Evaluation of an energy and exergybased generic modeling approach of combined heat and power plants
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Abstract
Reduced models of combined heat and power plants are required for different applications. Among other usages, they are implemented as mixed integer linear programs (MILP) in energy market models or pricebased unit commitment problems to study the economic feasibility and optimal operation strategies of different units. Generic models are particularly useful when limited information is available for each considered plant. This paper presents a MILP modeling approach for combined heat and power (CHP) plants. The approach is based on energy and exergy balances and a few typical plant characteristics for different operating conditions. The reduction of electrical power output due to heat extraction is estimated by the transferred exergy to the district heating network. Furthermore, the accuracy, strengths and limitations of this approach are investigated for various CHP plant types with extraction condensing turbines designed for district heating systems. Therefore, detailed thermodynamic cycle simulations of CHP plants including part load operations are used to obtain the real plant operating conditions to compare them to the results of the described generic approach. The validation of the reduced, generic model shows that the accuracy mainly depends on the effectiveness of the heat extraction from the CHP plant. In addition, it can be seen that the main advantage of the presented exergybased method is the inherent consideration of the feed flow temperature for the calculation of the power reduction due to heat extraction.
Keywords
CHP, combined heat and power Cogeneration MILP, mixed integer linear programs District heating Extraction condensing turbine Thermodynamic simulation Exergy Unit commitmentAbbreviations
 CCPP
Combined cycle power plant
 CHP
Combined heat and power
 DH
District heating
 HC
Heating condenser
 MILP
Mixed integer linear programs
Introduction
A lot of countries promote combined heat and power (CHP) plants as they offer a higher efficiency, compared to the separate production of heat and power [9, 15]. In Germany, for example, 15.4 % of the produced electricity in 2010 was generated in CHP plants [8] and the ambitious political target is a share of 25 % in 2020 [19]. The Federal Ministry for Economic Affairs and Energy will prepare the amendment to the CHP Act in 2015 [3]. In other countries, such as for example Denmark, the share is already higher.
CHP plants supply heat and power, so that the optimal operation management is not only influenced by the electricity market, but also by the requirements of the heat customers. This is important to consider in electricity market models which are often implemented as mixed integer linear programs (MILP). MILPs are also used for pricebased unit commitment problems, to study optimal operating strategies or economic feasibilities [7, 16, 20]. If studies for general indications for large geographical areas are performed, detailed information on the CHP plant characteristics are often unavailable. The generic approach presented here is based on data easy to access and guarantees physical consistency, therefore gives the opportunity to improve these models.
Exergetic considerations are not unusual in the field of CHP plants, as it is one of the possibilities to allocate the costs, fuel consumption or emissions to both products, thermal energy and electrical power [24, 25]. In [12] the legislative efficiency indicators from different countries are exergetically compared. The influence of the steam extraction rate on the exergy destruction in the district heaters is part of [17]. At the end of “Determination of the power loss coefficient”, the approach is compared to the “Dresdner Method” [2], which is based on the exergy of the district heating feedwater.
The thermodynamic characteristics of CHP plants are well described in [14] and typical design parameters can be found in [4, 18, 27].
This characteristic line is influenced by various design parameters of the CHP plant as well as operation conditions, like for example the district heating feed or return flow temperature.
Beside the steam extraction from the turbine, further alternatives are possible. In combined cycle power plants (CCPP) a heat exchanger can be installed in the flue gas duct which heats up the district heating water directly. This is especially used in plants with back pressure turbine as the exhaust gas temperature is high. Furthermore, hot condensate could be used for heating the district heating system.
After a short introduction of important key figures of a CHP plant, the approach and a possible integration in MILPs is presented. The approach is based on few basic parameters which are often available for each plant even during large scale studies and can therefore improve their results. To evaluate the approach, it is compared with thermodynamic simulations of CHP plants designed for district heating supply, see “Validation of the generic modeling approach”.
Characteristic figures of cogeneration plants
Description of the generic modeling approach
The generic model uses rudimentary input data of a CHP plant to define the feasible operational range and to link the power and heat output to the fuel consumption. It is based on the idea that the loss of electrical power due to heat extraction can be estimated by the increase of the exergy in the district heating flow (see “Determination of the power loss coefficient”).
For maximum fuel rate the maximum extractable heat is given by the power to heat ratio \(\sigma \), but for lower fuel rates the assumption of a constant power to heat ratio is an avoidable simplification. In “Determination of the maximum heat extraction” an approach to asses (\(\sigma \)) as a function of the boiler load is presented.
Determination of the maximum heat extraction
With reasonable assumptions it is possible to derive the maximum heat extraction from the energy balance [6]. If the reference point of the enthalpy is set to the ambient air conditions, the left hand side of (4) is reduced to \(\dot{H}_{\rm F}\). The losses to the environment can be estimated as follows. The losses with the flue gas \(\dot{H}_{\text{L,FG}}\) are between 5–10 % for steam power plants [21, 23] and the other losses \(\dot{H}_{\text{L,o}}\) are summarized for this work with 1.5 % and added to \(\dot{H}_{\text{L,FG}}\) in further considerations. For CCPPs the losses also depend on the number of pressure levels in the steam cycle. A CCPP with only one pressure level is unable to exploit as much thermal energy from the flue gas as a dual or triple pressure steam cycle. For triple pressure plants the losses are in the same range as for steam power plants, but can increase significantly for inefficient plants with only one pressure level.
Data of the simulated CCPP with 1 and 3 pressure stages and the steam PP
1 Pr. CCPP  3 Pr. CCPP  Steam PP  

Max. electrical power (MW)  436  455  535 
Min. electrical power (MW)  178  190  189 
Max. steam extraction (3 HC, 110 °C)(MW)  201  235  693 
Electrical efficiency (wo DH) (%)  57.4  59.9  41.6 
El. eff. minimum stable load (wo DH) (%)  48.1  51.4  37.4 
Main steam pressure (bar)  131  131  188 
Main steam temperature (°C)  561  570  540 
Hot reheat steam pressure (bar)  NA  31  48.5 
Hot reheat steam temperature (°C)  NA  563  540 
Determination of the power loss coefficient
Due to the steam extraction in the turbine the yield of electrical power is reduced, see (3) and (5). The reduction depends on the pressure of the steam extraction which is again influenced by the required feed flow temperature in the district heating system. The concept of the approach is that the loss of electrical power due to heat extraction can be estimated by the increase of the exergy in the district heating flow. Consequently the influence of the temperatures in the district heating system is accounted for. Even if the power loss coefficient (\(\beta \)) is available for a realized CHP plant, this influence is rarely available.
The Dresdner Method [2] was suggested to estimate the power loss coefficient based on the medium district heating feed temperature of a time period. It uses a typical German condensing temperature and a factor depending on the number of district heaters. In difference to the Dresdner method, the approach described in this work is flexible regarding the cooling water temperature, considers the influence of the district heating return temperature and is independent of the number of district heaters.
The generic MILP model
In this section the MILP model implementation derived from “Description of the generic modeling approach” is presented. It can be used to describe the steady state characteristics of a CHP plant in a unit commitment problem and should be complemented with dynamic constraints as startup costs, maximum ramp rates and minimum operation and down time intervals.
A common approach to model a load dependent electric efficiency in MILP models is used in (10) and expanded for heat extraction with the power loss coefficient (\(\beta \)) in (11), see “Determination of the power loss coefficient”. The binary status variable \(\mathbf{Y}\) equals 1, if the unit is committed or 0, when the unit is not in operation. Equations (12) and (13) account for the operational gap between minimum boiler load and unit shut down and describe the restrictions of minimum and maximum boiler load. The restriction of maximum heat extraction is given by (14), see also “Determination of the maximum heat extraction”.
One advantage of the utilization of more than just the two obvious variables P and \(\dot{Q}\) for the restriction in Eq. (14), is that the model stays linear for heat extractions at different feed flow temperatures simultaneously.^{5}
Validation of the generic modeling approach
In this section the results of the generic model are compared to the thermodynamic simulations executed with EBSILON^{®}Professional (Ebsilon) [10]. To analyze general correlations, first a combined cycle power plant (CCPP) with one pressure level is simulated. Afterwards, results for a CCPP with three pressure stages, which represents a modern high efficient plant, as well as a steam power plant are discussed and further limitations are shown. For the exergy calculation the ambient and cooling water temperature are set to 15 °C and the ambient pressure to 1 atm. Some main data of the simulated CHP plants are listed in Table 1.
Combined cycle power plant

The gas turbine is modeled according to the Siemens SGT 4000F characteristics from the gas turbine library distributed by VTU Energy GmbH [26].

The steam cycle is simulated with sliding pressure as the method for load control; the pressure of each steam turbine stage is defined by the ellipsis law, proposed by Stodola [22].

Minimum load of the gas turbine and minimum boiler load for the steam power plant is set to 40 % of nominal load.

A pressure control flap is installed in the overflow steam line from the IP to the LP turbine, which is often used to control the condensing temperature in the district heaters.

\(T_{\text{RF}}=60\, ^\circ{\rm C} \) ; \(T_\text{FF}=90\)/110/130 \(^\circ{\rm C} \)
Combined cycle power plant with singlepressure boiler Figure 5 shows the flow diagram for the CCPP with a singlepressure boiler. The CHP plant with three district heaters is presented, but similar models were analyzed with one and two district heaters as well.^{7} This basic example is used to analyze the influence of the number of heating condensers but also shows the influence of the energetic losses (here referred to as flue gas losses) for CCPPs.
Figure 6 shows the influence of a different number of heating condensers and compares the simulations with the approach.
A higher number of heating exchangers reduces the exergy destruction of the heat transfer, increases the utilization of steam in the steam turbine and reduces the throttling losses of the control flap. Therefore, the power loss coefficient decreases with the number of heating condensers. Nevertheless, for the analyzed cases the differences are small for different reasons. First, the small mass flow through the remaining stages of the turbine downstream the extractions reduces the pressure and enthalpy differences as well as the isentropic efficiency of the last turbine stages. Hence, the generated mechanical power in these stages is small compared to the overall power production. Second, in the models with two and three district heaters the pressure on the low pressure steam extraction is so low that the condensing temperature is only slightly above the district heating return flow temperature. Therefore, only a small amount of steam is fed to the first district heater. However, the installation of multiple heating exchangers offers the opportunity to satisfy lowtemperature heat demands only with the low pressure district heater(s), so that no steam has to be extracted from one of the high pressure extractions for this purpose.
Furthermore, the figure illustrates that the simulated CHP plants are unable to extract as much heat as the so far described approach estimates. The losses with the flue gas for nominal load without heat extraction are (\(\dot{H}_{\text{L,FG}}=16.8\,\,\%\)) of the fuel feed (\(\dot{H}_{\rm F}\)) in the simulation but it increases to 18.3 % for maximum district heating.
The difference can be explained by the increase of the condensate inlet temperature of the boiler which leads to higher losses with the flue gas.^{8}
Another noticeable difference is a significant reduction of the maximum heat extraction for minimum stable load, which results from increased losses with the exhaust gas (\(\dot{H}_{\text{L,FG}}\)) during part load operation of CCPPs independent of the district heating. Hence, in addition to the change of condensate temperature as for nominal load, the energetic losses increase up to 26 % of the fuel rate for minimum power generation and maximum heat extraction of the CCPP.
In summary, Fig. 6 shows, that the differences between the thermodynamical simulations with different number of heating condensers are small for maximum heat extraction in the analyzed example. In addition it illustrates, that the approach overestimates the possible heat extraction, if the relative exhaust gas losses in full load without heat extraction (16.8 %) are used for part load. If the increased flue gas losses at maximum heat extraction are considered (18.3 % in full load and 26 % in minimum load) the results of the detailed thermodynamic simulations and the approach with constant flue gas losses match for the particular load cases. Hence, more accurate results can be obtained, if load dependent flue gas losses are considered, given that this dependency is known.
It can be stated that the restriction for maximum heat extraction is strongly influenced by the flue gas losses, see also “Determination of the maximum heat extraction” and Fig. 4, which can vary with the gas turbine load and the amount of heat extraction, as shown above.
Combined cycle power plant with triple pressure boiler The flow sheet of the CCPP model with triple pressure boiler is shown in Fig. 7.
Figure 8 shows the effect of the district heating feed flow temperature (\(T_{\text{FF}}\)) on the possible operating conditions and compares the simulation results with the generic modeling approach. The energetic losses \(\dot{H}_{\text{L,FG}}\) for the generic approach are set to be equal to the losses of the thermodynamic model in nominal load without district heating (8.7 %). In nominal GT load and maximum heat extraction they increase to 9.9 %, due to the higher condensate temperature, as described in “Combined cycle power plant”. In addition to this effect, the approach predicts higher possible district heating capacities due to higher steam flows to the condenser in the simulation. In this example, the superheated lowpressure steam is fed to the LP turbine, which results in a sufficient flow through the last steam turbine stage even with maximum district heating. The limitation to be considered, is to ensure the minimum flow through the last intermediate pressure stage, so that the minimal flow to the condenser increases to 18% instead of 10% of the nominal flow.
Hence, the increased flue gas losses as well as the higher heat transfer in the condenser reduce maximum heat extraction, see also (7) where both parameters could be summated.
The power loss coefficient increases with rising feed flow temperature of the district heating system (\(T_{\text{FF}}\)), as it requires higher extraction pressures and shifts the load share between the heating condensers.
Compared to the detailed Ebsilon simulations, the approach indicates the general trends, but does not exactly meet the simulation results which always depend on the specific design parameters of the plant. Figure 8 includes some additional values for maximum fuel and medium steam extraction, which is why a slightly non linear trend of the upper map restriction can be seen for the Ebsilon results. The trend for 110 \(^\circ{\rm C} \) is well predicted, for 90 \(^\circ{\rm C} \) the simulation model has a higher power loss coefficient as the approach predicts for maximum steam extraction. For 90 \(^\circ{\rm C} \) and lower steam extraction the power loss coefficient fits the approach better. The steep drop is due to a load shift from the first to the second district heater, if the heat extraction is increased from 160 to 225 MW. For a feed temperature of 130 \(^\circ{\rm C} \) the results of the approach overestimate the power loss. The extracted heat for 130 \(^\circ{\rm C} \) is nearly the same as for 110 \(^\circ{\rm C} \), due to the limitation of the minimum flow through the IP stage which is not considered in the simple generic modeling approach. If the steam extraction for the highpressure district heater could have been increased, the heat extraction and the power loss would have been higher.
Steam power plant
In the steam CHP plant, the steam is similarly extracted from the steam turbine. In contrast to the CCPP, the feed water is heated by condensate preheaters with steam from the steam turbine and not by flue gas. The flow sheet of the simulated model is shown in Fig. 9.
The steam extraction for district heating reduces the steam which is available for the lowtemperature preheaters. For maximum boiler load the thermal energy transferred in the preheaters, which extract steam from the LP turbine, decreases from 147 to 18 MW, compared to the operation without heat extraction. The transferred thermal energy in the residual preheaters, including feed water tank, increases accordingly from 361 to 447 MW (also discussed by [17]).
The P–\(\dot{Q}\) diagram for the steam power plant in Fig. 10 shows the influence of \(T_{\text{FF}}\) and the comparison with the approach. For \(T_{\text{FF}}=90 \,^\circ{\rm C} \) and \(T_{\text{FF}}=110\,^\circ{\rm C} \) the operation field of the simulations concur well with the presented approach. For \(T_{\text{FF}}=130\,^\circ{\rm C} \) the model has a higher reduction of electrical power than the approach. The main factor is that compared to the load case with \(T_{\text{FF}}=110\,^\circ{\rm C}\), the \(T_{\text{FF}}=130\,^\circ{\rm C}\) case requires a higher pressure in the overflow line, which leads to higher exergy destruction due to throttling in the control valve and to a reduction of power generation in the two upstream steam turbine stages.
Further individual limitations for real plants

A flow limitation for each steam extraction.

High feed temperatures might not be reachable due to the pressure reduction on the extraction points; possible solution: installation of a control valve, additional tapping on the steam turbine.

The flow in the district heating system can be limited hydraulically, which leads to vertical restrictions in the P–\(\dot{Q}\) diagram.

Not all CHP plants can run on full load without steam extraction.

Not all plants are designed for maximum steam extraction \( (\dot{Q}_{\text{CW,min}}>10\,\,\%\)).
Conclusions
The presented modeling approach offers the opportunity to predict and describe the performance characteristics of a combined heat and power plant based on very few key figures and reasonable assumptions, see “Description of the generic modeling approach” and “The generic MILP model”. It respects thermodynamic limitations and considers the influence of seasonal fluctuations typical for CHP plants, as it incorporates changes in the temperatures of the district heating system and can be adjusted to different site conditions (e.g. cooling water temperature).
A way to derive the power to heat ratio from the energy balance of the plant is presented in “Determination of the maximum heat extraction”. Even if an individual plant might have a lower ratio, the introduced procedure guarantees that the first law of thermodynamics is not violated. The power loss coefficient is approximated with the exergy transfered to the district heating system, see “Determination of the power loss coefficient”. The comparison with simulations in “Validation of the generic modeling approach” shows that the characteristics for typical CHP plant designs for district heating supply can be well expressed with the presented approach.
Nevertheless, based on different local requirements a high variety of CHP plants exist, which can not be represented explicitly by a generic model. Therefore, detailed information of the examined plants should always be preferred, whenever available.
Footnotes
 1.
For gas turbine plants the pressure and temperature of the ambient air has to be considered for the performance.
 2.
Maximum allowable steam extraction flows can constitute further constraints.
 3.
The isentropic stage efficiency becomes negative.
 4.
Even if a small flow to the condenser leads to a reduction of the condensing temperature and pressure, the variation of the enthalpy of evaporation is still small.
 5.
If this restriction is empirically modeled by only using P and \(\dot{Q}\), the weighting factors \(\frac{\dot{Q}_T}{\sum _T \dot{Q}_T}\) for the superposition of the characteristic parameters for different maximum heat restrictions for the different feed flow temperatures become nonlinear and far “less powerful” MINLP solvers have to be used.
 6.
 7.
If only two district heaters are installed, the one with the highest extraction pressure is neglected and in case of one district heater only the intermediate one is considered.
 8.
This effect is only relevant for CCPPs and does not occur at steam power plants, because the condensate preheating in steam power plants is realized with steam extractions from the steam turbine instead of heat exchangers in the flue gas system.
Notes
Acknowledgments
This work was developed during the research project “Thermal storages for increased flexibility of CHP plants” [11], funded by the German Federal Ministry for Economic Affairs and Energy (BMWi) with the project reference number 03ET1188A.
Authors’ contributions
EM carried out the computational simulation and analysis and drafted the manuscript. AC proposed the concept of the approach, checked the analysis, revised the manuscript and drafted “Description of the generic modeling approach”. GT provided technical guidance and a review of the manuscript. All authors read and approved the final manuscript.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no competing interests.
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