th June 2012 in case of the MLR and IMLR models.
Fig. 5. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 3rd August 2012 in case of the MLR and IMLR models.
considered very well. The R2 values with respect to the measured and modelled data of four selected whole days (two identified days and two validated days) have been calculated in case of all models as additional information. Table 4 contains the results, which reinforce that the IMLR and the MPR models have higher precision and show that the precision of the IMLR and the MPR models are nearly the same. Remark 4.1 1. It should be noted that the collector field outlet temperature is measured on the outside surface of the outlet pipe of the SZIU collector and not inside the collector for technical reason. Because of the inaccuracy caused by this outside measuring and that the collector field is charged with such disturbances (shadowing effect of clouds), which are quite difficult to predict, the outlet temperature cannot be expected to be modelled perfectly. The relatively small volume of the collector field also contributes to the difficulties, since it causes relatively fast and high changes in the collector temperature under the
Fig. 6. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 5th August 2012 in case of the MLR and IMLR models.
disturbances. In view of these difficulties the accuracy of the IMLR and the MPR models can be called very well. 2. It is fair to apply the measured Tout(0) value as initial temperature in the identification and validation of the models, since applying known initial condition is normal and general in the field of mathematical modelling (see e.g. ODE models), otherwise, a mathematical construction cannot be generally expected to model real processes precisely. E.g. the current state (temperature) of a solar collector depends on not only the input variables but also on the former state of the collector. Nevertheless, based on simulations not fully detailed here because of limits in volume, the IMLR and MPR models are rather accurate even with a roughly estimated initial temperature. For example, if the initial temperature Tout(0) is not measured, it can be substituted with the measured value Ta(0) (which is generally close to Tout(0)) in the above identified IMLR and MPR models. In particular, the modelling error changes to 3.5% (from 3.0%) and 4.4% (from 3.9%) in case of the IMLR model and 3.4% (from 3.0%) and 4.2% (from 3.8%) in case of the MPR model on the validated days 3rd August and 5th August, respectively, if Ta(0) is used as initial value. This represents a better (smaller) modelling error than in case of the MLR model even with known initial condition (where these values are 4.5% and 5.2%, respectively). This shows the robustness of the IMLR and MPR models with respect to the uncertainties in relation to the initial condition. Although, initial conditions are often not used in black-box models, there are also counterexamples. For example, the layer temperatures in a solar storage is modelled with an ANN in Ref. [11], where the calculated layer temperatures of the previous time step is needed among the inputs at each time step except the first one, when only measured layer temperatures are available and thus must be used. 3. It should be mentioned that the above models, identified based on four summer days, have been applied to two spring months (April and May 2012) as well. The resulted modelling errors are 4.9% (instead of 4.1%) with the IMLR model and 5.0% (instead of 4.0%) with the MPR model for this time period. These values are still rather good, although, they are significantly higher than for the above modelled summer period (July and August). The following may explain these results: the orbit of the sun gets lower in spring than in summer, so the surrounding objects (e.g.
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Table 4 R2 values with the MLR and IMLR models. Date
2nd July 2012 (identification)
28th June 2012 (identification)
3rd August 2012 (validation)
5th August 2012 (validation)
MLR model IMLR model MPR model
0.976 0.989 0.990
0.972 0.985 0.985
0.978 0.987 0.988
0.968 0.982 0.983
trees) partly shade the collector field more often, which makes the precise modelling more difficult, since the measured irradiance corresponds not to the whole collector field in such cases. On the other hand, the range of the inputs (I,Ta, etc.) and other here unmeasured variables (wind velocity, sky temperature) are probably significantly different in spring. These circumstances may change the characteristics of the collector field to a great extent, which suggests identifying the models for each season separately. (It is normal that a black-box model can be effectively used only within similar ranges, where it was identified (see e.g. Ref. [11]).) It can be also seen from above that the error has increased more in case of the MPR model than in case of the IMLR model, based on which the latter model is less sensitive/more robust with respect to the change in the operation circumstances. 4. Unfortunately, some of the parameter values of the MLR model in Table 1 have been mixed up in Ref. [1]. This failure has been corrected in Table 1 in this paper. The measure of cI,A, cI,B, cI,C1 and cI,C2 has been also corrected.
5. Conclusion Based on studies in the literature, the author found that MLR is a relatively rare black-box modelling tool in the field of solar collectors in spite of its simplicity. In the present paper, the recent MLR model in Ref. [1] has been empirically improved to minimize the modelling error. On the basis of measured data of a real collector, two new, improved models (called IMLR model and MPR model) have been validated. The experiments have shown that the IMLR and the MPR models predict the collector field (outlet) temperature significantly more precisely than the MLR model. In particular, the modelling errors are 4.1% and 4.0% (instead of 4.6%). The improved models are easyto-apply with low computational demand. Based on studies in the literature, the gained precisions are even better than the ones of other more complicated black-box models with higher computational demand (e.g. ANN models) in many cases, so the improved models can be recommended for describing solar collectors (or collector fields) effectively. Based on many attempts to decrease the modelling error further, it empirically seems that the precision of the IMLR model cannot be significantly improved any more if the regression functions/equations are all linear in terms of the input variables. Based on the method of Section 4.1, it is easy to identify the proposed IMLR and MPR models for any particular collector after assigning Cases A1, A2, A3, A4, B, C1 and C2, so this general model can be applied to any type of collectors, which has the same inputs and output as in Sections 2 and 3 (after identifying tA, tB and the similar characteristic parameters). Four days chosen from different (smooth and intermittent) operations proved to be enough in the identification to gain a precise IMLR model for a given season. (With respect to the whole year, the model could be easily identified for the remained three seasons in a similar way.) The IMLR model is nearly the same accurate as the MPR model if
the circumstances are similar (from the same season) as in the identification but more accurate if the circumstances are significantly different (not from the same season) than in the identification, so the IMLR model can be considered more robust and may be more convenient for the practice. Because of the simple linear or polynomial relations (2a)-(2g) and (3a)-(3g), the computational demand is low, which may make the model useful in designing model-based controls [27]. The work carried out in the present paper has been set as a future research task in the Conclusion of Ref. [1]. Further future research may deal with checking if the IMLR and MPR models can be improved by taking into account further input variables (wind velocity, sky temperature) or by using higher-order polynomials or other nonlinear functions in the regression. Similarly precise MLRbased models may be also worked out for other working components of solar heating systems and model-based controls might be designed on the basis of the IMLR model or the MPR model. Acknowledgements The author thanks the Editor for the encouraging help in the submission process and the anonymous Referees for their valuable comments, which helped to improve this work significantly. The n Farkas and the Department of author also thanks Prof. Istva Physics and Process Control (SZIU) for the possibility of measuring szlo on the SZIU collector and his colleagues, especially Dr. La kely, in the Department of Mathematics in the Faculty of MeSze chanical Engineering (SZIU) for their contribution. nos Bolyai Research ScholThis paper was supported by the Ja arship of the Hungarian Academy of Sciences. References [1] R. Kicsiny, Multiple linear regression based model for solar collectors, Sol. Energy 110 (2014) 496e506. [2] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, third ed., John Wiley and Sons, New York, 2006. [3] J. Deng, Y. Xu, X. Yang, A dynamic thermal performance model for flat-plate solar collectors based on the thermal inertia correction of the steady-state test method, Renew. Energy 76 (2015) 679e686. s, I. Farkas, A. Biro , R. Ne meth, Modelling and simulation of a solar [4] J. Buza thermal system, Math. Comput. Simul. 48 (1998) 33e46. [5] R. Kumar, M.A. Rosen, Thermal performance of integrated collector storage solar water heater with corrugated absorber surface, Appl. Therm. Eng. 30 (2010) 1764e1768. s, I. Farkas, Solar domestic hot water system simulation using block[6] J. Buza oriented software, in: The 3rd ISES-Europe Solar World Congress (Eurosun 2000), Copenhagen, Denmark, CD-ROM Proceedings, 2000, p. 9. [7] R. Kicsiny, New delay differential equation models for heating systems with pipes, Int. J. Heat Mass Transf. 79 (2014) 807e815. [8] R. Kicsiny, Transfer functions of solar heating systems for dynamic analysis and control design, Renew. Energy 77 (2015) 64e78. [9] S.A. Kalogirou, S. Panteliou, A. Dentsoras, Modelling of solar domestic water heating systems using artificial neural networks, Sol. Energy 65 (6) (1999) 335e342. [10] S.A. Kalogirou, Applications of artificial neural-networks for energy systems, Appl. Energy 67 (2000) 17e35. czy-Víg, I. Farkas, Neural network modelling of thermal stratification in a [11] P. Ge solar DHW storage, Sol. Energy 84 (2010) 801e806. € zen, T. Menlik, S. Ünvar, Determination of efficiency of flat-plate solar [12] A. So collectors using neural network approach, Expert Syst. Appl. 35 (2008) 1533e1539. czy-Víg, Neural network modelling of flat-plate solar collectors, [13] I. Farkas, P. Ge Comput. Electron. Agric. 40 (2003) 87e102.
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[14] S.A. Kalogirou, Prediction of flat-plate collector performance parameters using artificial neural networks, Sol. Energy 80 (2006) 248e259. [15] S. Fischer, P. Frey, H. Drück, A comparison between state-of-the-art and neural network modelling of solar collectors, Sol. Energy 86 (2012) 3268e3277. [16] B. Perers, An improved dynamic solar collector test method for determination of non-linear optical and thermal characteristics with multiple regression, Sol. Energy 59 (4e6) (1997) 163e178. [17] EN 12975-2:2006, Thermal Solar Systems and Components e Solar Collectors e Part 2: Test Methods, 2006. [18] W. Kong, Z. Wang, J. Fan, P. Bacher, B. Perers, Z. Chen, S. Furbo, An improved dynamic test method for solar collectors, Sol. Energy 86 (2012) 1838e1848. [19] V. Sabatelli, D. Marano, G. Braccio, V.K. Sharma, Efficiency test of solar collectors: uncertainty in the estimation of regression parameters and sensitivity analysis, Energy Convers. Manag. 43 (2002) 2287e2295. [20] Y.C. Maxwell Li, S.M. Lu, Uncertainty evaluation of a solar collector testing system in accordance with ISO 9806-1, Energy 30 (2005) 2447e2452. [21] E. Andersen, S. Furbo, Theoretical variations of the thermal performance of
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different solar collectors and solar combi systems as function of the varying yearly weather conditions in Denmark, Sol. Energy 83 (2009) 552e565. D.M. Etter, D. Kuncicky, H. Moore, Introduction to MATLAB 7, Springer, 2004. A. Buonomano, F. Calise, A. Palombo, Solar heating and cooling systems by CPVT and ET solar collectors: a novel transient simulation model, Appl. Energy 103 (2013) 588e606. M. Kovarik, P.F. Lesse, Optimal control of flow in low temperature solar heat collectors, Sol. Energy 18 (1976) 431e435. V. Badescu, Optimal control of flow in solar collector systems with fully mixed water storage tanks, Energy Convers. Manag. 49 (2008) 169e184. s, A. L nyosi, I. Kalma r, E. Kaboldy, L. Nagy, A combined I. Farkas, J. Buza agyma solar hot water system for the use of swimming pool and kindergarten operation, in: B. Frankovic (Ed.), Energy and the Environment, vol. I, Croatian Solar Energy Association, Opatija, 2000, pp. 81e88. E.F. Camacho, F.R. Rubio, M. Berenguel, L. Valenzuela, A survey on control schemes for distributed solar collector fields. Part II: advanced control approaches, Sol. Energy 81 (2007) 1252e1272.
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Improved multiple linear regression based models for solar collectors rd Kicsiny Richa n University, Pa ter K. u. 1., 2100 Go € do €llo }, Hungary Department of Mathematics, Institute for Mathematics and Informatics, Szent Istva
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 November 2015 Received in revised form 9 January 2016 Accepted 19 January 2016 Available online xxx
Mathematical modelling is the theoretically established tool to investigate and develop solar thermal collectors as environmentally friendly technological heat producers. In the present paper, the recent and accurate multiple linear regression (MLR) based collector model in Ref. [1] is empirically improved to minimize the modelling error. Two new, improved models called IMLR model and MPR model (where MPR is the abbreviation of multiple polynomial regression) are validated and compared with the former model (MLR model) based on measured data of a real collector field. The IMLR and the MPR models are significantly more precise while retaining simple usability and low computational demand. Many attempts to decrease the modelling error further show that the gained precision of the IMLR model cannot be significantly improved any more if the regression functions are linear in terms of the input variables. In the MPR model, some of the regression functions are nonlinear (polynomial) in terms of the input variables. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Solar collectors Mathematical modelling Black-box model Multiple linear regression Polynomial regression
1. Introduction It is highly important nowadays to study and develop solar thermal collectors within the framework of environmental protection. The theoretical tool for this purpose is mathematical modelling. Two main types of mathematical models exist for collectors: physically-based (or white-box) models describe exact and known physical laws, while black-box models represent some experienced or measured correlations empirically. In the literature, there are numerous physically-based models. One of the most important and still often used ones is the HottelWhillier-Bliss model [2,3], which is also among the earliest models. The collector temperature is determined as a function of s et al. time and a space coordinate in this distributed model. Buza [4] proposed a simpler model based on the piston flow concept assuming that the collector temperature is homogeneous in space. This model is a linear ordinary differential equation (ODE) validated in Ref. [1] and is probably the simplest such (physically-based and ODE) model used in the practice (see e.g. Refs. [5e8]), which describes the transient processes of a collector with a proper accuracy. The greatest advantage of black-box type models is that it is not needed to know precisely the physical laws of a collector in order to create an usable model. Nevertheless, the model may be rather precise even if it is mathematically simple as in the case of Ref. [1].
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.renene.2016.01.056 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
In the scope of solar thermal systems, the most widely used blackbox model type may be the artificial neural network (ANN). In Ref. [9], the useful heat gained from a solar heating system as well as the temperature rise of the storage water were predicted with an ANN with an error of 7e10%, which is considered proper accuracy with respect to such systems [10]. The layer temperatures are modelled in a solar storage by means of an ANN elaborated in Ref. [11]. In particular, ANNs are frequently applied to model solar collectors separately as well. Generally speaking, ANNs are precise modelling approaches but quite troublesome to use because of the necessary training/learning process. A lot of measured data must be collected under various working conditions to train the ANN for gaining a satisfactory precision. E.g. in Ref. [12], the training requires the measured data of three months or in Ref. [13], the measurements of 17 days are needed to work out an ANN, which models a collector under similar circumstances as during the measurements. Only one so-called back-propagation algorithm (which is needed in the training) from the 11 available ones provides a proper approximation in the latter work, based on which, it can be easily concluded that the success of an ANN, and its computational demand, depend highly on the user's expertise. Also, the convergence of the algorithm indicating the end of the training can be time-consuming. Ref. [14] demonstrates the problem of uncertainty as well, where six separate ANNs are used to identify several collector parameters, so there is no general and exact instruction for designing appropriate ANNs. In fact, an
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Nomenclature t Tout
I Ta Tin V v
tA tB t1 t2
time, s homogeneous temperature inside the collector (assumed to be the same as the outlet collector temperature), C global solar irradiance on the collector surface, W/ m2 ambient temperature of the collector, C inlet collector (fluid) temperature, C volume of the collector, m3 (constant) flow rate inside the collector, m3/s time delay before Case A or A3, s time delay before Case B, s time of flowing inside the collector from the inlet to the outlet when the pump is switched on permanently, s length of time between successive measurements on the collector, s
universal applicable algorithm, which ensures a reliable and fast design of a proper ANN for a collector is still missing based on Fischer at al [15]. Because of the above difficulties on complexity/uncertainty and time/computational demand, a precise and general but still simple black-box model has been proposed recently in Ref. [1], which can be applied fast and easily for many types of solar collectors. The model (called MLR model in short) is founded on standard methods from mathematical statistics, in particular, on multiple linear regression (MLR). Based on studies in the literature, the author found that MLR is a relatively rare black-box modelling tool in the field of solar collectors in spite of its simplicity. MLR can be applied to identify collector parameters in white-box models (see Refs. [16e18]) and to model collector efficiency in collector test methods [19,20]. In Ref. [21], the annual thermal performance of collectors is modelled as a function of the annual solar radiation by means of linear regression. In the mentioned MLR model, a typical day has been divided into sub-cases, for which, separate regression equations have been proposed to reach higher precision. The regression equations describe empirical relations immediately between input and output variables of collectors. Considering the high precision (with an error of 4.6%), simple usability and low computational demand of the MLR model, it is definitely worth improving further to maximize its precision while retaining simple usability and low computational demand. Accordingly, this improvement has been set as a future research task in the Conclusion of Ref. [1]. The following are the contributions of the present paper in details: the MLR model is empirically improved by means of inserting new operating sub-cases in a rather natural way. On the basis of measured data of a real collector, two new, improved models called IMLR model and MPR model (where MPR is the abbreviation of multiple polynomial regression) are identified (based on four days), validated (based on two months) and compared with the MLR model in view of accuracy. In the MPR model, some of the regression functions/equations are nonlinear (polynomial) in terms of the input variables. Polynomial regression is generally considered as a special linear regression, since the regression functions are linear in terms of the constant parameters (which are to be identified), although not linear in the input variables. If the regression functions are all linear, the regression can be
225
called simple linear regression. Matlab [22] has been used to carry out the needed calculations numerically. This software is widely used in the field of solar engineering to simulate different systems (see e.g. Ref. [23]). The paper is structured as follows: Section 2 recalls the details of the recent MLR model for the Reader's convenience. In Section 3, the new IMLR and MPR models are worked out. These models are validated in Section 4 by means of measured and simulated data. Finally, conclusions and future research suggestions can be found in Section 5.
2. MLR model For the Reader's convenience, the MLR model [1] is recalled in details in this section. Fig. 1 shows the studied solar collector. The inputs in the MLR model are from appropriate Tin, I, Ta and Tout values. The output is from appropriate Tout values. The flow rate v is a prefixed positive constant or 0 according to the differential control, which is not only the most frequent control method but also the optimal or nearly optimal one many times [24,25]. Since the flow rate is bounded, only Tin(t t1) can function as an input in the MLR model, where t1 is a (positive) time delay and the formed output is Tout(t). Similarly, only former I(t t2) and Ta(t t2) values can function as inputs corresponding to the output Tout(t) because these effects have bounded propagation speed. (For simplicity, the same delay (t2) is assumed for I and Ta.) Of course, an adequate former value of Tout affects Tout(t) itself and functions essentially as the initial value in the MLR model at the time (t t2). In the (black-box type) MLR model, distinct sub-models were identified for separate operating conditions. It is clear, for example, that the collector behaves very differently if the pump is off (v ¼ 0) or on (v > 0) permanently. Under the same conditions, including a high enough solar irradiance, the collector temperature and thus Tout increases much more fast when the pump is off. The effect of Tin was neglected in permanently switched off case, since there is no flowing from the inlet to the outlet in the collector. Assuming a typical day, when the increase of Tout is relatively high, three separate operating conditions were distinguished in accordance with Fig. 2. The exact specification of each case is the following: Case A: The pump is switched off permanently. Case A contains the term started at the beginning of the day and finished, when the pump is first switched on. All the terms, which begin at a time when the pump is permanently off for exactly tA time and finish at the next switch-on or at the end of the day, also belong to this case. (tA is the time, which is generally enough for Tout to become not fluctuating but permanently monotone, since, intentionally, frequent fluctuations are characteristics of Cases C1 and C2.)
Fig. 1. The studied solar collector.
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Case C1 :
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ þ cout;C1 Tout ðt t2 Þ þ cC1 (1c) Case C2 :
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ þ cout;C2 Tout ðt t2 Þ þ cC2 (1d) The constant parameters cI,A, ca,A, cout,A, cA, cin,B, cI,B, ca,B, cout,B, cB, cin,C1, cI,C1, ca,C1, cout,C1, cC1, cin,C2, cI,C2, ca,C2, cout,C2, cC2 are to be identified in the model. The measurements take place at times t ¼ 0, t2, 2t2, 3t2,... Accordingly, Tout,mod that is the modelled value of Tout is calculated at times t ¼ t2, 2t2, 3t2,… from corresponding values of I(t t2), Ta(t t2), Tout(t t2) and Tin(t t1) based on Eqs. (1a)e(1d). t1 occurs in Eqs. (1b)e(1d), where t > t1 naturally holds because of the relatively small collector volume and the time schedule of the pump operation (see Fig. 2).
3. IMLR and MPR models
Fig. 2. Outlet temperature, solar irradiance and pump operation on a typical day.
Case B: The pump is switched on permanently. Case B contains all the terms, which begin at a time when the pump is permanently on for exactly tB time and finish at the next switch-off. (tB is the time, which is generally enough for Tout to become not fluctuating but permanently monotone.) Case C relates to frequent switch-ons and -offs. It can be admitted that two further cases can be significantly distinguished in Case C: in general, Tout is increasing before and decreasing after the solar noon, respectively, so this case is divided into Cases C1 and C2. Case C1: Terms not within Case A and Case B before the solar noon (12:30 according to summer time). Case C2: Terms not within Case A and Case B after the solar noon. For making the modelling accuracy as well as possible, different sub-models were elaborated based on MLR for all operating cases. Tin is an input in the sub-models of Case B, Case C1 and Case C2, since some fluid is flowing from the inlet to the outlet of the collector in accordance with that the pump is permanently or intermittently on in these cases. For lack of flowing, Tin is neglected in the sub-model of Case A as the pump is permanently switched off. The below linear equations form the sub-models of the operating cases in the MLR model:
Case A :
Tout;mod ðtÞ
¼ cI;A Iðt t2 Þ þ ca;A Ta ðt t2 Þ þ cout;A Tout ðt t2 Þ þ cA (1a) Case B :
Tout;mod ðtÞ
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ þ cout;B Tout ðt t2 Þ þ cB
(1b)
The MLR model is empirically improved in this section. The resulted new models will be called IMLR and MPR models. The steps of the improvements are the following (see also Fig. 2): 1. The (largest) operating case Case A is divided into four sub-cases as follows: Case A1: This case consists of the time period from the beginning of the day to the time when the solar irradiance is first greater than 10 W/m2. This case practically belongs to the term of no irradiance in the first half of the day. Clearly, the effect of the irradiance can be neglected here. This makes the corresponding mathematical relation simpler and refines the model globally. Based on experiments, this step decreases the modelling error. Case A2: This case consists of the time period from the end of Case A1 to the time when the solar irradiance is first greater than 100 W/m2. Usually, this time is followed by a very intensive increase in the irradiance, so this is apparently the time of sunrise, when the irradiance changes from (mostly) diffuse to (mostly) direct. Based on experiments, it is worth distinguishing this case in the model to decrease the modelling error. Case A3: Time periods besides Cases A1, A2 and A4 (see below) within Case A. Case A4: The last three hours of the day. In essence, the term after Case C2 corresponds to the free cooling of the collector from a relatively high temperature. Based on experiments, this section cannot be modelled well with a single relation, so it should be divided into sub-sections. Empirically, the problem can be solved well with only two sub-parts if the last three hours are separated. This may be caused by that the temperature is rather high in the beginning of the cooling process, where the highly nonlinear effect of heat radiation (to the colder environment) is probably considerable. After 21:00, the temperature is always much lower (and decreases much more slowly), so the heat radiation is more negligible. In addition, the temperature range after Case C2 is still rather big, which makes for a single linear relation impossible to model the cooling process precisely. Clearly, the effect of the irradiance can be neglected here because of the night.
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2. To avoid unneeded overestimates in Case A3, the minimum of Tout,mod(t) from Eq. (2c) below and the previously modelled Tout,mod(t t2) is considered as the modelled outlet temperature at time t in Case A3 if I(t 2t2) I(t t2)10 W/m2 and the pump stagnates for at least 20 min. This inequality means in essence that the solar irradiance has not increased since the last measured time (10 W/m2 compensates the accidental measuring error), so it is meaningful to assume that the collector temperature has neither increased considering also that the pump stagnates for at least 20 min. This time is assumed to be enough for the collector (with a rather small volume) to warm up during a stagnation (cf. Case A in the MLR model). This correction is more important in the IMLR and MPR models since Case A3 has much less measured data for identification than Case A in the MLR model because of its much shorter length. In addition, the temperature range occurring during Case A3 is usually rather wide, which makes the precise modelling difficult. It is also suggested to apply the maximum of the measured Tout temperature occurring in the identification as an upper limit for the modelled Tout temperature. Based on experiments, this upper limit does not improve significantly the overall modelling precision for a long time (for several months) but rarely occurring rough overestimates can be avoided in this way. 3. The coefficients of the zeroth-order members (cf. cA, cB, cC1 and cC2 in Eqs. (1a)e(1d) in the MLR model) are set zero in Eqs. (2a)e(2g) and (3a)e(3g) below. (The corresponding identification of each case can be carried out easily with a suitable statistical or spreadsheet program (SPSS, Excel, etc.).) This set is in line with the physical phenomenon that the collector (outlet) temperature must be zero if all the inputs Tin, I, Ta and the previous collector temperature are zero. Based on experiments, this natural constraint results in a bit lower precision in the identification but higher precision in the validation, that is the modelling error decreases.
Case C1 :
Tout;mod ðtÞ ¼ ca;A1 Ta ðt t2 Þ þ cout;A1 Tout ðt t2 Þ
þ cout;C1 Tout ðt t2 Þ (2f) Case C2 :
Case A2:
þ cout;C2 Tout ðt t2 Þ (2g) ca,A1, cout,A1, cI,A2, ca,A2, cout,A2, cI,A3, ca,A3, cout,A3, ca,A4, cout,A4, cin,B, cI,B, ca,B, cout,B, cin,C1, cI,C1, ca,C1, cout,C1, cin,C2, cI,C2, ca,C2, cout,C2 are constant parameters to be identified. The following second-order equations correspond to the above operating cases:
Case A1 :
Case A3 :
Tout;mod ðtÞ
¼ ca;A1 Ta ðt t2 Þ þ cout;A1 Tout ðt t2 Þ þ ca;A1;2 Ta2 ðt t2 Þ (3a) Case A2 :
Tout;mod ðtÞ
¼ cI;A2 Iðt t2 Þ þ ca;A2 Ta ðt t2 Þ þ cout;A2 Tout ðt t2 Þ þ cI;A2;2 I 2 ðt t2 Þ þ ca;A2;2 Ta2 ðt t2 Þ (3b) Case A3 : Tout;mod ðtÞ ¼ cI;A3 Iðt t2 Þ þ ca;A3 Ta ðt t2 Þ þ cout;A3 Tout ðt t2 Þ þ cI;A3;2 I 2 ðt t2 Þ þ ca;A3;2 Ta2 ðt t2 Þ (3c) Case A4 :
Tout;mod ðtÞ
¼ ca;A4 Ta ðt t2 Þ þ cout;A4 Tout ðt t2 Þ þ ca;A4;2 Ta2 ðt t2 Þ (3d) Tout;mod ðtÞ
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ 2 ðt t1 Þ þ cI;B;2 I 2 ðt t2 Þ þ cout;B Tout ðt t2 Þ þ cin;B;2 Tin
Tout;mod ðtÞ ¼ cI;A2 Iðt t2 Þ þ ca;A2 Ta ðt t2 Þ þ cout;A2 Tout ðt t2 Þ
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ
Case B : (2a)
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ
The following linear equations correspond to the above operating cases:
Case A1 :
227
þ ca;B;2 Ta2 ðt t2 Þ
(2b)
(3e) Case C1 :
Tout;mod ðtÞ
Tout;mod ðtÞ
¼ cin;C1 Tin ðt t1 Þ þ cI;C1 Iðt t2 Þ þ ca;C1 Ta ðt t2 Þ
¼ cI;A3 Iðt t2 Þ þ ca;A3 Ta ðt t2 Þ þ cout;A3 Tout ðt t2 Þ
2 ðt t1 Þ þ cout;C1 Tout ðt t2 Þ þ cin;C1;2 Tin
(2c)
þ cI;C1;2 I2 ðt t2 Þ þ ca;C1;2 Ta2 ðt t2 Þ Case A4 :
Tout;mod ðtÞ ¼ ca;A4 Ta ðt t2 Þ þ cout;A4 Tout ðt t2 Þ
(3f) (2d)
Case B :
Tout;mod ðtÞ
¼ cin;C2 Tin ðt t1 Þ þ cI;C2 Iðt t2 Þ þ ca;C2 Ta ðt t2 Þ
Tout;mod ðtÞ
2 ðt t1 Þ þ cout;C2 Tout ðt t2 Þ þ cin;C2;2 Tin
¼ cin;B Tin ðt t1 Þ þ cI;B Iðt t2 Þ þ ca;B Ta ðt t2 Þ þ cout;B Tout ðt t2 Þ
Case C2 :
(2e)
þ cI;C2;2 I2 ðt t2 Þ þ ca;C2;2 Ta2 ðt t2 Þ (3g)
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ca,A1, cout,A1, ca,A1,2, cI,A2, ca,A2, cout,A2, cI,A2,2, ca,A2,2, cI,A3, ca,A3, cout,A3, cI,A3,2, ca,A3,2, ca,A4, cout,A4, ca,A4,2, cin,B, cI,B, ca,B, cout,B, cin,B,2, cI,B,2, ca,B,2, cin,C1, cI,C1, ca,C1, cout,C1, cin,C1,2, cI,C1,2, ca,C1,2, cin,C2, cI,C2, ca,C2, cout,C2, cin,C2,2, cI,C2,2, ca,C2,2 are constant parameters to be identified. In the IMLR model, all regression functions are linear with respect to the input variables, that is, the regression equations are Eqs. (2a)e(2g). To improve the precision of the IMLR model, the linear regression equations of Cases A4, C1 and C2 have been changed to the polynomial equations Eqs. (3d), (3f) and (3g), since polynomial regression generally provides more flexible curvefitting functionality and thus higher precision. The gained model is the MPR model. (See Note 2 of Remark 3.1 below for more details.) Remark 3.1 1. The above second-order equations (Eqs. (3a)e(3g)) may be the simplest higher order relations, where each measured input occurs at first and second orders. Tout(tt2) occurs at first order only, since it works as an initial function, which has a very direct and simple connection with Tout,mod(t) based on the following simple assumption: Tout,mod(t) would be equal to Tout(tt2) if the other inputs took no effect. 2. Many other attempts have been made in the present research to decrease the modelling error further. Because of limits in volume, only a few of them can be mentioned here. For example, Cases B, C1 and C2 have been also divided into sub-cases and/or more and more days have been used for identification. Based on experiments, these complexities have not significantly improved the precision. Similarly, the precision of the MPR model cannot be improved significantly if Eqs. (3a)e(3c) and (3e) are used instead of (2a)(2c) and (2e). That is why the simpler linear relations are remained for Cases A1eA3 and B in the MPR model in accordance with the high R2 values of the IMLR model in these Cases (see Table 2).
4. Validation and comparison In this section, the IMLR model (Eqs. (2a)e(2g)) and the MPR model (Eqs. (2a)e(2c), (3d), (2e), (3f), (3g)) are identified and validated based on simulation and measured data then they are compared with the MLR model in view of precision. The results and figures of the MLR model used in the comparison are from Ref. [1]. The identification and the validation of the IMLR and MPR models are based on the same days as in case of the MLR model in Ref. [1]. The used real flat plate collector field of 33.3 m2 [26] at the Szent n University (SZIU) in Go € do €llo }, Hungary (SZIU collector in Istva short) is also the same. Tout, Tin, I, Ta and v are measured once a minute at the SZIU collector. The measured value of Tout(0) is used as initial condition in the models. The later measured values of Tout is used only in the identification and for comparison purposes. The measured inputs are Tin, I, Ta in the IMLR model (see Eqs. (2a)e(2g)). 2 , I2 and T 2 can be considered as further In the MPR model, Tin a measured inputs (see Eqs. (3d) and (3f)e(3g)), in terms of which, the regression equations are linear, so standard MLR routines can be used in the identification below. tB is set 4 min in all of the three models, while tA has been changed to 10 min in the IMLR and the MPR models (from 20 min in the MLR model). This shorter time for considering the pump already permanently switched off removes naturally some terms from Cases C1 and/or C2. This change results in less measured data for Cases C1 and C2, which makes their identification less accurate but also refines the model on a day
globally. Empirically, the second effect is more dominating and the average daily modelling error decreases. The other technical details of the identification and validation of the IMLR and MPR models are very similar as in case of the MLR model [1], so they are not fully specified below.
4.1. Identification In the identification, four measured days are chosen from a selected season (summer) so that they cover a wide variety of possible operating conditions. Two of them (2nd July 2012, 24th June 2012) are with relatively few pump switches (smooth operation) and two others (28th June 2012, 8th June 2012) are with relatively many pump switches (intermittent operation) because the operating conditions can be characterized well with the pump operation (if the pump is switched on or off). According to many computer experiments, such four days are enough to gain a rather precise identified model. These four days are chosen from the first third of the summer. In this manner, the identified model can be already applied in the remained part of the summer (two months). Ten standard MLR routines are used independently to identify parameters ca,A1, cout,A1, …, ca,C2,2 in the IMLR and MPR models (see Eqs. (2a)e(2g), (3d) and (3f)e(3g)) based on the measured data of the operating cases (Cases A1, A2, A3, A4, B, C1, C2). According to the minute-based measuring, t2 is set 1 min v ¼ 0.98 m3/h if the pump is switched on and V ¼ 0.027 m3, from which t1 z 1.5 min. The measured value of Tin(tt1) should be used in the equations in the identification (and in the validation). t1 ¼ 1.5 min is not suitable for the minute-based measuring, so Tin(tt1) is replaced with (Tin(tt2) þ Tin(t 2t2))/2 in the identification. The well-known, standard MLR routine (using least squares method) is available and easy-to-apply in most spreadsheet or statistical programs (Excel, SPSS, etc.) with low computational demand, so it is not detailed here. The parameters of the IMLR model identified with the ten independent MLR routines can be seen in Table 1 along with the corresponding values of the MLR model (rounded to four decimals). Usually, the results of the MLR routine is evaluated with the square of the corresponding correlation coefficient R2. Table 2 contains the R2 values resulted from the whole identification for each case (rounded to three decimals). The IMLR and the MPR models can be called reasonable as R2 > 0.75 in most cases, even, it is close to 1 in Cases A1, A2, A3, B and C2 for both models and close to 1 in Case A4 for the MPR model (see Table 2). Nevertheless, the indices of Table 3 are more expressive and important in our case, especially, in light of the comparison with the MLR model accepted already in the literature. Table 3 presents the average of error (time average of the difference between the modelled and measured outlet temperatures) and the average of absolute error (time average of the absolute difference between the modelled and measured outlet temperatures) values for two days (2nd July 2012, 28th June 2012) of the identification of all models. The average of absolute error values are also shown in proportion to the difference between the daily maximal and minimal measured outlet temperature values (in %). Table 3 also contains the mean of these % values with respect to all of the four days of the identification (3.2% for the IMLR model). Figs. 3 and 4 compare the measured and modelled outlet temperatures in case of the identification of the MLR and the IMLR models for a day with smooth and intermittent pump operation, respectively. The figures also show the operating state of the pump.
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229
Table 1 Parameter values of the MLR, IMLR and MPR models.
cI,A, m2K2/W ca,A, e cout,A, e c A, C ca,A1, e cout,A1, e cI,A2, m2K2/W ca,A2, e cout,A2, e cI,A3, m2K2/W ca,A3, e cout,A3, e ca,A4, e cout,A4, e ca,A4,2, 1/ C cin,B, e cI,B, m2K2/W ca,B, e cout,B, e cB, C cin,C1, e cI,C1, m2K2/W ca,C1, e cout,C1, e cin,C1,2, 1/ C cI,C1,2, m4K3/W2 ca,C1,2, 1/ C cC1, C cin,C2, e cI,C2, m2K2/W ca,C2, e cout,C2, e cC2, C cin,C2,2, 1/ C cI,C2,2, m4K3/W2 ca,C2,2, 1/ C
MLR model
IMLR model
MPR model
0.0048 0.0191 0.9815 0.0543 e e e e e e e e e e e 0.0354 0.0024 0.0086 0.9552 0.5324 0.0277 0.0056 0.2289 0.7736 e e e 2.9295 0.0655 0.0021 0.0823 0.8916 0.3457 e e e
e e e e 0.0048 0.9945 0.0008 0.0054 0.9939 0.0062 0.0298 0.9723 0.0004 0.9978 e 0.0436 0.0021 0.0036 0.9451 e 0.0493 0.0093 0.3361 0.7225 e e e e 0.1714 0.0026 0.0935 0.7981 e e e e
e e e e 0.0048 0.9945 0.0008 0.0054 0.9939 0.0062 0.0298 0.9723 0.0002 0.9978 0.00003 0.0436 0.0021 0.0036 0.9451 e 0.1818 0.0179 0.2666 0.6926 0.0013 0.00002 0.00004 e 0.3629 0.0163 0.1961 0.7593 e 0.002 0.00002 0.0054
Table 2 R2 values of the IMLR and MPR models from the identification.
IMLR model MPR model
A1
A2
A3
A4
B
C1
C2
1.000 1.000
1.000 1.000
0.999 0.999
0.631 1.000
0.994 0.994
0.793 0.794
0.961 0.962
Fig. 3. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 2nd July 2012 in case of the MLR and IMLR models.
temperature is to be modelled in the validation of course and not to be measured. The modelled days are from 3rd July 2012 to 31st August 2012. It means 56 days in the validation because of minor technical interruptions during the operation. The measured and modelled outlet temperatures are compared and evaluated in case of each model. Table 3 presents the average of error and average of absolute error values for the above two days for the validation in case of all models. The average of absolute error values are also shown in proportion to the difference between the daily maximal and minimal measured outlet temperature values as well (in %). Table 3 also contains the mean of these % values with respect to the whole time period for the validation 3rd July e 31st August in case of all models (4.6% for the MLR model, 4.1% for the IMLR model and 4.0% for the MPR model). Figs. 5 and 6 present the measured and modelled outlet temperatures in case of the MLR and the IMLR models for a day with smooth (3rd August 2012) and intermittent operation (5th August 2012), respectively. The figures also show the operating state of the pump.
Table 3 Average of error values and average of absolute error values with the models. MLR model Identification
2nd July (smooth operation) 28th June (intermittent operation)
Validation
Mean % value for the whole identification (four days) 3rd August (smooth operation) 5th August (intermittent operation) Mean % value for the whole validation (3rd July e 31st August)
Average Average Average Average Average Average Average Average Average Average
of of of of of of of of of of
error absolute error absolute absolute error absolute error absolute absolute
error error error error error error
0.47 C 2.79 C; 4.6% 0.23 C 3.01 C; 5.2% 4.7% 1.31 C 2.85 C; 4.5% 1.58 C 3.07 C; 5.2% 4.6%
IMLR model
0.53 C 1.64 C; 2.7% 0.80 C 1.84 C; 3.2% 3.2% 1.18 C 1.92 C; 3.0% 1.90 C 2.30 C; 3.9% 4.1%
MPR model 0.64 C 1.59 C; 2.6% 0.55 C 1.80 C; 3.1% 3.1% 1.41 C 1.89 C; 3.0% 1.86 C 2.22 C; 3.8% 4.0%
4.2. Validation
4.3. Comparison and assessment
All identified models are applied with the related measured inputs from the remaining part of the summer in the validation. One input is changed in comparison with the inputs in the identification that is the modelled value Tout,mod(tt2) is used as Tout(tt2) in the models (in Eqs. (1a)e(1d), (2a)e(2g), (3d) and (3f)e(3g) (not Tout,meas(tt2))) according to that the outlet
It has been shown that the collector field (outlet) temperature can be modelled precisely with the IMLR and the MPR models, in particular, more precisely than with the also accurate MLR model. On the basis of the validation, the mean modelling error is 4.1% with the IMLR model, 4.0% with the MPR model and 4.6% with the MLR model, so the accuracy of the IMLR and the MPR models can be
230
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Fig. 4. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 28th June 2012 in case of the MLR and IMLR models.
Fig. 5. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 3rd August 2012 in case of the MLR and IMLR models.
considered very well. The R2 values with respect to the measured and modelled data of four selected whole days (two identified days and two validated days) have been calculated in case of all models as additional information. Table 4 contains the results, which reinforce that the IMLR and the MPR models have higher precision and show that the precision of the IMLR and the MPR models are nearly the same. Remark 4.1 1. It should be noted that the collector field outlet temperature is measured on the outside surface of the outlet pipe of the SZIU collector and not inside the collector for technical reason. Because of the inaccuracy caused by this outside measuring and that the collector field is charged with such disturbances (shadowing effect of clouds), which are quite difficult to predict, the outlet temperature cannot be expected to be modelled perfectly. The relatively small volume of the collector field also contributes to the difficulties, since it causes relatively fast and high changes in the collector temperature under the
Fig. 6. Measured Tout,meas and modelled Tout,mod collector field (outlet) temperatures on 5th August 2012 in case of the MLR and IMLR models.
disturbances. In view of these difficulties the accuracy of the IMLR and the MPR models can be called very well. 2. It is fair to apply the measured Tout(0) value as initial temperature in the identification and validation of the models, since applying known initial condition is normal and general in the field of mathematical modelling (see e.g. ODE models), otherwise, a mathematical construction cannot be generally expected to model real processes precisely. E.g. the current state (temperature) of a solar collector depends on not only the input variables but also on the former state of the collector. Nevertheless, based on simulations not fully detailed here because of limits in volume, the IMLR and MPR models are rather accurate even with a roughly estimated initial temperature. For example, if the initial temperature Tout(0) is not measured, it can be substituted with the measured value Ta(0) (which is generally close to Tout(0)) in the above identified IMLR and MPR models. In particular, the modelling error changes to 3.5% (from 3.0%) and 4.4% (from 3.9%) in case of the IMLR model and 3.4% (from 3.0%) and 4.2% (from 3.8%) in case of the MPR model on the validated days 3rd August and 5th August, respectively, if Ta(0) is used as initial value. This represents a better (smaller) modelling error than in case of the MLR model even with known initial condition (where these values are 4.5% and 5.2%, respectively). This shows the robustness of the IMLR and MPR models with respect to the uncertainties in relation to the initial condition. Although, initial conditions are often not used in black-box models, there are also counterexamples. For example, the layer temperatures in a solar storage is modelled with an ANN in Ref. [11], where the calculated layer temperatures of the previous time step is needed among the inputs at each time step except the first one, when only measured layer temperatures are available and thus must be used. 3. It should be mentioned that the above models, identified based on four summer days, have been applied to two spring months (April and May 2012) as well. The resulted modelling errors are 4.9% (instead of 4.1%) with the IMLR model and 5.0% (instead of 4.0%) with the MPR model for this time period. These values are still rather good, although, they are significantly higher than for the above modelled summer period (July and August). The following may explain these results: the orbit of the sun gets lower in spring than in summer, so the surrounding objects (e.g.
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231
Table 4 R2 values with the MLR and IMLR models. Date
2nd July 2012 (identification)
28th June 2012 (identification)
3rd August 2012 (validation)
5th August 2012 (validation)
MLR model IMLR model MPR model
0.976 0.989 0.990
0.972 0.985 0.985
0.978 0.987 0.988
0.968 0.982 0.983
trees) partly shade the collector field more often, which makes the precise modelling more difficult, since the measured irradiance corresponds not to the whole collector field in such cases. On the other hand, the range of the inputs (I,Ta, etc.) and other here unmeasured variables (wind velocity, sky temperature) are probably significantly different in spring. These circumstances may change the characteristics of the collector field to a great extent, which suggests identifying the models for each season separately. (It is normal that a black-box model can be effectively used only within similar ranges, where it was identified (see e.g. Ref. [11]).) It can be also seen from above that the error has increased more in case of the MPR model than in case of the IMLR model, based on which the latter model is less sensitive/more robust with respect to the change in the operation circumstances. 4. Unfortunately, some of the parameter values of the MLR model in Table 1 have been mixed up in Ref. [1]. This failure has been corrected in Table 1 in this paper. The measure of cI,A, cI,B, cI,C1 and cI,C2 has been also corrected.
5. Conclusion Based on studies in the literature, the author found that MLR is a relatively rare black-box modelling tool in the field of solar collectors in spite of its simplicity. In the present paper, the recent MLR model in Ref. [1] has been empirically improved to minimize the modelling error. On the basis of measured data of a real collector, two new, improved models (called IMLR model and MPR model) have been validated. The experiments have shown that the IMLR and the MPR models predict the collector field (outlet) temperature significantly more precisely than the MLR model. In particular, the modelling errors are 4.1% and 4.0% (instead of 4.6%). The improved models are easyto-apply with low computational demand. Based on studies in the literature, the gained precisions are even better than the ones of other more complicated black-box models with higher computational demand (e.g. ANN models) in many cases, so the improved models can be recommended for describing solar collectors (or collector fields) effectively. Based on many attempts to decrease the modelling error further, it empirically seems that the precision of the IMLR model cannot be significantly improved any more if the regression functions/equations are all linear in terms of the input variables. Based on the method of Section 4.1, it is easy to identify the proposed IMLR and MPR models for any particular collector after assigning Cases A1, A2, A3, A4, B, C1 and C2, so this general model can be applied to any type of collectors, which has the same inputs and output as in Sections 2 and 3 (after identifying tA, tB and the similar characteristic parameters). Four days chosen from different (smooth and intermittent) operations proved to be enough in the identification to gain a precise IMLR model for a given season. (With respect to the whole year, the model could be easily identified for the remained three seasons in a similar way.) The IMLR model is nearly the same accurate as the MPR model if
the circumstances are similar (from the same season) as in the identification but more accurate if the circumstances are significantly different (not from the same season) than in the identification, so the IMLR model can be considered more robust and may be more convenient for the practice. Because of the simple linear or polynomial relations (2a)-(2g) and (3a)-(3g), the computational demand is low, which may make the model useful in designing model-based controls [27]. The work carried out in the present paper has been set as a future research task in the Conclusion of Ref. [1]. Further future research may deal with checking if the IMLR and MPR models can be improved by taking into account further input variables (wind velocity, sky temperature) or by using higher-order polynomials or other nonlinear functions in the regression. Similarly precise MLRbased models may be also worked out for other working components of solar heating systems and model-based controls might be designed on the basis of the IMLR model or the MPR model. Acknowledgements The author thanks the Editor for the encouraging help in the submission process and the anonymous Referees for their valuable comments, which helped to improve this work significantly. The n Farkas and the Department of author also thanks Prof. Istva Physics and Process Control (SZIU) for the possibility of measuring szlo on the SZIU collector and his colleagues, especially Dr. La kely, in the Department of Mathematics in the Faculty of MeSze chanical Engineering (SZIU) for their contribution. nos Bolyai Research ScholThis paper was supported by the Ja arship of the Hungarian Academy of Sciences. References [1] R. Kicsiny, Multiple linear regression based model for solar collectors, Sol. Energy 110 (2014) 496e506. [2] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, third ed., John Wiley and Sons, New York, 2006. [3] J. Deng, Y. Xu, X. Yang, A dynamic thermal performance model for flat-plate solar collectors based on the thermal inertia correction of the steady-state test method, Renew. Energy 76 (2015) 679e686. s, I. Farkas, A. Biro , R. Ne meth, Modelling and simulation of a solar [4] J. Buza thermal system, Math. Comput. Simul. 48 (1998) 33e46. [5] R. Kumar, M.A. Rosen, Thermal performance of integrated collector storage solar water heater with corrugated absorber surface, Appl. Therm. Eng. 30 (2010) 1764e1768. s, I. Farkas, Solar domestic hot water system simulation using block[6] J. Buza oriented software, in: The 3rd ISES-Europe Solar World Congress (Eurosun 2000), Copenhagen, Denmark, CD-ROM Proceedings, 2000, p. 9. [7] R. Kicsiny, New delay differential equation models for heating systems with pipes, Int. J. Heat Mass Transf. 79 (2014) 807e815. [8] R. Kicsiny, Transfer functions of solar heating systems for dynamic analysis and control design, Renew. Energy 77 (2015) 64e78. [9] S.A. Kalogirou, S. Panteliou, A. Dentsoras, Modelling of solar domestic water heating systems using artificial neural networks, Sol. Energy 65 (6) (1999) 335e342. [10] S.A. Kalogirou, Applications of artificial neural-networks for energy systems, Appl. Energy 67 (2000) 17e35. czy-Víg, I. Farkas, Neural network modelling of thermal stratification in a [11] P. Ge solar DHW storage, Sol. Energy 84 (2010) 801e806. € zen, T. Menlik, S. Ünvar, Determination of efficiency of flat-plate solar [12] A. So collectors using neural network approach, Expert Syst. Appl. 35 (2008) 1533e1539. czy-Víg, Neural network modelling of flat-plate solar collectors, [13] I. Farkas, P. Ge Comput. Electron. Agric. 40 (2003) 87e102.
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