Monitoring energy efficiency of condensing boilers via hybrid first-principle modelling and estimation

The operating principle of condensing boilers is based on exploiting heat from ﬂ ue gases to pre-heat cold water at the inlet of the boiler: by condensing into liquid form, ﬂ ue gases recover their latent heat of vaporization, leading to 10 e 12% increased ef ﬁ ciency with respect to traditional boilers. However, monitoring the energy ef ﬁ ciency of condensing boilers is complex due to their nonlinear dynamics: currently, (static) nonlinear ef ﬁ ciency curves of condensing boilers are calculated at quasi-stationary regime and ‘ a posteriori ’ , i.e. from data collected during chamber tests: therefore, with this static approach, it is possible to monitor the energy ef ﬁ ciency only at steady-state regime. In this work we propose a novel model-based monitoring approach for condensing boilers that extends the operating regime for which monitoring is possible: the approach is based on a hybrid dynamic model of the condensing boiler, where state-dependent switching accounts for dynamically changing condensing/non condensing proportions. Monitoring the energy ef ﬁ ciency over the boiler's complete dynamic regime is possible via switching estimators designed for the different condensing/non condensing modes. By using real-world boiler ef ﬁ ciency data we show that the proposed approach results in a (dynamic) nonlinear ef ﬁ ciency curve which gives a more complete description of the condensing boilers operation than static nonlinear ef ﬁ ciency curves: in addition, the dynamic curve can be derived ‘ a priori ’ , i.e. from ﬁ rst principles, or from data collected during normal boiler operation (without requiring special chamber tests).


Introduction
Many reports and data confirm that in both Europe and US energy used by buildings accounts for over one third of energy consumption and CO 2 emissions [1]. Among the possible ways to improve energy efficiency in the building sector, developing better control and energy monitoring strategies can result in 10e40% energy savings [2]. The most accurate control and energy monitoring strategies are model-based: this means that mathematical models of the energy and heat transfer dynamics of the building equipment are developed and used to design better operational strategies (for energy-efficient control) or to monitor deviations of the energy consumptions from nominal patterns (for monitoring of energy efficiency). In this work we will focus on monitoring the energy efficiency of condensing boilers, which are becoming a more and more crucial equipment inside heating, ventilating and air conditioning (HVAC) systems: in fact, boiler operation has been estimated in around 85% of the HVAC energy consumption and 67% of the HVAC CO 2 emissions [3]. Nowadays condensing boilers are replacing less energy-efficient traditional boilers [4,5]. The operating principle of condensing boilers is based on exploiting heat from flue gases to pre-heat cold water at the inlet of the boiler. When flue gases condense into liquid form, they recover their latent heat of vaporization (see Fig. 1). The condensing mode can result in as much as 10e12% increase in efficiency with respect to traditional boilers. For the condensing mode to be activated, return water temperature at the boiler inlet should be low and below the dew temperature of the flue gas: when this condition is not maintained, the boiler will operate in the traditional noncondensing mode [6].
External conditions and ageing (wearing of materials, isolation, limescale, etc.) will lead to major deviations from the nominal energy efficiency of condensing boilers, thus calling for a constant monitoring of efficiency. Due to their bimodal (condensing/noncondensing) behavior, model-based monitoring of condensing boilers is more complex than model-based monitoring of traditional boilers: this is due to the complexity in modelling boiler dynamics over their entire operating range. Currently, the efficiency of condensing boilers is calculated via nonlinear efficiency curves, which are derived 'a posteriori', i.e. from data collected during special tests in adiabatic chambers, performed by the manufacturer at static or quasi-stationary regime (the interested reader can consult the technical libraries of many boiler manufacturers). However, static boiler operation in adiabatic chambers can be very different than dynamic boiler operation in buildings [8]. In other words, the range of validity of models derived from experimental tests is limited to the domain of the experimental data, which is the main limitation of the current state of the art [9]: therefore, it not appropriate to use the efficiency curve derived from static data to design monitoring strategies for dynamic regimes. New model-based monitoring methods are needed, which should be derived 'a priori', i.e. involving parameters derived from physical considerations, or from data collected during dynamic operation without requiring special chamber tests. By capturing the dynamic operation of condensing boilers instead of the static one, energy efficiency could then be monitored in the entire range of condensing boiler operation. Achieving this goal is at the core of the presented work, which overcomes the state of the art as summarized in the next section.

Related work
Mathematical models of boilers are meant to estimate the boiler efficiency as a function of certain design parameters. For traditional boilers several dynamic models have been developed, e.g. firstprinciple models [10,11], fuzzy models [12], Markovian jump models [13], and nonlinear models [14]. However, the situation for condensing boilers is less rich: static models of condensing boilers are dominating literature: in Refs. [15,16] the static efficiency is computed or measured as a function of the return water temperature, while in Ref. [17] the static seasonal efficiency of condensing boilers is normalized with respect to efficiency at full load. Static models are also used to calculate flue gas exit temperature and condensation rate of water vapor as a function of return water temperature: in Ref. [18] a payback period for retrofitting a conventional boiler into a condensing boiler is calculated based on static combustion and heat transfer calculations; in Ref. [19] a static model of a condensing heater is developed to evaluate the impact of relative humidity on the efficiency; Ref. [20] derives static charts for boiler combustion efficiency according to different natural gas blends characteristics parameters. On the other end of the spectrum are models based on computational fluid dynamic [21] that, due to their complexity, can be used to study new materials, but they cannot be used for real-time monitoring or control purposes [22].
The simplest way to describe some dynamical behavior of condensing boilers is the lumped element model [23], whose main limitation is assuming that the heat exchange occurs in a single point: this does not allow to differentiate between the wet exchange of condensing mode and the dry exchange of noncondensing mode. For this reason, a more common approach is to couple the lumped element model with a nonlinear efficiency curve [24]: unfortunately, as the nonlinear efficiency curve is obtained from steady-state operation, there is no guarantee that the same efficiency curve is valid also in dynamic regime: actually, the boiler efficiency during transient behavior is typically lower than at steady state [8]. The approach in Ref. [25] proposes a set of equations based on steady-state operation and two point heat exchange which describe the main physical processes inherent to the boiler sub-components; in Ref. [26] the heat transfer between the flue gases and the water is calculated by the classical ε-NTU method and a fixed distribution of dry/wet heat exchange; in Ref. [27] the dynamic behavior of the model is obtained by extending the nonlinear efficiency curve (obtained) from steady-state data with thermal mass considerations; in Ref. [28], an analytical heat transfer model in a secondary heat exchanger was proposed to calculate the heat transferred from flue gas to cooling water and the condensation rate of water vapor in the flue gas. Unfortunately, by relying on the lumped element model idea, all these approaches neglect that the heat transfer in condensing boilers is spatially distributed and time dependent: the proportion of dry/wet exchange in condensing boilers change dynamically in space and time. Furthermore, in most works mentioned above, heat transfer is considered only through water and gas, while a more complex and realistic heat exchange model should include the heat transfer via the extended surface and the tube wall. Despite the numerous modelling approaches which have been listed, we can clearly identify a series of shortcomings in existing condensing boiler models: Heat transfer dynamics are oversimplified to a static nonlinear efficiency curve. The efficiency curve is calculated by installers and specifiers of condensing boilers, at steady-state (e.g. in special adiabatic rooms). Therefore, current models are not able to capture the true heat transfer dynamics. The bimodal condensing/non-condensing behavior is oversimplified with two heat exchangers, one for dry and one for wet heat exchange, always in a fixed proportion. A model is required that can capture dynamical changes in space and time of dry/ wet heat exchange.
With this work we will bridge these gaps and arrive to a novel monitoring approach. First, we exploit some preliminary ideas by the authors [29] to develop a model with state-dependent switching triggered by the temperature of the combustion gas: the switching mechanism is able to describe highly dynamic behavior in heat transfer and distribution of dry/wet heat exchange. Then, we show that monitoring the energy efficiency over the boiler's complete dynamic regime is possible via switching estimators designed for the different condensing/non condensing modes. By using real-world boiler efficiency data we show that the proposed approach results in a (dynamic) nonlinear efficiency curve which gives a more complete description of the condensing boilers operation than static nonlinear efficiency curves: in addition, the dynamic curve can be derived 'a priori', i.e. from first principles, or from data collected during normal operation (without requiring special chamber tests). Due to its switching nature, the model can be suited for most of the hybrid control algorithms developed in recent years in the field of smart heating systems [16,23,30e32]. In addition, the proposed monitoring approach can potentially be combined with other space heating components, to the purpose of dynamic analysis of hybrid heat pump generators in residential [33] and district heating [34] systems. In particular, the monitoring algorithm can be integrated as a module of rule-based, model predictive control or multiobjective energy management systems to predict energy supply and demands in buildings The rest of the paper is organized as follows: Section 2 gives the basics of dynamic boiler operation and develops a hybrid dynamical model of the condensing boiler, while Section 3 presents the dynamic monitoring architecture. In Section 4 the results coming from real-world boiler efficiency data are presented. Section 5 concludes the work.

Condensing boiler operation
Let us first present the basics of condensing boiler dynamic operation. Fig. 1 highlights some of the key components of the boiler: the burner with combustion chamber, two heat exchangers (primary and secondary) and the stack. The burner's task is to mix fuel (natural gas) and oxygen and to enable combustion via an ignition device. The reaction taking place in the combustion chamber with 15% excess air is [39].
On the left-hand side of (1) we have methane, oxygen and nitrogen. The combustion gas are a mix of the products on the righthand side of (1): 8.17% of CO 2 (carbon dioxide), 16.33% of H 2 O (water vapor), 2.88% of O 2 (oxygen) and 72.62% of N 2 (nitrogen).
The gas right after the reaction in (1) has the same temperature as the flame (constant pressure adiabatic flame temperature): as the combustion gas moves through the heat exchanger, heat is transferred from gas to water, passing through different layers as sketched in Fig. 2. Let us identify four layers: combustion gas, extended surface, tube wall and water. The combustion gas temperature T g and water temperature T w in [ C] evolve dynamically according to [40]: where the first term on the right-hand side of (2) comes from the heat exchange within the gas and the second term comes from the heat exchange with the extended surface at temperature T s . In (2) where the first term on the right-hand side of (4) comes from the heat exchange with water and the second term comes from the heat exchange with the extended surface. In (4) , and the thermal resistance between tube wall and extended surface core in [kW/m 2 C]. The first term on the right-hand side of (5) comes from the heat exchange with gas and the second term comes from the heat exchange with tube wall. In (5), r s is the density in [kg/m 3 ], and c s the specific heat capacity in [kJ/kg C] of extended surface material, respectively. Similar equations as (2)e(5) have been derived by the authors in Ref. [29]: however, differently from Ref. [29], here we have further increased the flexibility of the model because the parameter D m , which is the perimeter at the interface between tube wall and the extended surface, can be used to finely regulate (with D i < D m < D o ) the heat exchange through conduction. Note that (2)e(5) are equations related to sensible heat [41], i.e. they do not include any latent heat, as it will be explained hereafter.

Hybrid dynamics of latent heat
By spatially discretizing the partial differential equations (2) and (3) (gas and water side) we obtain ordinary differential equations. As a consequence, (4) and (5) must be spatially discretized as well: Fig. 2 shows that the four layers of the heat exchanger are discretized into n elements. The symbols T gx;wx;tx;sx , x2 1,2 …, n represent the temperature of gas, water, tube wall and extended surface in section x, each one modelled as a separate state. The evolution of such temperatures in dry exchange is discretized as follows: where r gx in [kg/m 3 ] is the density of combustion gas in the element x, and Dl is the length of the element in [m]. Note that in the first equation of (6) the density r gx of combustion gas varies from element to element as the boiler is a constant-pressure combustion system (at atmospheric pressure). Thus, the density will change according to the ideal gas law pV ¼ mRT=M, where P is the pressure of the gas in . It results that r g ¼ m=V ¼ pM=ðRTÞ. In order to take into account the counter flow nature of the condensing boiler, a backward discretization has been used in the first equation of (6), while a forward discretization has been used in the second equation of (6). Equations in (6) are valid till the moisture en-trained in flue products as water vapor remains vaporized. The heat transfer phenomenon occurring under this condition is called dry exchange and the boiler is said to operate in non-condensing mode. Noncondensing mode occurs approximately above the dew point of the combustion gas, which is about 54.4 C if the combustion occurs at 15% excess air [42], as the reaction (1). Below 54.4 C, the water vapor will change phase and condense [42]. The condensation will occur at constant temperature, and in order to be sustained it is necessary that the return water temperature is low enough (in particular, lower than 54.4 C). The heat transfer phenomenon occurring under this condition is called wet exchange and the boiler is said to operate in condensing mode. Wet exchange can be described by where the term W heat in [kJ/s] due to phase change comes from making the first equation of (6) equal to zero Because of the condensing regime, the temperature of the gases exiting the flue of a condensing boiler is typically 50e60 C, as compared with 120e180 C in a non-condensing boiler. Differently from the model derived by the authors in Ref. [29], the term (8) provides an extra feature, since it allows to determine the quantity of condensate as a function of the design parameters (note that not all the gas will become liquid). The quantity of condensate can be calculated from latent heat flow W heat ¼ c c w c , where c c and m c are the specific heat and mass flow rate of condensate. The mass of condensate can be calculated by integrating over time the mass flow rate of the condensate w c ¼ W heat =c c .
Remark 1. The modelling approach (6)- (8) is modular in the sense that it is possible to add more or less sections n to increase or decrease modelling precision. In every section the heat exchange equations can be governed by either dry (6) or wet exchange (8), depending on the gas temperature T gx in each section. The resulting dynamical model is hybrid with state-dependent switching, thus capturing the spatialdependent and time-dependent bimodal behavior exhibited by condensing boilers.
Due to the monotonic properties of the temperature in a boiler (the water temperature is increasing from inlet to outlet and the gas temperature is decreasing from inlet to outlet), the boiler can work in at most n þ 1 configurations, depending on how many sections are operating in wet regime (from 0 sections to n sections). A representation of possible boiler modes as a function of the number of sections is given in Table 1: the table illustrates how the modes can go from all sections in dry exchange to all sections in wet exchange.

Dynamic monitoring
The hybrid model given by dry heat exchange (6) and wet heat exchange (7) and (8) can be exploited for monitoring the online possible changes in the efficiency. To this purpose, let us describe one section x, in dry or wet regime respectively, with the statespace formulation where T x ¼ ½T gx T wx T tx T sx 0 2ℝ 4 is the state, d x ¼ ½T gxÀ1 T wxþ1 0 2ℝ 2 comes from neighbor sections, and u x ¼ ½T g0 T wnþ1 0 2ℝ 2 are the inputs to the boiler (only for the first and last sections) i.e. the gas temperature and return water temperature at the inlet. The matrices A d x , L d x and B d x , of appropriate dimensions, can be derived based on (6). The matrices A w x , L w x and B w x have a different structure than A d x , L d x and B d x , and can be derived based on (7). For example, Table 1 Possible boiler modes as a function of the number of sections (D stands for dry or non-condensing exchange, W stands for wet or condensing exchange).

No. of sections
By putting together the n þ 1 configurations, we obtain where T2ℝ 4n is the collection of all temperatures in all sections, u ¼ ½T g0 T wnþ1 02ℝ 2 are the gas temperature in the first section and the return water temperature at in the last section, and the matrices A i 2ℝ 4nÂ4n and B i 2ℝ 4nÂ2 are derived accordingly (the effect of neighbor sections disappears after coupling the sections together). In (14), the switching from one configuration i to another is driven by T gx in the different sections. Finally, q represents a set of parameters which is crucial to monitoring efficiency: in our case we assume that the following parameters influence efficiency. h s , h t and R ts whose value might change due to limescale deposit and aging; w w is also affected by limescale deposit, and most importantly, usually not measured in practice.
Limescale can build up on the water pipes of the heat exchanger and create an insulating layer which inhibits heat transfer to the water. It has been calculated that a 1 mm layer of limescale causes a 7% increase in boiler energy to meet the same heat demand, thus significantly modifying the boiler efficiency curve [43]. Limescale phenomena can be regarded as a combination of degradation of the boiler efficiency curve and changes in the mass flow rate, which have to be detected and diagnosed by the monitoring tools.
All values related to fluid properties (mass, density, specific heat capacity) and boiler dimensions (perimeter, area, volume) are assumed not to change with time: since h s , h t , R ts , w g and w w appear in a linear fashion in (14), a linear-in-the-parameters estimator can be used to monitor their value [44].
where A mi 2ℝ 4nÂ4n are Hurwitz matrices and the different estimators are activated synchronously depending on T gx in the different sections. Other least-squares or gradient-based estimators can be used in place of (15) [44]. By comparing b A i and b B i with their nominal values, one can monitor trends in parameter changes: the proposed dynamic monitoring algorithm can be sketched as in Fig. 3

Reducing the need for measurements
The monitoring algorithm in (15) exploits the underlying assumption that the entire state T can be measured. This can be quite a strong assumption, as temperatures of tube wall and extended surface might be quite difficult to measure. There are several approaches to relax this assumption. The first one is observer-based monitoring, which requires only measurements water and/or gas temperature. Consider the following descriptions of one section x in dry and wet exchange regime where the matrix C x is used to isolate the measurable variables T gx and T wx . The following adaptive observer can be adopted with C2ℝ 2nÂ4n , y2ℝ 2n collecting the measurements of gas and water temperature in all sections, and K i 2ℝ 4nÂ2n has to be designed, eventually at every time step, is such a way that ð b A i ðqÞ À K i CÞ is a Hurwitz matrix. Provided that the couple ð b A i ðqÞ; CÞ is detectable, using classical results from adaptive control [44], stability of the observer can be proven. If the inputs to the boiler are persistently exciting, then the estimates b A i ðqÞ and b B i ðqÞ (updated via a gradient-based algorithm) can converge to their true values.
The resulting dynamic monitoring architecture can also be sketched as in Fig. 3, where the different estimators, one for each regime, are activated based on state-dependent conditions. The activation of the different observers can be based on active measurements of the modes or on mode-identification mechanisms. Active measurements of the modes require that T gx , the state that is responsible for the switching, is measurable. This means that the number of sections under consideration cannot be greater than the number of sensors measuring T gx along the heat exchanger (at most one section for each sensor). Since measurements of water temperature are quite common in boilers, an alternative approach is to have T wx as the state that is responsible for the switching: this is motivated by the fact that nonlinear efficiency curves are given as a function of water temperature. Therefore, different sensors of water or gas temperature along the heat exchanger should be available in order to accurately identify the modes: in the absence of many sensors (e.g. in the presence of two sensors to measure inlet/outlet water temperature and a sensor to measure gas flue temperature), one should resort to mode-identification techniques, which include nonlinear estimation techniques [45,46], identification methods for hybrid systems [47,48], and estimation via multiple-models [49e51]. Note that for all these techniques, in general, a trade-off exists between available measurements and accuracy of the estimation of the active mode. Therefore, the complexity of the monitoring mechanism (the n þ 1 possible configurations) is always driven by the number of available sensors.

Simulation and real-world results
The efficiency of the boiler is the ratio between output power (water side) and input power (gas side): the input power P in in [kW] is calculated from where w ng is the mass flow rate in [kg/s] and H v the low heating value of natural gas in [kJ/kg], respectively. The output power is given by Ref. [52]: where T sup in [ C] is the supply water temperature at the boiler outlet and T ret in [ C] the return water temperature at the boiler inlet. As a consequence Efficiency ¼ P out P in $100: To draw the efficiency curve stemming from the proposed modelling approach, simulations have been performed on a MAT-LAB ® implementation of the condensing boiler with n ¼ 5 sections. The parameters w ng , w w and T ret are kept constant during the simulation till steady state is reached. Then, efficiency is obtained from (20). The steady-state simulations are run over different values of T ret , ranging from 20 C to 70 C. Fig. 4 shows the resulting efficiency curve as a function of T ret for three different flow rates (w ng ). It can be observed that the efficiency curve resembles the typical condensing boiler efficiency curve appearing in literature [41]. Note that, as expected, greater efficiency is attained when return water temperature is below the dew point.

Comparisons with real-world efficiency curves
It is important to validate the proposed approach against realworld boilers. In order to do so, we use a CREST condensing boiler by Lochinvar, whose efficiency curve and parameters can be found in Refs. [53] and [54].  which for a heating value of 55,500 kJ/kg gives approximately a methane mass flow rate of 0.0170e0.0001 kg/s). Fig. 5 shows a good match between the efficiency curves of the CREST boiler and of the proposed model. The values for the identified parameters are shown in Table 2. The error between the real efficiency curve and our proposed model is around 0.5%. To see how the accuracy of the model changes for changing number of sections, Fig. 6 shows a better match with the CREST boiler by using n ¼ 7 sections. Furthermore, Fig. 7 shows that the accuracy is a decreasing function of the number of sections: in practice, increasing the number of sections also requires more sensors, in order to be able to identify all the modes in Table 1. Therefore, as explained in Section 3.1, a trade-off should be made between accuracy and available sensors.
From literature we know that steady-state efficiency can be different than transient efficiency [8]: Fig. 8 shows that, as expected, transient efficiency is lower than steady-state efficiency. Note that no state-of-the-art approach based on static nonlinear efficiency curve can provide the transient efficiency of a boiler. Therefore, Fig. 8 shows that what we gain with the proposed     approach is a monitoring procedure over the entire dynamic operating range of the boiler: in particular, the two transient efficiency curves in Fig. 8 are calculated from (20) in two instants of time before the steady-state is reached. In order to show the effect of degradations on the efficiency curve, we decrease the nominal h i , h o and R tf in Table 2, and plot again the efficiency curves with such values. Fig. 9 shows that the performance curve is indeed degraded with respect to the nominal one in Fig. 5, thus indicating that the proposed model can be used to monitor efficiency degradation. Only steady-state degradation is shown in Fig. 9 for better visibility.

Conclusions
Developing accurate and dynamic models of condensing boilers is a key enabler to energy efficiency via better controls and monitoring strategies. In contrast with the state of the art, where modelbased monitoring algorithms for energy efficiency are on simplified nonlinear efficiency curves calculated at static (or quasi-stationary) regimes, in this work we proposed a novel monitoring algorithm relying on a hybrid dynamic first-principle modelling. Because of the fact that such model accounts for dynamic heat transfer phenomena and for a time-varying distribution of condensing/non condensing heat exchange, dynamic monitoring of the energy efficiency of condensing boilers during their complete dynamics regime is possible by appropriately designing a set of observers. Interestingly, the proposed approach not only recovers (at steadystate) the static nonlinear efficiency curve, but it also results in a dynamic nonlinear efficiency curve that state-of-the-art approaches cannot provide. The efficiency curve has been shown both in the nominal and degraded case, where degradation in efficiency has been modelled as changes in heat transfer coefficients due to limescale. Comparisons with real-life boiler efficiency data have been presented.