The objective of the design optimization is the minimization of the operational cost of a CHP system through the determination of the optimal displacement volume of the ICE. The CHP system is utilized to cover the needs of a residential 10-house district over a 24- hour period. The design optimization framework of this work complies with the following rules/assumptions:
1. The demand of the residential 10-house district follows the demand pattern of an average day (see Figure 3.14)
2. The demand in electrical power is covered entirely either by the CHP system, the electricity grid or a combination of both.
3. The rate of the natural gas for the CHP operation follows the “British Gas Standard gas charges”.
4. The rate of the electrical power bought from the grid follows the “British Gas Standard domestic single rate electricity charges”.
5. The temperature of the produced hot water is equal to 70oC.
6. The volume of the hot water production is set in steady state and equal to 35 gallons per occupant1. It is assumed that an insulated sufficiently large storage tank is the
end point of the external circuit. More specifically, the system is assumed to produce hot water with a fixed mass flow rate throughout the day, regardless of the demand. The hot water is stored and used according to the needs of the 10-house district. 7. Heating power shortages are covered either by an electrical or a natural gas boiler. The
efficiency of the boilers is assumed 70% and 90%, respectively.
During the optimization procedure two PI controllers are utilized that force the power output and water temperature into certain set-points. In the case of the power output the
set-point is equal to the electrical power demand minus any electrical power obtained by conventional means. The conventionally generated electrical power is an optimization vari- able. The temperature set-point for the water is, according to rule 5, 70oC. The optimized
PI controllers are adequate to force the system into steady state in significantly less time than Figure 3.7 suggests. The main purpose of modeling the dynamic behavior during the optimization is to ensure continuity and smooth simulation. Also, the dynamic modeling of any process system ensures a more accurate depiction of the system behavior during change and allows the modeler to ensure more precise optimization results. A power demand pro- file of high discreteness results into more control intervals and, as a result, a more precise optimization outcome. The reason for the later lies in the fact that a more discrete profile would be able to describe the power demand more adequately and allow the optimization to account for rapid demand change. It is assumed that the resulting 864 intervals adequately describe the power demand fluctuations of the ten house district. The optimization is per- formed in gPROMS•[282] with the implementation of a control vector parameterizationR
algorithm via single shooting (see Chapter 6 for more details regarding the MIDO algorithm and available MIDO algorithms in literature).
Time of the day (hr)
0 4 8 12 16 20 24
Electrical power demand (W)
1500 2000 2500 3000 3500 4000
Figure 3.14: Electrical power demand profile
The potential use of electrical power from the grid, according to rule/assumption 2, poses the need to optimize the amount of power and the time of the day that the electrical power is bought. Consequently, the result of the design optimization is not only the optimal ICE displacement volume, as described in the beginning of this section, but also the optimal profile of power bought from the grid. The heating power shortages of rule/assumption 7,
lead into considering and optimizing two case scenarios. The first case scenario features an electrical boiler as the energy source for supplementary heating power, while the second case scenario features a natural gas boiler that serves the same purpose.
In the first case, an electrical boiler is utilized to cover the heating power shortages. As a result, the electrical power is bought from the grid (i) as part of the operational cost minimization strategy and (ii) as a supplement for water heating.
Figure 3.15 (left) presents the optimal electrical power produced from the CHP plant for the partial coverage of the electrical power demand of the 10-house district while Figure 3.15 (right) presents the optimal electrical power profile utilized for supplementary water heating.
Figure 3.15: Electrical boiler case: Optimal (cost minimization) electrical power production profile (left) and electrical power consumption for the water boiler (right)
The optimal displacement volume for the ICE of the electrical boiler case is 832cc and the annualized operational cost of the system is $1075 per household.
The annualized operational cost of the optimized system, compared to the annualized cost of electrical power for covering the electrical and thermal needs of the 10-house district is 75.1% lower, based on the “British Gas Standard domestic single rate electricity charges”. The use of the CHP system compared to the use of conventional power generates $3250 of revenue per household, annually.
In the second case, a natural gas boiler is utilized to cover the heating power shortages. As a result, the electrical power bought from the grid serves only as part of the operational cost minimization strategy.
The power profile for the minimization of the operational cost of the system and the power profile for supplementary water heating are presented in Figure 3.16 (left) and (right), respectively.
Figure 3.16: Natural gas boiler case: Optimal (cost minimization) electrical power produc- tion profile (left) and electrical power consumption for the water boiler (right)
Annualized cost per household ($) Electrical boiler Natural gas boiler CHP and conventional power sources 1075 1068
Conventional power sources 4325 1490
Table 3.4: Operational cost summary. Comparison between the usage or not of a CHP in tandem with an electrical or a thermal boiler
The optimal displacement volume for the ICE of the natural gas boiler case is 738cc and the annualized operational cost is $10680.
The daily cost of covering the electrical power and heating needs of a 10-house district is calculated to $1490 per household, annually. The aforementioned amount is calculated on the assumption that the district utilizes natural gas for heating purposes and electrical power provided from the grid. The rates of the natural gas and electrical power are set according to the “British Gas Standard gas charges” and the “British Gas Standard domestic single rate electricity charges”. The generated revenue per household, from the use of the CHP is $422.
It is clear from Table 3.4 that the use of a CHP system is profitable when compared to conventional power sources. More specifically, it is shown that the use of only electrical power is the least cost effective, followed by the use of a combination of electrical power and natural gas. The utilization of a CHP plant with the parallel use of electrical boilers appears to be the most profitable among all four cases, a result which is primarily attributed to the highly
efficient CHP system and secondarily to the rate policy of electrical power, especially during the night hours. The later contributes greatly to the form of the profiles of electrical power generation in both scenarios as shown in Figures 3.15 and 3.16 (left). Between 00 : 30am and 07 : 30am the lowest operation level of the CHP plant is monitored. During these time intervals the highest consumption occurs for both the electrical power (Figure 3.15 (right)) and natural gas (Figures 3.16 (right)) acquired from conventional power sources. According to rules 3 and 4, during the aforementioned hours of the day, the rates for both electrical power and natural gas drop to the day’s minimum.
The form of the CHP electrical power production and the power for water heating from conventional power sources seem to follow a very similar profile regardless the case scenario (Figure 3.15 and Figure 3.16). These profiles seem to defy the electrical demand profile of the residential district (Figure 3.14) but rather they tend to prefer covering a base electrical load and allowing the rest to be covered by the electrical grid electricity, with the exception of the time intervals between 00 : 30am and 07 : 30am as described in the previous paragraph. The reason behind that is that in terms of cost minimization, the optimization process recognizes the need to rely solely on the CHP operation for the production of hot water and use the produced electricity as a supplement to the electrical power demands of the residential district. Furthermore, the different levels of electrical power generation in the two case scenarios and especially the different levels of utilized power from conventional sources are in agreement with the previous argument as they identify that the minimization of the amount of power acquired from conventional sources for heating purposes as the main factor that affects the operational cost of the district. The previous statement is evidently the reason why the size of the ICE used in both cases is relatively large compared to the load for the two case scenarios. As shown in Figure 3.12, an ICE with a larger displacement volume working in low loads results into great thermal efficiency, which a-posteriori appears to contribute greatly to the minimization of the operational cost.
3.5 Concluding remarks
In this chapter a modeling approach to a CHP system for covering the thermal and electrical needs of a small domestic area was presented. Furthermore, a framework for the design optimization of the size of the prime mover was shown. The optimization is performed for operational cost minimization under specific conventional power sources utilization hypothe- ses. The optimal use profile of conventional energy sources is simultaneously acquired. It is shown that the use of the CHP system is the most cost effective strategy among the tested hypotheses, a result consistent with the literature. The mathematical model and the optimal
Chapter 4
Decentralized multi-parametric Model
Predictive Control for a residential
scale cogeneration unit
How can we efficiently operate a residential cogeneration unit?
Portions of this chapter have been submitted for publication in:
• Diangelakis, N.A.; Avraamidou, S.; Pistikopoulos E.N.; Decentralized multi-parametric Model Predictive Control for a residential scale cogeneration unit (2016) Industrial & Chemical Engineering Research, 55(12), 3313-3326
4.1 Introduction
From the perspective of a residential power resources end-user it is easily understood that the two main operational objectives are the minimum cost of power and minimum outage. Until today, the provision of power in the residential sector takes place in a centralized manner where a central grid also provides power based on the needs of the households, upon demand. The provision of thermal power happens in a similar way where a central natural gas grid provides the fuel needed to cover the thermal need of the households independently. Individual boilers are then utilized to transform the chemical power of the fuel into heat in the form of hot water, suitable for space heating and hot utility usage. The concept of the centralized approach is depicted in Figure 3.1.
Although the aforementioned approach provides a robust way of providing heat and power to the domestic sector in terms of power and heat shortages, its potential drawbacks, namely (a) the environmental impact from the use of fossil fuels and (b) the end user cost, have led
to the consideration of alternative solutions. These include alternative power sources, such as domestic cogeneration units, as well as renewable resources, such as solar, photovoltaic and wind power, for covering the heat and electrical power demand (Figure 3.2). Although the aforementioned resources act as a supplement to the central power and natural gas grids, not only do they provide a more environmentally friendly alternative but also the potential for a more autonomous, efficient and more cost effective power provision.
Among the technologies presented in Figure 3.2, combined heat and power (CHP) systems have the potential to replace the conventional processes used so far for the production of usable heat and electricity. CHP systems utilize the same amount of fuel for the generation of both electrical power and usable heat, emitting consequently lower amount of exhaust gases. Therefore, owing to their environmentally friendly and cost effective nature, CHP systems can play a dominant role in the emission reduction strategies [265].
In this chapter, we present a framework for the design of decentralized advanced model predictive control scheme based on the model developed in Chapter 3.
4.1.1 Decentralized Model Predictive Control
Figure 4.1 presents helpful control problem structures and decentralization policies for Model Predictive Control (MPC) problems. Earlier works on decentralization were mainly fo- cused in the stability of decentralized linear control of large scale systems with interac- tions between subsystems, and various control design structures and algorithms had been developed[55, 236, 237, 314, 354]. After the 90’s, due to the advances in convex optimization along with other computational techniques, interest in decentralized MPC control was raised again. Many efforts were then devoted to develop design methods guaranteeing stability and performance. Examples of methods developed include those based on optimization [84, 318], on overlapping decompositions [162, 164, 165], on vector Lyapunov functions [163] and on sequential design [159, 317]. Even though the problem of decentralized MPC, can just be solved with regular MPC algorithms neglecting any interaction between subsystems, a small number of decentralized MPC algorithms have been developed so far that can guarantee stability and performance. A stabilizing decentralized MPC algorithm for nonlinear discrete time systems was presented by [224]. This algorithm obtains closed loop stability, relying on the inclusion of contractive constraints in the formulation of the MPC problem. Other approaches developed to stabilize D-MPC controllers for nonlinear discrete-time systems, treat the interactions between subprocesses as disturbances [284].
In our work, we take advantage of the inherent dual nature of the domestic CHP system in order to partition it into two subsystems based on its operational and modeling principles,
(a) Centralized (b) Decentralized
(c) Distributed (d) Hierarchical
Figure 4.1: Alternative control structures overview. Multiple subsystems are controlled via (a) centralized, (b) decentralized and (c) distributed/coordinated control structures while (d) presents a hierarchical application of controllers [69, 317]
which are discussed in the next section. The rest of the chapter discusses the decentralized explicit control approach and framework that was applied to the system. The results are presented for the two distinct operating modes.