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Hamburg University of Applied Sciences
Faculty of Life Sciences
SUPERVISORY CONTROL OF A COMBINED HEAT AND
POWER PLANT BY ECONOMIC OPTIMIZATION
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING (M.ENG)
IN
RENEWABLE ENERGY SYSTEMS
BY
MUSTAFA GÖKSEL DELİKAYA
Submitted on the 31st of March 2015
Supervisors
Prof. Dr.-Ing. Gerwald Lichtenberg (HAW Hamburg)
Mehmet Elci, M.Sc. (Fraunhofer ISE)
This master thesis has been completed at the Fraunhofer Institute for Solar
Energy Systems (ISE) in Freiburg, Germany.
ABSTRACT
Using combined heat and power (CHP) units within district heating systems (DHS) has
been an effective way of meeting residential energy demand in Germany. Generally
speaking, electricity fed in to the grid by the CHP is usually sold at a fixed price in today’s
electricity market. Assuming that the share of renewable energies will be higher in the near
future, it can be anticipated that the electricity prices will highly fluctuate due to the
uncertainties within the renewable energy sources, such as wind speed and solar irradiance.
Therefore, control mechanisms for heat and power producing plants are expected to switch
their operation strategy from heat-driven to power-driven operation. A power-driven operation
makes sure that the CHPs are shut down when the electricity market is not competitive
enough to produce electricity. In this master’s thesis, a power-driven operation is achieved
through an economic optimization. The optimization problem, which is formulated as a
discrete optimization problem, is to find out the best ON/OFF operation trajectory of the units
involved in a DHS; namely a CHP, a boiler and a storage tank. A simplified model capturing
the power-based dynamics of a physical DHS model is implemented at simulation and
modeling tool Dymola (Dynamic Modeling Laboratory). Optimization tool GenOpt (Generic
Optimization Program) with particle swarm optimization (PSO) algorithm is used to solve the
discrete optimization problem. The implementation of the model is verified by several test
cases. Finally, a future scenario of the year 2023 is approximated in order to compare the
financial gains and grid interactivity of the power-driven and the heat-driven operation. In
addition, the effect of varying the storage size on plant gains and grid interactivity is
investigated and discussed.
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ACKNOWLEDGEMENTS
I am deeply grateful to my supervisor Prof. Dr.-Ing. Gerwald Lichtenberg for taking me his
student and providing me all the supervision and encouragement I needed. His guidance
was mentoring me on my first steps at my academic career. I especially thank him in making
me realize the keywords and the underlying problem of this master’s thesis, which were not
clear in the beginning. It always amazed me how he approached to the problems and
proposed creative solutions. It has always been enlightening to have a conversation with
him. I would like to thank all his advices on academic writing and corrections for my
preliminary report as well.
In addition, I would like to thank to my second supervisor PhD student Mr. Mehmet Elci
for letting me involved in such an interesting project. I would like to thank him for spending
his valuable time with me anytime I needed his supervision. It was very inspiring to have
discussions and trying different approaches to overcome the problems that are faced during
this study. I am also thankful for all his corrections, which helped me a lot to give the final
form of this thesis.
I would like to thank to PhD student Mr. Kai Kruppa for his advices and corrections, which
helped me to structure the modeling chapter of this thesis in a more understandable way.
Lastly, I would like to thank to my colleagues, Ms. Sunah Park, Mr. Tom Cordes, Ms.
Friederike Rautenberg, Mr. Kai Iking, Mr. Marc Eisenbarth and Mr. Max Walch for the great
working environment, the friendship and their emotional support throughout my work.
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Table of Contents
List of Figures..................................................................................................................... ix
List of Tables ...................................................................................................................... xi
List of Abbreviations ........................................................................................................ xiii
List of Symbols.................................................................................................................. xv
1. INTRODUCTION ............................................................................................................ 1
1.1. Motivation .................................................................................................................... 2
1.2. Literature Survey.......................................................................................................... 2
1.2.1. Deterministic Methods............................................................................................... 5
1.2.2. Stochastic Methods ................................................................................................... 5
1.3. Thesis Objectives......................................................................................................... 6
2. SYSTEM ......................................................................................................................... 7
2.1. Structure of a District Heating System.......................................................................... 7
2.2. District “Gutleutmatten” ................................................................................................ 8
2.3. System Boundaries ...................................................................................................... 8
3. MODELING .................................................................................................................. 11
3.1. Operating States ........................................................................................................ 11
3.2. Storage Tank Model ................................................................................................... 11
3.3. Mathematical Model of the System ............................................................................ 13
3.4. Formulation of the Cost Function ............................................................................... 17
3.5. Model Identification .................................................................................................... 20
4. IMPLEMENTATION...................................................................................................... 21
4.1. Implementing minimum operation time constraint ...................................................... 22
4.2. Implementing storage tank model .............................................................................. 22
5. OPTIMIZATION ............................................................................................................ 27
5.1. Optimization Tool ....................................................................................................... 27
5.2. Optimization Algorithm ............................................................................................... 28
6. TEST CASES ............................................................................................................... 31
6.1. Reducing input variables ............................................................................................ 31
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6.2. Verifying Optimization Result by Enumeration Method ............................................... 34
6.3. Testing minimum operation time constraint ................................................................ 35
6.4. Testing Temperature Constraint................................................................................. 37
6.5. Selection of Penalty Weights...................................................................................... 39
6.6. Effect of prediction horizon on optimization results..................................................... 45
7. REFERENCE MODEL AND SCENARIO ...................................................................... 49
7.1. Heat-driven model...................................................................................................... 49
7.2. Reference Scenario ................................................................................................... 51
8. RESULTS ..................................................................................................................... 53
8.1. Comparison with respect to overall plant gains .......................................................... 53
8.2. Comparison with respect to grid interactivity .............................................................. 55
8.3. Comparison with respect to size of the storage tank .................................................. 57
9. CONCLUSION.............................................................................................................. 59
9.1. Interpretation of results .............................................................................................. 59
9.2. Limitations of the thesis.............................................................................................. 59
9.3. Recommendation for further work .............................................................................. 60
REFERENCES .................................................................................................................... 61
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List of Figures
Fig. 1.1 State trajectory of a DES .......................................................................................... 1
Fig. 1.2 Ramade Wonham (RW) Framework ......................................................................... 1
Fig. 2.1 Schematic representation of a DHS .......................................................................... 7
Fig. 2.2 Site plan of District “Gutleutmatten” .......................................................................... 8
Fig. 3.1 Single node representation of the storage tank ....................................................... 12
Fig. 3.2 Energy flow within the system boundary ................................................................. 17
Fig. 3.3 Input-outputs and disturbances of the system ......................................................... 20
Fig. 4.1 Implementation of the system at “Dymola” .............................................................. 21
Fig. 4.2 Implementation of the minimum operation time constraint ...................................... 22
Fig. 4.3 Representation of the storage tank with a single node ............................................ 23
Fig. 4.4 Operation range of the storage tank ....................................................................... 24
Fig. 4.5 Hysteresis band for the storage tank ...................................................................... 24
Fig. 5.1 Coupling a simulation program with GenOpt ........................................................... 27
Fig. 5.2 PSO algorithm ........................................................................................................ 30
Fig. 6.1 Load and prices for Test Case 1 ............................................................................. 32
Fig. 6.2 Optimization results of Test Case 1 ........................................................................ 32
Fig. 6.3 Model with 8 input variables.................................................................................... 33
Fig. 6.4 Model with 2 input variables.................................................................................... 34
Fig. 6.5 Results of enumeration and optimization ................................................................ 35
Fig. 6.10 Optimization results of 1st half-day (1) ................................................................... 41
Fig. 6.11 Optimization results of 1st half-day (2) ................................................................... 41
Fig. 6.12 Illustration of the constraint violation ..................................................................... 42
Fig. 6.13 Optimization results of whole day (1) .................................................................... 44
Fig. 6.14 Optimization results of whole day (2) .................................................................... 44
Fig. 6.15 Load and prices for Test Case 6 ........................................................................... 46
Fig. 6.16 Optimization results of 1st case ............................................................................. 46
Fig. 6.17 Optimization results of 2nd case ............................................................................ 47
Fig. 7.1 Decision Tree for selection of the operating states at heat-driven operation ........... 50
Fig. 7.2 Operation state distribution in the tank with respect to temperature levels .............. 51
Fig. 7.3 Electricity prices and residual load of year 2011 ..................................................... 52
Fig. 7.4 Approximated electricity prices for the future scenario ............................................ 52
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Fig. 8.1 Results of the future scenario (2) ............................................................................ 54
Fig. 8.2 Comparison of LGMCabs values ............................................................................... 57
Fig. 8.3 Optimization results of the same summer scenario with different storage sizes ...... 58
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List of Tables
Table 3.1 Operating States .................................................................................................. 11
Table 3.2 Parameters of the system .................................................................................... 13
Table 3.3 Disturbances of the system.................................................................................. 14
Table 3.4 Outputs of the system .......................................................................................... 14
Table 3.5 Active equations depending on the operating state .............................................. 16
Table 6.1 Optimization parameters for Test Case 1 ............................................................. 31
Table 6.3 Optimization Parameters for Test Case 3 ............................................................ 38
Table 6.6 Comparison of overall gains ................................................................................ 47
Table 7.1 Power Capacities ................................................................................................. 51
Table 8.1 Optimization parameters for comparison scenario ............................................... 53
Table 8.2 Parameters values for initialization ...................................................................... 53
Table 8.3 Result of the future scenario (1) ........................................................................... 54
Table 8.4 Comparison of specific gains based on gas consumptions and operation hours .. 55
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List of Abbreviations
Abbreviation
Description
BB
Branch and Bound
CHP
Combined Heat and Power
DE
Differential Evolution
DES
Discrete Event Systems
DHS
District Heating System
Dymola
Dynamic Modeling Laboratory
GA
Genetic Algorithm
GenOpt
Generic Optimization Algorithm
GLM
Gutleutmatten (name of a project)
HVAC
Heating Ventilation and Air Conditioning
IP
Integer Programming
ISE
Institute for Solar Energy Systems
KP
Knapsack Problem
MIP
Mixed Integer Programming
PSO
Particle Swarm Optimization
SA
Simulated Annealing
TRNSYS
Transient Systems Simulation Program
TSP
Travelling Salesman Problem
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List of Symbols
Symbol
𝐴
𝑏CHP
Unit
𝑚2
Description
Heat Transfer Area
Boolean signal showing the CHP’s state (on/off)
𝑏Boiler
Boolean signal showing the Boiler’s state (on/off)
𝑐c
Cognitive velocity constant
𝑏St
Boolean signal showing the Storage Tank’s state (on/off)
𝑐el
𝑐𝑐.⁄𝑘𝑘ℎ
Electricity price
𝑐heat
Heat price
𝑐fuel
Fuel price
𝑐p
𝐽⁄(𝑘𝑘 ∙ 𝐾 )
𝑑𝑑
𝐽
Change of internal energy
𝑑𝑑e
𝐽
Change of potential energy
𝑊
Mean residual load over an evaluation period
𝑐s
𝑑𝑑k
𝐺
𝐺̅
𝐺k (𝑡)
𝐻p
Specific heat capacity
Social velocity constant
𝐽
Change of kinetic energy
𝑊
Residual Load
the 𝑘th component of the best position vector ever reached
Prediction horizon
𝐽CHP
𝑐𝑐.
Gain of the CHP
𝐽St
𝑐𝑐.
Gain of the Storage Tank
𝐽Boiler
𝐽T
𝐿𝐿𝐿𝐿abs
𝑐𝑐.
Gain of the Boiler
𝑐𝑐.
Total System Gains
𝑛i
𝑝
Absolute load grid matching coefficient
Number of input variables
Number of time steps
𝑃el
𝑊
Electrical power production
𝑃load
𝑊
National heat load
𝑃wind
𝑊
𝑃ik (𝑡)
𝑃PV
𝑃1
𝑊
the 𝑘th component of the best position vector of particle 𝑖
Power production by photovoltaic systems
Power production by wind turbines
Penalty value for the minimum operation time constraint
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𝑃2
𝑄
Penalty value for the temperature constraint
𝐽
Heat exchange within a control mass
𝑊
CHP nominal thermal power
𝐽
Heat discharged from the storage tank
𝑊
Amount of fuel consumption at Boiler
𝑄loss,Boiler
𝑊
Amount of heat loss at Boiler
𝑄loss,St
𝑊
Amount of heat loss at Storage Tank
𝐽
Net heat transfer
𝑊
Nominal discharge power of the storage tank
𝑄ch
𝑊
Heat charged to the storage tank
𝑄CHP,el
𝑊
CHP nominal electrical power
𝑄fuel,CHP
𝑊
Amount of fuel consumption at CHP
𝑄loss
𝐽
Amount of heat loss
𝑊
Amount of heat loss at CHP
𝑊
Amount of heat loss per unit time
𝑊
Net heat transfer per unit time
𝑄CHP,th
𝑄disch
𝑄fuel,Boiler
𝑄loss,CHP
𝑄̇loss
𝑄net
𝑄̇net
𝑄St,disch
𝑟1 (𝑡) , 𝑟2 (𝑡)
𝑠CHP
𝑠St
Random coefficients between 0 and 1
Boolean signal for the minimum operation time constraint
Boolean signal for the temperature constraint
𝑇amb
℃
Ambient temperature
𝑇low
℃
Low temperature level
𝑇high
𝑇s
𝑡min
𝑈
𝑢b
𝑢CHP
℃
High temperature level
℃
Temperature of the storage tank
ℎ𝑜𝑜𝑜𝑜
Minimum operation time
𝑊
Heat supply by the boiler
𝑊
Heat supplied to the storage tank
𝑊
Heat supply by the storage tank
𝑊 ⁄(𝑚2 ∙ 𝐾)
Overall heat transfer coefficient
𝑊
Heat supply by the CHP
𝑊
Heat load
𝑉
𝑚3
Volume of the storage tank
𝑊
𝐽
𝑢CHP,St
𝑢load
𝑢st
𝑣ik (𝑡)
𝑊el
𝑥ik (𝑡)
𝜌
𝑘𝑘ℎ
𝑘𝑘⁄𝑚3
𝑘th velocity vector component of the particle 𝑖
Work done on a control mass
Total electricity production for an evaluation period
𝑘th position vector component of the particle 𝑖
Density
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∆𝑇
𝜂CHP,th
𝜂CHP,el
𝜂b
℃
Temperature difference
Thermal efficiency of the CHP
Electrical efficiency of the CHP
Efficiency of the boiler
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Supervisory Control of a Combined Heat and Power Plant by Economic Optimization – Delikaya (2015)
1. INTRODUCTION
Supervisory control theory has been initiated in 1989, which deals with comprehensive
ways of solving control problems with discrete event systems (DES) (Ramadge and Wonham
1989, p. 81). These systems can be encountered at logistic systems, traffic systems,
manufacturing workcells and many other industrial processes (Thistle 1996, p. 25). Possible
event sequence of such systems is illustrated in Fig. 1.1.
Fig. 1.1 State trajectory of a DES (Ramadge and Wonham 1989, p. 82)
As shown in Fig. 1.1, such systems include state transitions at discrete time periods.
These states indicate certain events to be performed for a certain amount of time. At many
industrial processes including the DESs, optimizing the order of these events make
enormous changes in terms of timing and efficiency of the process. According to Ramadge
Wonham (RW) framework as represented in Fig 1.2, a supervisor controls the events
generated by the plant and terminate them when needed.
Fig. 1.2 Ramade Wonham (RW) Framework (Morgenstern and Schneider n.d., p. 3)
In this thesis a supervisory control of a combined heat and power (CHP) plant is achieved
by an economic optimization. Thus, an optimization tool acts as a supervisor and optimizes
the best trajectory of possible discrete events rather than controlling them.
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1.1. Motivation
Concurrent production of electrical and thermal energy by the CHPs has been accepted
as a good way of increasing the overall efficiency of district heating systems (DHS). German
parliament has approved a new CHP law in 2008, which aims to increase the total share of
electricity production by the CHP units up to 25 % by 2020 (BMWi 2012, p. 2). Considering
this information one can claim that the CHPs will have a significant role in electricity
production in the near future. On the other hand, Germany is one of the European leaders in
renewable energies and many supporting schemes and incentives for renewable energy
based plants are in force in Germany. Therefore, it can be anticipated that wind and solar
power production will also take a higher share in electricity production, thus playing a
significant role in variation of electricity prices in the future. Thus, producing electricity at
peak hours is desired for financial reasons (Streckiene et al. 2009, p. 2308). Therefore, an
optimal operation of a DHS will be significant in future’s highly fluctuating electricity market in
Germany. By an optimal operation of a DHS it is meant that the system responds to the
changes in electricity prices and distributes the load among the heat production units within
the system, so that the highest financial gain is obtained and the heat demand is fully
covered. The DHS should be also able to cover the heat load while producing electricity. This
is why combining a heat storage with the CHP units is a widely preferred solution for a
flexible operation of a DHS, since the heat storage is able to store the excess heat produced
by the CHP unit especially at peak hours, when producing electricity is more favorable than
other time periods of a day (Zhao et al. 1998). Therefore, an economic optimization has been
carried out in this thesis in order to achieve an optimal operation of a DHS.
1.2. Literature Survey
In order to analyze all subcomponents of the DHS, it is common to model such systems
within a simulation environment for carrying out an optimization process. Simulation-based
optimization is a promising way of analyzing the performance of a system with complex
natures. There are many simulation tools, such as “EnergyPlus”, “TRNSYS” and “Dymola”,
which can be coupled with an optimization tool to optimize various parameters in heating,
ventilation and air conditioning (HVAC) systems of buildings and district heating systems in
general. Parametric simulation method is an alternative technique to optimization, at which
each single variable is manually parameterized to see the effect on the performance of the
systems while all other variables are kept constant (Nguyen et al. 2013, p. 1044).
Nevertheless without doing an optimization, achieving a good performance of such systems
might be a cumbersome task considering the effort and the time spent. A simulation-based
optimization can be summarized in following steps:
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•
mathematical modeling and formulation of the optimization problem, setting
constraints and design variables
•
determining a tool for implementation and simulation of the model
•
determining an optimization tool and an algorithm to be coupled with the
simulation tool and executing the optimization process
•
controlling whether the global optima is found
Arguably, while planning a DHS, it must be analyzed what kind of heat production units
should be utilized and how they should share the heat demand. Furthermore, temperature
levels, at which heat should be transferred, must be well specified in order to operate the
DHS in a cost-effective way. Until today, many parameters have been investigated and
several approaches have been taken to minimize the operational costs of the DHS. Those
can be categorized into three groups (Benonysson et al. 1995, p. 299):
•
optimal distribution of heat load between different heat producing units and a heat
storage unit, if applicable
•
minimization of supply temperatures of each heat production unit, so that the cost
savings through less fuel consumption can be achieved
•
optimizing the DHS dynamics, such as start-stop times of pumps, the temperature
levels at each heat production unit and as well as load distribution among the
units concurrently
Modeling and expressing all physical behaviors of the components in a DHS brings also
complexities, i.e. extra constraints to be handled, together. That’s why actual models should
be replaced with simplified models for optimization purposes. A simplified model is a reduced
order model, which only captures main behaviors of the actual model. It can be then used for
special optimization purposes. It is asserted that optimizing physically expressed actual
models require undesirably large execution times with respect to simplified models (Chandan
et al. 2012, p. 3070). Furthermore, it is pointed out that simplified models perform similar
results to those obtained by optimizing actual models (Nguyen et al. 2013, p. 1045).
However, it is also necessary to note that simplified models lead to an uncertainty within the
model. Therefore, the assumptions being made to simplify the actual models should be
realistic to get accurate and reliable simplified models. To meet this end, in this master’s
thesis a reduced order model, which represents the main features of a detailed DHS, is
modeled and the aforementioned first optimization approach, which focuses on sharing the
heat demand optimally among the units involved in the system, is taken to minimize the
overall system costs.
As mentioned, it is aimed to share out the heat demand optimally among the heat
production units while considering the fluctuations in electricity prices. This optimization
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problem has been modeled as a discrete optimization problem, where the discrete variables
define the modes of operation of the DHS. These modes or operating states are introduced
in Section 3.1. With this in mind, in literature there are two well-known problems that are
linked to the discrete optimization problems:
First one is the Travelling Salesman Problem (TSP) that is related to discrete optimization
problems. There are many articles written and solution methods are proposed for the TSP.
Consider a region of n cities. A salesman is supposed to stop by each of n cities, aiming to
take the shortest route and return to the starting point and each random route is a possible
solution of the TSP (Laporte 1992, pp. 231-232). Similarly the Knapsack Problem (KP) is the
second typical example of discrete optimization problems. The optimization problem is
finding the best combination of items to be packed in a knapsack, in which the total weight
must be lower than or equal to a certain limit and usually each item has a monetary value
disproportional to its weight (Suzuki 1978, p. 162). Both of these problems can be subject to
some other constraints.
A literature study has shown that aforementioned problems can be generally called as
Integer Programming (IP) problems or 0-1 IP problems more specifically. An unconstrained
linear 0-1 IP problem can be mathematically represented as
𝑛
𝑚𝑚𝑚 � 𝑐j ∙ 𝑥j ,
(𝟏. 𝟏𝟏)
𝑗=1
subject to
𝑛
� 𝑠j ∙ 𝑥j ≤ d ,
𝑗=1
𝑥j 𝜖 {0,1}𝑛 .
(𝟏. 𝟏𝟏)
(𝟏. 𝟏𝟏)
Considering the KP, 𝑐j would be the monetary value of each item; 𝑠j could be interpreted
as the weight of the item; and 𝑑 would be the maximum allowed weight in the knapsack; 𝑥j
would stand for whether the item is selected or not and 𝑛 would be the number of items.
There are many different formulations IP problems depending on the constraints of the
problem. When 𝑥j can take both integer and non-integer values, then this particular problem
is called mixed integer programing (MIP).
There are many methods to solve IP and MIP problems. These can be grouped under
two main headlines: stochastic and deterministic ones. Stochastic methods provide a
solution of coping with generally nonlinear, high dimensional complex systems, which are not
suitable for other classical deterministic optimization methods (Spall 2004, p. 170). Many of
them are nature-inspired, and they make use of iterative trial and error processes to
converge an optimal solution. Thus, stochastic methods do not assure finding the global
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optima. Deterministic methods, on the contrary, make use of the analytical properties of the
problem to converge to the global solution (Lin et al. 2012). There are no stochastic elements
and assumptions within the algorithms, they rely on linear algebra and are mainly based on
computation of gradient of the variables for convergence (Cavazzuti 2013, p. 77). One
advantage of deterministic methods is that they guarantee finding the global optima
eventually. However, in comparison to stochastic methods, they are more compute-intensive
and therefore require higher execution times (Collet and Rennard 2006).
Some examples of both these methods are explained below.
1.2.1. Deterministic Methods
One of the well-known algorithms to solve the IP problems is Branch and Bound (BB)
method. It is an efficient algorithm for systematically enumerating of discrete problems. The
algorithm mainly includes two steps. First step is branching, which is a relaxation of the
feasible region by dividing it into several sub-regions. Over these sub-regions upper and
lower bounds are determined, which is known as the bounding step of the algorithm. The
idea is eliminating the branches by comparing upper and lower bound values of each subregion, thus reducing the search space.
Cutting Planes method aims simply to tighten the feasible solution region by adding linear
inequality constraints, which cut the search plane into two groups and omit some of the noninteger values. By doing this recursively, an integer solution is found. Although some
problems are solved using cutting planes method, it is asserted that it behaves differently on
similar problems and shows weak performance in reducing the size of search space.
Nevertheless when it’s combined with BB, the method is known as “branch and cut”, it
demonstrates a good performance on solving IP problems (Sherali and Driscoll 2000).
Benders Decomposition is another solution method for solving especially integer
programming problems. It is a class of multistage optimization algorithms. This method,
unlike the other traditional approaches, divides variables in the optimization problem into two
sets: master problem variables and sub-problem variables. First a first-stage master problem
with an arbitrary number of variables is solved. Then some of the set of solutions are
eliminated by the information interpreted from the results of sub-problem. This elimination
procedure, which is also known as Bender’s cut, is done iteratively, thus leading to the
optimality (Taskin 2010).
1.2.2. Stochastic Methods
Simulated Annealing (SA) method is a local search meta-heuristic method, which is used
to solve generally discrete optimization problems. It is known that a preheated material, after
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slow cooling, reaches its best lattice energy state by avoiding the crystal defects. The name
itself stems from annealing in metallurgy. The SA method inspires from this thermodynamic
behavior and establishes a connection in the algorithm (Henderson et al. 2003, p. 288).
Genetic Algorithm (GA) finds solution to optimization problems using principles inspired
by genetics and natural selection, such as crossover, selection, and mutation. Each possible
solution or chromosome is assigned to a fitness value. Then, relatively “fit” chromosomes are
selected for reproduction of new individuals. A crossover operator is used for exchanging the
genetic information between each chromosome, and mutation operator is used to keep
enough diversity among the chromosomes to avoid premature convergence (Deep et al.
2009, p. 506). In literature many applications of the GAs are available for both integer and
mixed integer programming problems. A similar algorithm to the GA is Differential Evolution
(DE) algorithm. The main difference in finding better solution is that, the GA makes use of
selection operator, whereas the DE relies on mutation operator (Karaboga and Ökdem 2004,
pp. 53-54).
Among all of these methods, it is not intended to show pros or cons of each algorithm,
but to give a general overview on solution methods for IP problems to the reader. There is no
clear consensus that any of the algorithms mentioned above outperforms the other one
though. However, a greater emphasis will be placed on explaining Particle Swarm
Optimization (PSO) algorithm, which is a type of stochastic algorithms, in Section 5.2.
Because it is the algorithm being used for solving discrete optimization problem described
within this thesis.
1.3. Thesis Objectives
Main objective of this thesis is to carry out an economic optimization of a district heating
system. It concerns following tasks:
•
mathematical modeling of the economic optimization problem
•
implementing constraints and improving the already existing DHS model, which
included the main units of the DHS
•
constructing the cost function of constrained optimization problem (The coupling of
the model with the optimization tool had already been carried out.)
•
plausibility analysis of the model via several test cases
•
comparing the power-driven operation with the heat-driven operation with respect to
overall financial gains and grid interactivity
•
analyzing the effect of storage size on system gains and grid interactivity
6
2. SYSTEM
2.1. Structure of a District Heating System
DHSs are highly utilized systems for space heating and water heating in industrial and
residential buildings. The DHS consists of heat production units, which could be either heatonly plants or CHP plants coupled with a boiler and a storage tank. The CHPs are power
generation plants, which make use of the thermal energy in exhaust gases, cooling systems
or other thermal energy waste streams to enhance the overall efficiency of the system. CHPs
can be powered by fossil fuels, such as gas, oil or coal. Wood or biogas in the case of
biomass boilers or heat from earth of in the case of geothermal energy can be used to heat
up the water in the boiler. The water is then channeled to the various places of consumption,
which are industrial, public, or private buildings. A district heating system can be designed to
serve only a few residences, as well as a small village or a part of a city. Using CHP plants
within the DHS is a prevalent way of supplying local energy demand for public authorities or
industries. A schematic representation of a DHS is given in Fig. 2.1.
Fig. 2.1 Schematic representation of a DHS
As seen in Fig. 2.1, the CHP generates two types of energy from a single authority:
electricity and heat for local use. The principle is as follows: the boiler produces a
superheated steam. This steam then powers a turbine, which is connected to a generator to
produce electricity. Similarly a CHP can be powered by a combustion engine to generate
energy. Electrical energy produced is sent to the power grid to supply local users. The heat
given off by the turbine, which is usually lost, is recovered to provide heat for a local
7
community or an industrial facility. Thus, the CHP offers a higher level of efficiency than that
of solely heat or power producing plants.
2.2.
District “Gutleutmatten”
In this thesis, “Gutleutmatten” (GLM), which is a part of the city Freiburg, is the district in
question. The project will be completed by 2016 with the joint partnership of Fraunhofer
Institute for Solar Energy Systems (ISE), Badenova Wärmeplus GmbH, and Solites. To meet
residential heating demand, decentralized solar thermal systems and a district heating
system will be utilized. On a total of 40,000 𝑚2 heated floor area, 32 buildings including
around 500 households will be built (Oliva et al. 2014, para. 1). All buildings will have a
decentralized storage unit, with a total capacity of 200 𝑚3 (Elci et al. 2014, para. 7). Based
on the energy standard “KfW-Effizienzhaus 55” as cited in (Elci et al. 2014, para. 6), specific
heating demand of 35 kWh/(m2 a) is assumed for each building. Main power supply units will
be a biogas-fired CHP with an electrical nominal power of 600 𝑘𝑘el and thermal capacity
of 654 𝑘𝑘th and a natural gas-fired boilers with total capacity of 4000 𝑘𝑘𝑠 (Elci et al. 2014,
para. 6). The district heating station for District GLM already in operation and it is located
close to the district, and it is also capable of supplying heat to other buildings outside the
district GLM. A site plan of the district GLM is shown in Fig. 2.2.
Fig. 2.2 Site plan of District “Gutleutmatten”(Elci et al. 2014)
2.3. System Boundaries
Before describing the model, system boundaries are clearly defined and they stay
consistent for any of the test cases, which will be discussed in the next sections. The system
consists of a boiler, a CHP and a storage tank unit. First and foremost fuel burned at the
boiler and at the CHP is regarded as consumptions, while the electricity fed into the grid and
the supplied heat to consumer are regarded as gains in financial terms.
8
It is also necessary to address following assumptions that define the system boundaries:
•
The electricity produced by the CHP unit is sold at the energy exchange. However,
the electricity consumption of pumps and other auxiliary units in the DHS is
neglected.
•
Gas, electricity and heat prices as well as heat demand and technical specifications
of the CHP and the boiler are assumed to be known. They are the disturbances of the
system, which are neither controlled nor predicted.
•
Investment and maintenance costs are ignored and they are not playing any role for
the economic optimization. Both the boiler and the CHP are expected to operate at
any periods of a year. Decommission of the heat production units due to maintenance
is ignored.
•
There is a continuous fuel supply available to run the CHP and the boiler.
•
The DHS must be able to cover the heat demand under any circumstances.
9
10
3. MODELING
3.1. Operating States
As mentioned before, discrete variables of the optimization problem are the modes of
operation or the operating states of the DHS.
Maintenance services for the CHPs are regularly performed depending on the yearly
operation time of the units. Maintenance costs per unit operation time is generally fixed, thus
the CHPs are generally operated at nominal power in order to minimize the proportion of the
maintenance costs with respect to the gain of the CHP per unit operation time. That’s why; it
is assumed that there are two operating possibilities for the CHP, either running at nominal
power or “shutdown state”. On the other hand, the boiler and the storage tank are flexible to
supply heat at any level. Considering there are three main units, one can argue that there are
eight possible combinations that these three units can share the heat demand. Table 3.1
shows all possible states, at which a DHS can operate. “ON” stands for that the unit supplies
heat or electricity, whereas “OFF” stands for that the unit is shut down.
Table 3.1 Operating States
States
1
2
3
4
5
6
7
8
CHP
ON
ON
OFF
ON
ON
OFF
OFF
OFF
Boiler
OFF
ON
OFF
OFF
ON
ON
ON
OFF
Storage Tank
OFF
OFF
ON
ON
ON
OFF
ON
OFF
3.2. Storage Tank Model
Storage tanks, which are operated properly, are generally stratified with respect to the
temperature levels of the water in the tank. The water, which is fed into the tank, is placed to
the corresponding part of the storage tank depending on its temperature. Generally, the
upper part of the tank stores relatively hotter water, which is then supplied to provide the
needed heat for domestic use. Since only input-output relations are considered and all
hydraulic parts of the DHS are neglected for the simplicity, the storage tank is simplified and
represented as a single mass rather than a stratified model. The storage tank is basically
charged by the CHP. This charging power depends on the heat load. As long as the heat
11
load is less than the CHP nominal power, then excess heat produced by the CHP is stored in
the storage tank. Fig. 3.1 shows a single node representation of the storage tank. Additional
to the charging and the discharging affect, a heat loss through its outer wall is considered.
It’s assumed that the temperature of the storage tank, ambient temperature, and properties
of the water are known to calculate the temperature in the next time step.
Fig. 3.1 Single node representation of the storage tank
Following formulations show how the charging and the discharging processes affect the
temperature of the storage tank. Amount of net heat transfer 𝑄net into the body can be
calculated as
𝑄net = 𝑄ch − (𝑄disch + 𝑄loss ) ,
(𝟑. 𝟏)
where 𝑄ch and 𝑄disch stand for the charging and the discharging powers respectively and
𝑄loss stands for the amount of the heat lost to the environment.
Assume that the temperature of the single mass represented in Fig. 3.1 at time 𝑡 is
denoted by 𝑇 (𝑡) . When this mass is subject to a heat transfer that amounts to 𝑄net until time
(𝑡 + 1) , then the temperature at that time 𝑇 (𝑡+1) can be calculated by using 1st law of
thermodynamics, which is
𝑄 − 𝑊 = 𝑑𝑑 + 𝑑𝐸k + 𝑑𝑃e ,
(𝟑. 𝟐)
where 𝑄 is the heat given to the system; 𝑊 is the work done on the system; 𝑑𝑑 is the change
of internal energy; 𝑑𝐸k is the change of kinetic energy and 𝑑𝑃e is the change of potential
energy.
The system boundary in this case is the mass including all heat transfers. In
thermodynamics, everything within a closed system is called a control mass. In this case,
there is no work done on the system, and 𝑑𝐸k and 𝑑𝑃e are generally negligible for a
stationary control mass. Internal energy change of the control mass at constant pressure can
be represented as
𝑄 = 𝜌 ∙ 𝑉 ∙ 𝑐𝑝 ∙ (∆𝑇) ,
12
(𝟑. 𝟑)
where 𝜌 is the density [𝑘𝑘⁄𝑚3 ] of the control mass; 𝑉 is the volume [𝑚3 ]; of the control
mass 𝑐p is the specific heat capacity [𝐽⁄(𝑘𝑘 ∙ 𝐾)] of the control mass and ∆𝑇 is the
temperature difference [𝐾] due to heat transfer.
It is now possible to apply the 1st law of thermodynamics to the control mass. The net
heat transfer would be then
𝑄net = 𝜌 ∙ 𝑉 ∙ 𝐶p ∙ �𝑇 (𝑡+1) − 𝑇 (𝑡) � .
(𝟑. 𝟒)
When the unknown term is get alone on the left side, the final form of the equation will be
𝑇 (𝑡+1) =
𝑄net
+ 𝑇 (𝑡) .
𝜌 ∙ 𝑉 ∙ 𝑐p
(𝟑. 𝟓)
Let us now derive the term 𝑄̇loss . According to the Newton’s law of cooling it can be
written as
𝑄̇ = 𝑈 ∙ 𝐴 ∙ ∆𝑇 ,
(𝟑. 𝟔)
where 𝑄̇ is the heat removed or gained [𝑊] by convection per unit time; 𝑈 is the overall heat
transfer coefficient [𝑊 ⁄(𝑚 2 ∙ 𝐾) ]; 𝐴 is the heat transfer surface area [𝑚2 ] and ∆𝑇 is the
temperature difference [𝐾] between ambient and the control mass.
When the corresponding temperatures are placed in Eq. (3.6), following formulation can
be derived as
𝑄̇loss = 𝑈 ∙ 𝐴 ∙ �𝑇 (𝑡) − 𝑇amb � ,
where 𝑇amb is the ambient temperature.
(𝟑. 𝟕)
3.3. Mathematical Model of the System
For simplicity, Table 3.2, Table 3.3, and Table 3.4 are given to describe the parameters,
the outputs, and the disturbances of the system, which will then be used in the equations
describing the mathematical model. Additional to the parameters (𝜌, 𝑈, 𝐴, 𝑉 and 𝑇amb) which
are explained in the previous section, other parameters are also listed in Table 3.2.
Table 3.2 Parameters of the system
Parameters
Description
𝑸𝐂𝐂𝐂,𝐭𝐭
Nominal thermal power of CHP
𝑸𝐒𝐒,𝐝𝐝𝐝𝐝𝐝
Maximum discharge power of storage tank
𝑸𝐂𝐂𝐂,𝐞𝐞
Nominal electrical power of CHP
13
Table 3.3 Disturbances of the system
Disturbances
Description
𝒖𝐥𝐥𝐥𝐥
Heat Load
𝒄𝐞𝐞
Electricity price
𝒄𝐟𝐟𝐟𝐟
𝒄𝐡𝐡𝐡𝐡
Fuel price
Heat Price
Table 3.4 Outputs of the system
Outputs
Description
𝒖𝐂𝐂𝐂
Heat supply by CHP (only to consumer)
𝒖𝐂𝐂𝐂,𝐒𝐒
Heat supply to storage tank
𝒖𝐬𝐬
Heat supply by storage tank
𝑻𝐬
Temperature of storage tank
𝒖𝐛
Heat supply by boiler
𝑷𝐞𝐞
Electricity production
The outputs of the system depend on the operating states of the system. In order to
formulate the whole system mathematically, the outputs of the system are separately defined
for each operating state and unit.
Let us first consider the CHP. As known, it can charge the storage tank, supply heat to
the consumer, do both actions simultaneously, or it can be shut down.
i.
When CHP charges storage tank and supplies to the consumer ;
𝑢CHP = 𝑚𝑚𝑚(𝑄CHP,th , 𝑢load ) 1 ,
𝑢CHP,St = 𝑚𝑚𝑚((𝑄CHP,th − 𝑢load ) , 0)
ii.
When CHP is shut down;
𝑃el = 𝑄CHP,el .
1
2
,
(𝟑. 𝟗)
(𝟑. 𝟏𝟏)
𝑢 CHP = 0 ,
(𝟑. 𝟏𝟏)
𝑃el = 0 .
(𝟑. 𝟏𝟏)
𝑢 CHP,St = 0 ,
iii.
2
(𝟑. 𝟖)
When CHP supplies heat only to the consume;
min(a,b) outputs the minimum of a and b.
max(a,b) outputs the maximum of a and b.
14
(𝟑. 𝟏𝟏)
𝑢CHP = 𝑄CHP,th
and Eq. (3.12) and Eq. (3.13) also hold true in this case.
(𝟑. 𝟏𝟏)
The boiler can either supply heat or not. The supplied heat is the remaining heat demand
after the CHP and the storage tank supplied to the consumer. The two equations for those
actions are
𝑢 b = 𝑢load − (𝑢CHP + 𝑢st ) ,
𝑢b = 0 .
(𝟑. 𝟏𝟏)
(𝟑. 𝟏𝟏)
In order to derive the equations for the storage tank, three cases are separately
considered. Temperatures are calculated based on the formulations derived in Section 3.2.
i.
Storage tank is charged, but no discharge is allowed at the same time.
The net heat transfer is then the difference between supplied heat by the CHP and heat
loss to the environment. The corresponding equations are
𝑢 st = 0 ,
(𝑡)
𝑄̇net = 𝑢CHP,St − 𝑄̇loss = 𝑢CHP,St − 𝑈 ∙ 𝐴 ∙ �𝑇s − 𝑇amb �.
(𝟑. 𝟏𝟏)
(𝟑. 𝟏𝟏)
When 𝑄net in Eq. (3.18) is substituted into Eq. (3.5), then following equation is obtained:
(𝑡+1)
ii.
𝑇s
=
(𝑡)
�−𝑈 ∙ 𝐴 + 𝜌 ∙ 𝑉 ∙ 𝑐p � ∙ 𝑇s + 𝑢CHP,St + 𝑈 ∙ 𝐴 ∙ 𝑇amb
.
Storage tank is neither charged nor discharged.
(𝟑. 𝟏𝟏)
Eq. (3.6) holds true in this case as well. The net heat transfer is basically only the heat
loss, which is
(𝑡)
𝑄̇net = −𝑄̇loss = −𝑈 ∙ 𝐴 ∙ �𝑇s − 𝑇amb �.
(𝟑. 𝟐𝟐)
When 𝑄𝑛𝑛𝑛 in Eq. (3.20) is substituted into Eq. (3.5), then following equation is obtained:
(𝑡+1)
iii.
𝑇s
=
(𝑡)
�−𝑈 ∙ 𝐴 + 𝜌 ∙ 𝑉 ∙ 𝑐p � ∙ 𝑇s + 𝑈 ∙ 𝐴 ∙ 𝑇amb
.
Storage tank is not charged, but discharged.
(𝟑. 𝟐𝟐)
In this case the net heat transfer is sum of the discharge power and the heat loss, which
can be written as
(𝑡)
𝑄̇net = −�𝑢st + 𝑄̇loss � = −𝑢st − 𝑈 ∙ 𝐴 ∙ �𝑇s − 𝑇amb �.
15
(𝟑. 𝟐𝟐)
The discharge power of the storage tank is the remaining heat demand after the CHP
supplied to the consumer provided that it is less than the maximum discharge power of the
storage tank. Heat supply by the CHP (𝑢CHP ) can also be “0”. The discharge power can be
written as follows:
(𝟑. 𝟐𝟐)
𝑢st = min�(𝑢load − 𝑢CHP ) , 𝑄St,disch �.
When 𝑄net in Eq. (3.23) is substituted into Eq. (3.5), then following equation is obtained:
(𝑡+1)
𝑇s
=
(𝑡)
�−𝑈 ∙ 𝐴 + 𝜌 ∙ 𝑉 ∙ 𝑐p � ∙ 𝑇s − 𝑢𝑆𝑆 + 𝑈 ∙ 𝐴 ∙ 𝑇amb
.
(𝟑. 𝟐𝟐)
As can be anticipated, these equations are state dependent. Therefore, depending on the
operating state some of the equations might not be valid. It is shown in Table 3.5, when
these equations hold true. Each column has 6 green boxes, indicating that the corresponding
equation on the left column is used for calculations. It is worth noting that these 6 equations,
which are active at each state, are used to calculate the system outputs given in Table 3.4.
Table 3.5 Active equations depending on the operating state
States
Eq. (3.8)
1
2
3
4
Eq. (3.9)
Eq. (3.10)
Eq. (3.11)
Eq. (3.12)
Eq. (3.13)
Eq. (3.14)
Eq. (3.15)
Eq. (3.16)
Eq. (3.17)
Eq. (3.19)
Eq. (3.21)
Eq. (3.22)
Eq. (3.24)
16
5
6
7
8
3.4. Formulation of the Cost Function
Before formulating the cost function, it is useful to show some fundamental mathematical
expressions of individual units. Fig 3.2 shows all the inputs and outputs (power-based) of
each single unit. 𝑄fuel,CHP and 𝑄fuel,Boiler are fuel consumptions by the CHP and the boiler
respectively. 𝑄loss,CHP , 𝑄loss,Boiler and 𝑄loss,St are heat losses of individual units as shown.
Other variables are already described in Table 3.3.
Fig. 3.2 Energy flow within the system boundary
General energy balance equations of each single unit can be listed as
𝑄CHP,th = 𝑢CHP + 𝑢CHP,St ,
(𝟑. 𝟐𝟐)
𝑄fuel,Boiler = 𝑢b + 𝑄loss,Boiler ,
(𝟑. 𝟐𝟐)
𝑄fuel,CHP = 𝑄CHP,th + 𝑄loss,CHP ,
𝑢CHP,St = 𝑢st + 𝑄loss,St .
(𝟑. 𝟐𝟐)
(𝟑. 𝟐𝟐)
As already shown in Section 3.2, heat loss within the storage tank has been
mathematically formulated. However for the boiler and the storage tank efficiencies are taken
into account to consider losses. The amount of energy that needed to be extracted can be
found by the equations
𝑄fuel,CHP =
𝑄CHP,th
𝑃el
=
𝜂CHP,th 𝜂CHP,el
17
(𝟑. 𝟐𝟐)
and
𝑄fuel,Boiler =
𝑢b
,
𝜂b
(𝟑. 𝟑𝟑)
where 𝜂CHP,th is the thermal efficiency of the CHP; 𝜂CHP,el is the electrical efficiency of the
CHP and 𝜂b is the efficiency of the boiler.
Based on these expressions, the gain of each unit can be formulated as follows:
𝐽CHP = � [𝑃el ∙ 𝑐el + 𝑢CHP ∙ 𝑐heat ] − 𝑄fuel,CHP ∙ 𝑐fuel � ∗ 𝜏 ,
𝐽Boiler = �𝑢b ∙ 𝑐heat − 𝑄fuel,Boiler ∙ 𝑐fuel � ∗ 𝜏 ,
𝐽St = [𝑢st ∙ 𝑐heat ] ∙ 𝜏 ,
(𝟑. 𝟑𝟑)
(𝟑. 𝟑𝟑)
(𝟑. 𝟑𝟑)
where 𝜏 is the operation time in hours [h]; 𝑐el , 𝑐heat and 𝑐fuel are the previously described
prices in cents per kilowatt-hour [ct./kWh]; 𝑃el , 𝑢CHP , 𝑄fuel,CHP , 𝑢b , 𝑄fuel,Boiler and 𝑢st are in
kilowatts [kW]; 𝐽CHP , 𝐽Boiler and 𝐽St are the gains of the CHP, the boiler and the storage tank
in cents [ct.].
As can be understood, the gains are basically calculated on the basis of extracting
consumption by production. It is necessary to note that heat supply to the consumer (𝑢CHP,St )
is not counted as a gain of the CHP, since it is not directly sold to the consumer. That’s why;
the actual heat sold to the consumer 𝑢st is counted as the gain of the storage tank.
There are mainly two constraints within the model that should be considered before
deriving the cost function.
•
First one is the temperature constraint for the storage tank. The temperature of the
storage tank must not be charged any more when it is full.
•
Secondly, a minimum operation time for the CHP should be taken into consideration.
This is to make sure that the CHP is warmed up and, is running at optimal
temperature for a certain minimum time before it is shut down again. Similarly when
CHP is shut down, it should wait at least for minimum operation time before operating
again. Moreover, with minimum operation time is desired since the life cycle of the
CHP can be increased with a proper operation schedule.
Then the final form of the cost function is written as follows:
𝐻𝑝
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
𝐽T = 𝑚𝑚𝑚 �( 𝐽CHP ∙ 𝑏CHP + 𝐽St ∙ 𝑏St + 𝐽Boiler ∙ 𝑏Boiler ),
𝑡=1
(𝟑. 𝟑𝟑𝐚)
which is subject to the temperature constraint
(𝑡)
𝑇s
< 𝑇high
18
(𝟑. 𝟑𝟑𝐛)
and the minimum operation time constraint, which is only active
(𝑡)
(𝑡−1)
𝑤ℎ𝑒𝑒 𝑏CHP − 𝑏CHP ≠ 0,
𝑡ℎ𝑒𝑒
(𝑡+𝑡min)
(𝑘)
(𝑘−1)
� �𝑏CHP − 𝑏CHP � = 0 ,
𝑘=𝑡
(𝟑. 𝟑𝟑𝐜)
where 𝐽T is the overall gain over a prediction horizon; 𝑇high is the allowed maximum
temperature level for storage tank; 𝑇s is the temperature of the storage tank; 𝑡min is the
minimum operation time period, at which the CHP should maintain its mode of operation;
𝑏CHP , 𝑏St , 𝑏Boiler ∈ {0,1}𝑝 are binary variables, 𝑝 is the number of time steps in the prediction
horizon and 𝐻p is the prediction horizon.
As can be deduced, the optimization problem is maximizing the overall system gains over
a prediction horizon. It is worth mentioning that the combination of binary variables 3 𝑏CHP , 𝑏St
and 𝑏Boiler are indicating whether the unit is operating, thus forming operating states of the
system. Consider i.e. {𝑏CHP , 𝑏St , 𝑏Boiler } = {1,0,0}, which would be the “Operating State 1”
according to the Table 3.1. Therefore, optimization problem can be also regarded as finding
the best sequence of operating states over a prediction horizon so that overall system gains
are maximized.
Every time when these binary variables change, then the operating state and the related
equations change as well. Although the gains of each single unit are second-based, the
binary variables are not expected to change at any second. Thus, a number of time steps
should be defined to fix the maximum number of input changes over a prediction horizon.
As can be understood from Eq. (3.34c), the minimum operation time constraint is not
always active. The constraint is only active when 𝑏CHP ∈ {0,1} changes its value. That would
mean that the CHP switches its operation mode from “ON” to “OFF” or vice versa. When it is
the case, then the constraint is active and Eq. (3.34c) makes sure that, the CHP maintains its
operation for a certain minimum time period. After that, the constraint is inactive again until
𝑏CHP variable takes another value. There might be cases, where one constraint must be
violated in order not to violate the other constraint. In those cases, the temperature constraint
will be regarded as a soft constraint, which can be violated in some rare occasions.
3
Binary variables are representing the Boolean parameters (1 = True & 0 = False)
19
3.5. Model Identification
Figure 3.3 shows the input-output relation of the model.
Fig. 3.3 Input-outputs and disturbances of the system
The operating state, which is the function of binary variables, is the input variable. It is
assumed that the operating states of the system are discrete variables and they are
assumed to be constant for a certain time step. Likewise, electricity price and heat load
information can only be obtained in discrete-time by authorities. That’s why all the outputs of
the system are in discrete-time as well.
It is also necessary to mention that it is a time-dependent linear model. The gains of the
system 𝐽CHP , 𝐽St and 𝐽Boiler as derived previously are in the following form:
𝐽(𝑡) = 𝑎1 (𝑡) ∙ 𝑢1 (𝑡) + 𝑎2 (𝑡) ∙ 𝑢2 (𝑡) + ⋯ ,
(𝟑. 𝟑𝟑)
where 𝑎1 and 𝑎2 can are time-variant constants (prices); and 𝑢1 and 𝑢2 are the outputs
(power production and heat consumption or production) which form the gains. As can be
deduced, the gains of the system are linear combinations of these variables.
Thus, it can be concluded that the described model is a discrete-time linear model.
20
4. IMPLEMENTATION
For the implementation of the mathematical model, Modelica based tool Dymola
(Dynamic Modeling Laboratory) has been utilized. Modelica is a widely used programming
language for modeling of large and complex systems, for instance, process oriented
systems, air conditioning systems, and many other applications involving electrical, hydraulic,
thermal and control subsystems (Otter 2009, p. 7). In Modelica, systems can be
mathematically expressed by algebraic, differential or discrete equations. For graphical
editing and visualizing simulation results, commercial Modelica environment Dymola is used.
Dymola has two main windows: modeling and simulation. One can very quickly make
changes in the model, and see the results by plotting or animating them at simulation
window.
As explained before the model is a discrete-time model with discrete input variables.
Implementation of the model is achieved by defining each operation state and related
mathematical formulations. A screenshot from the main window of the implemented model is
given in Fig. 4.1.
Fig. 4.1 Implementation of the system at “Dymola”
As seen, three main units, the CHP, the boiler, and the storage tank are implemented.
The operating states of the system are stored in the table “Operating_States”. At each time
step, information about the operating state is conveyed to the main units as integer variables.
21
All parameters, variables and formulations are written as code within these units.
“Heat_Load” block enables all the units to communicate each other. The information of how
much power each single unit supplies is gathered at this block and then distributed to the
single units. By this arrangement, an energy balance is set up between the heat demand and
the total heat supply. Electricity and the heat demand are given to the model as text (txt.)
files. Electricity prices are stored in the table “Electricity_Price”, whereas the heat demand is
stored in “Heat_Load” block. “Overal_Gain” operator within the blue box on the top sums up
the overall system gains continuously.
Constraints are implemented within the model since the optimization tool, which is going to
be explained in Section 5, does not support constraint handling within the tool. Thus,
constraints are handled with a penalty function approach within the model. For each
constraint, a corresponding Boolean signal is defined. By this way, one can see in the
simulation window whether a constraint is violated based on the signal value becoming
“True” or “False”. In the next sections, how the temperature and minimum operation time
constraint has been implemented is explained.
4.1. Implementing minimum operation time constraint
Two timers are used to implement the constraint. Fig. 4.2 illustrates how the minimum
operation time is implemented.
Fig. 4.2 Implementation of the minimum operation time constraint
4.2. Implementing storage tank model
As seen in Fig. 4.2, the operating state is given as input to “Integer_to_Boolean” block. At
this block integer input is converted to a Boolean variable, where “True” stands for that the
CHP is running and “False” stands for that the CHP is shut down. Then these signals are
conveyed to the timers. The timer “ON” starts counting, when the CHP operates. On the
contrary, the timer “OFF” starts counting, when the CHP is shut down. Then the information
22
from the timers is conveyed to the next block to check whether the minimum operation time
is reached. Boolean signals “On_Timer” and “Off_Timer” show initially “False” as default.
When the threshold value, which is the minimum operation time, is reached, then the
corresponding Boolean output signal switches to “True”. By this, one can easily figure out
whether minimum operation time is reached. For instance, if the Boolean signal “Off_Timer”
shows “False” while the CHP is running, then that would mean the minimum operation time is
not reached, and the constraint is violated. A proper operation would be running the CHP
after “Off_Timer” switches to “True”.
Fig. 4.3 Representation of the storage tank with a single node
It is necessary to define high and low temperature levels to indicate whether there is heat
available in the storage tank. The output and the input of the storage tank model are powerbased signals. The temperature of the storage should be interpreted as follows:
It is assumed that the discharge temperature of the storage tank is constant. However,
the temperature which is defined at this heat capacitor is just an indicator, showing the
availability of the storage tank. For instance, when the temperature of the heat capacitor is
equal to a low limit, then it is assumed that the storage tank is empty. On the other hand,
when the temperature of the heat capacitor reaches to a high limit, then this would mean that
the storage tank is full and should not be charged any more.
Therefore, as seen in Fig. 4.4, a low temperature level 𝑇low is defined to cut off the
discharge power. Likewise, a high temperature level 𝑇high is defined to avoid charging the
storage tank.
23
Fig. 4.4 Operation range of the storage tank
As seen in Fig. 4.4, the storage tank is assumed to be empty, when its temperature is
less than low temperature level. Whenever the temperature drops below this level, the
storage tank does not supply heat. However that’s not regarded as a constraint but a usual
occasion that can happen in real plants, as the ambient temperature mainly determines the
temperature of the storage tank in winter. The interval between the high and the low
temperature levels is the normal operation region for the storage tank. When the temperature
exceeds the high temperature level, then the Boolean signal shows “True” as output for a
certain time. Fig. 4.5 illustrates how it is defined.
Fig. 4.5 Hysteresis band for the storage tank
When the temperature of the heat capacitor exceeds the level "Thigh ", the Boolean output
becomes “True”. As soon as the temperature drops to the level "Thigh − ∆𝑇" , then the
Boolean operator shows output as “False” again. The reason defining such a hysteresis band
is to make sure that the temperature of the storage tank drops to a certain level (𝑇high − ∆𝑇).
24
This ∆𝑇 value is lower than 0.1℃ and it is implemented due to the following reason. When
temperature of the storage tank reaches to the level 𝑇high, then the temperature constraint is
violated and the system should automatically drop the temperature of the storage tank based
on the control input given by the optimizer. However; it is not the case in simulation
environment. The solver at Dymola is not able to capture the changes happening in a very
short time span (milliseconds level). That’s why the storage tank is repetitively charged and
discharged (CHP operates intermittently), when the temperature of the storage tank reaches
to the level 𝑇high. That’s why this very small hysteresis band is implemented to make sure
that the CHP does not change its mode of operation so quickly that simulator do not
recognize. Therefore, by this arrangement, the CHP is decommissioned for a short time (until
the storage tank temperature drops by ∆𝑇) then the simulation programs already captures
that the CHP is shut down and the minimum operation time constraint makes sure that the
CHP does not charge right after the Boolean signal shows “False” again.
Computation of the cost function is done within the model. To do that, the constraints are
combined with the cost function in such a way:
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
(𝑡)
𝐽 = 𝐽CHP ∙ 𝑏CHP + 𝐽Storage ∙ 𝑏St + 𝐽Boiler ∙ 𝑏Boiler − 𝑃1 ∙ 𝑠CHP − 𝑃2 ∙ 𝑠St ,
(𝟒. 𝟏)
where 𝑡 is the subscript denoting the time instant; 𝑃1 and 𝑃2 are penalty functions for the
CHP and the storage tank respectively; 𝑠CHP is the Boolean signal for minimum operation
time constraint; 𝑏CHP , 𝑏St , and 𝑏Boiler are signals indicating whether the CHP unit operates
and 𝑠St is the the Boolean signal for the temperature constraint.
It can be seen in Eq. (4.1) that the first three terms are already defined instantaneous
gains from each individual unit. As can be understood, the constraints 𝑃1 and 𝑃2 are not
always active. It depends on the previous actions (states) of the system whether a constraint
is going to be active or not. Therefore when a Boolean signal becomes “True”, then the
corresponding constraint in the Eq. (4.1) is active. Without doubt, the penalties 𝑃1 and 𝑃2
decrease the total gains of the system. This is why violating a constraint would mean that the
system gain drops; therefore a violation of one of the constraints leads to an unfeasible
result. How to select proper values for 𝑃1 and 𝑃2 is discussed in Section 6.5.
25
26
5. OPTIMIZATION
5.1. Optimization Tool
There are many projects at Fraunhofer ISE, which are conducted by different
subdivisions. All ongoing works should be coordinated and organized in a way that everyone
can get involved in them and be able to contribute to them when needed. It is common that
most of the student works are part of a large-scale project. Thus, at Fraunhofer ISE each
member of the project groups is supposed to work more or less with the same simulation,
optimization and programming tools. As an optimization tool GenOpt, generic optimization
program, is selected, as it can be easily coupled with Dymola and it is generally adopted tool
at Fraunhofer ISE.
GenOpt is a tool that is used for minimization or maximization of cost functions calculated
by an external simulation program. Being a freely downloadable java program, GenOpt is
mainly an interface between any text-based simulation program and an optimization
algorithm (Coffey et al. 2010, p. 1086). Any simulation program can be coupled with GenOpt,
provided that the simulation program is able to read text (txt.) files and calculated cost
function must be also written in a text (txt.) file by the simulation program (Wetter 2009, p. 9).
GenOpt is not suitable for optimization problems, in which cost function is quadratic, or has a
gradient (Wetter 2015). The standard library of GenOpt offers many optimization algorithms,
and it’s also possible to extend library by the user with new optimization algorithms (Wetter
2009, p. 70). Fig 5.1 shows how GenOpt can be coupled with a simulation program.
Fig. 5.1 Coupling a simulation program with GenOpt (Wetter 2015)
As seen in Fig. 5.1, GenOpt consists of two parts: simulation and optimization. The
simulation part reads the input files, stores the simulation results, and writes necessary
27
output files, while the optimization part contains the algorithms, controls the optimization
process and stops it when the convergence criteria is fulfilled. There are several steps to do
an optimization with GenOpt (Wetter 2009, pp. 69-70):
Firstly, to carry out an optimization it should be defined, where the input and the output
files should be stored and overwritten. That’s the initialization part. Then user should select
the optimization algorithm and related parameters in a “command” file. User can specify in a
“configuration” file, how the simulation should end, what the error indicators should be etc.
Lastly user should define an input template, which in turn will be overwritten by GenOpt. The
input template will be the real input file after the optimization stops. GenOpt overwrites the
keywords assigned to the parameters in the template file by replacing them with numerical
values obtained from optimization results.
It is worth mentioning that the optimization constraints can only be implemented as
penalty or barrier functions at GenOpt (Wetter 2009, p. 15).
5.2. Optimization Algorithm
The debate over how to select an optimization algorithm for district heating systems has
drawn a large amount of attention. It has been asserted one should make a detailed analysis
of the optimization problem before selecting the algorithm. Type of input variables (discrete,
continuous or both) and constraints (equality or inequality) are the main decision factors.
Moreover, nature of the cost function, whether it is linear, nonlinear, continuous,
discontinuous, or has a local minima etc. should be analyzed (Nguyen et al. 2013, p. 1050).
As mentioned earlier, there are many ways of solving discrete optimization problems.
However at GenOpt, only Particle Swarm Optimization (PSO) algorithm can be used for
solving discrete optimization problems. It is one of the stochastic optimization methods,
which is based on simulation of a social behavior of fish schools, bird flocks or other
biological groups, which, in general, take action by information exchange among members in
a group (Nema et al. 2008, p. 1412). As other stochastic algorithms the PSO does not
require gradient of the cost function.
Scientist interested in investigating how a bird swarm can flock synchronously, how they
change direction, scatter and get together suddenly etc. Research in this area showed that,
the behavior of a fish school or a bird swarm is not a coincidently but a deliberately taken
action. As cited in (Kennedy and Eberhart 1995, p. 1943), a sociobiologist E.O Wilson has
stated, “In theory at least, individual members of the school can profit from the discoveries
and previous experience of all other members of the school during the search for food items,
whenever the resource is unpredictably distributed in patches.” It is further claimed that birds
are trying to keep the optimum distance between them, so that they could communicate and
28
orient themselves quickly to a location where they can find food or where they can escape
from a predator (Kennedy and Eberhart 1995, p.1943).
In 1995 Kennedy and Eberhart have been inspired by these findings and introduced the
PSO, which has roots in swarming intelligence, particularly bird flocking and fish schooling
(Pal et al. 2011, p. 663). It is a population based optimization algorithm. In the PSO,
population is called swarm, and the individuals are called particles. A swarm is regarded as a
disorganized collection of individuals, which tend to cluster together.
In the PSO, each particle has a velocity and a position vector. Position vectors can be
regarded as solution vectors of the optimization. The algorithm aims to find the best position
vector that is reached throughout all iterations. The positions vectors are function of both
velocity vectors and the previous position vector of the particles.
Consider e.g. n-dimensional optimization problem and let the set of solutions be 𝑥 ∈ ℝ𝑛
(particles’ position vectors), with 𝑥𝑖𝑖 denoting 𝑘th position vector component of the particle 𝑖.
Similarly 𝑣𝑖𝑖 represents the velocity vector component of the particle with the same notation.
At each 𝑡th iteration, the particles update their velocity and position vectors according to
following formulations found in (Nema et al. 2008) and (Pal et al. 2011):
(𝑡)
(𝑡−1)
𝑣ik = 𝑣ik
(𝑡−1)
+ 𝑐c ∙ 𝑟1
(𝑡−1)
∙ �𝑃ik
(𝑡)
(𝑡)
(𝑡−1)
− 𝑥ik
(𝑡−1)
𝑥ik = 𝑥ik
(𝑡−1)
� + 𝑐s ∙ 𝑟2
(𝑡−1)
+ 𝑣ik
(𝑙)
,
𝑃ik = 𝑎𝑎𝑎 𝑚𝑚𝑚 �𝑓 �𝑥j �� 4 ,
(𝑡)
(𝑙)
𝑥j :0≤𝑙≤𝑡−1
(𝑡)
𝐺k = 𝑎𝑎𝑎 𝑚𝑚𝑚 �𝑓 �𝑃ik �� ,
(𝑡)
𝑃ik :1≤𝑖≤𝑁
(𝑡−1)
∙ �𝐺ik
(𝑡−1)
− 𝑥ik
�,
(𝟓. 𝟏)
(𝟓. 𝟐)
(𝟓. 𝟑)
(𝟓. 𝟒)
where 𝑐c 𝑎𝑎𝑎 𝑐s are acceleration constants; 𝑟1 𝑎𝑎𝑎 𝑟2 are random numbers distributed in [0,1]
that attain the stochastic swarm behavior; 𝑓 is the cost function;𝑃ik is the 𝑘th component of
the best position vector that particle 𝑖 reaches by its own experience; 𝐺k is the 𝑘 th
component of the best position vector reached in the entire swarm of N particles. Fig 5.2
indicates how the algorithm works based on the equations above.
4
arg 𝑚𝑚𝑚𝑥 𝑓(𝑥) is the value of x for which f(x) gets its minimum.
29
Fig. 5.2 PSO algorithm (own figure inspired by (Nema et al. 2008, p. 1412))
As can be seen optimization is terminated when maximum number of iterations is
reached. Then the best position vector 𝐺 (𝑡) is specified as the output of the PSO algorithm.
30
6. TEST CASES
In this section some test cases are illustrated in order to show whether the
implementation of the model and constraints are successful. Some test cases are very
straightforward so that the results can be approximately anticipated and be compared with
the formally obtained optimization results.
6.1. Reducing input variables
It is discussed before that there are 8 different operating states for the DHS. Input
variables for optimization are also fixed to 8 in a way that each operating state is assigned to
an input variable. However, following test case has shown that it can be possible to reduce
decision or input variables from 8 to 2.
Test Case 1: As known, the boiler and the storage tank both able to supply heat. The
boiler burns the fuel in order to supply heat, whereas the storage tank supplies heat
depending on availability. In this test case, which of these units should be prioritized to
supply heat is questioned.
Table 6.1 shows the optimization parameters for Test Case 1. They are free parameters
that one can specify within optimization tool “GenOpt” and simulation tool “Dymola”. As
already explained, 𝐻p and 𝑡min are the prediction horizon and the minimum operation time for
the CHP respectively. Additionally, 𝑝 is the number of time steps in the prediction horizon,
and 𝑛𝑖 is the number of input variables.
Table 6.1 Optimization parameters for Test Case 1
𝑯𝐩
𝒑
Step size
𝒕𝐦𝐦𝐦
𝒏𝐢
8 ℎ𝑜𝑜𝑜𝑜
8
1 ℎ𝑜𝑜𝑜
1 ℎ𝑜𝑜𝑜
8
Fixing the prediction horizon 𝐻p to 24 ℎ𝑜𝑜𝑜𝑜 and step size 𝑝 to 24 means that optimizer is
only able to change the operating state at every 1 ℎ𝑜𝑜𝑜, which is the step size for this test
case. As the 𝑡min is already fixed to 1 ℎ𝑜𝑜𝑜, minimum operation time constraint will not be
violated in this test case. In Fig. 6.1, heat load and prices are given. As seen, CHP nominal
power is chosen as 215 𝑘𝑘 and heat load is higher than that except in the 1st hour.
31
400
350
300
Price [ct./kWh]
Heat Load and CHP Nominal Thermal Power [kW]
Q_CHP,th
u_load
c_el
c_heat
250
200
150
10
100
8
6
50
4
2
0
0
0
1
2
3
4
5
6
7
8
Time [h]
Fig. 6.1 Load and prices for Test Case 1
It is worth mentioning that the start temperature of the storage tank is high enough that it
could discharge and cover the heat demand throughout the prediction horizon. Therefore,
operating the boiler or the storage tank are both feasible actions after 1st hour.
Optimization results of Test case 1 are given in Fig. 6.2.
u_load
u_CHP
u_st
u_CHP,St
State
350
300
250
Operating State
Heat load disribution among the units [kW]
400
200
150
100
50
0
4
1
0
1
2
3
4
5
Time [h]
Fig. 6.2 Optimization results of Test Case 1
32
6
7
8
As seen in Fig. 6.2, operating state is “1” (only CHP operates) in the 1st hour. Since the
nominal thermal power 𝑄CHP,th is greater than the heat load 𝑢load , the rest of the heat is
transferred to the storage tank as can be seen by following the line for 𝑢CHP,St . At the end of
the 1st hour 𝑢load becomes greater than 𝑄CHP,th; this is why, the CHP starts operating at full
capacity (𝑢CHP = 𝑄CHP,th) after the 1st hour. At the same time operating state switches to “4”
(CHP and Storage Tank operates) and rest of the heat, which is not covered by the CHP, is
covered by the storage tank. Another possibility would be covering it by the boiler; however,
that is not chosen by the optimizer.
Test Case 1 and other similar test cases have shown that it would be unreasonable to
operate the boiler, while the storage tank could supply the amount of required heat at the
same time. When Eq. (3.32) and Eq. (3.33) are studied, it could be understood that the gain
of the storage tank is always greater than the gain of the boiler when the same amount of
heat (𝑢b = 𝑢st ) is supplied.
Now that it is known priority can be given to the storage tank, one can reduce the input
variables. Fig. 6.3 shows flow diagram of model with 8 input variables. As seen, optimizer is
free to choose all possible operating states by determining the operation (ON/OFF) of each
single unit.
Fig. 6.3 Model with 8 input variables
However, it is indeed enough to decide whether the CHP operates or not. Because then it
is possible to set a hierarchy in the model in a way that firstly the storage tank covers the
heat demand then the boiler. Operation principle of the model with 2 input variables is given
in Fig. 6.4.
33
Fig. 6.4 Model with 2 input variables
As seen in Fig. 6.4, optimizer does not need to do any selections for the operation modes
of the storage tank and the boiler. As long as heat demand is not covered by the CHP or its
capacity is not enough to cover the entire heat demand then the storage tank (1) supplies
heat first. When even more heat should be supplied that exceeds the capacity of the storage
tank (or when it is already empty) then the boiler (2) supplies heat as a backup unit.
By reducing input variables from 8 to 2, it is also possible to get more flexible operation in
terms of switch periods of operating states. When the optimizer controls all the 8 states, then
all the units can only change their mode of operation when the simulation time crosses the
step size of the optimization. However, model with 2 input variables allows that the storage
tank and boiler can freely change their mode of operation without having any dependencies
on optimization parameters. Moreover, model with 2 input variables is also beneficial in terms
of optimization complexity and execution time. Consider i.e. number of input variables 𝑛i = 8
and number of step size in the prediction horizon is 𝑝 . One can agree that there are
altogether 8𝑝 possible results. By decreasing the input variables to 2, the same optimization
result can be found between 2𝑝 possibilities. Arguably, this arrangement shortens the
execution time for the same problem. Considering the mathematical relation, (8𝑝 = 23𝑝 ), one
can further claim that the prediction horizon can be enlarged by factor 3, when the same
optimization is done with 2 input variables. That’s why for the next test cases, only the model
with 2 input variables is used.
6.2. Verifying Optimization Result by Enumeration Method
In order to understand whether the optimization result is feasible, the optimization result
of the “Test Case 1” has been compared to that of enumeration method. An enumeration
method makes sure that all possible inputs are simulated and the optimum input is
determined by comparing all possible results. The enumeration method is performed with 2
input variables; so the number of possible solutions for the Test Case 1 are altogether
28 = 256. All these possible solutions are listed in a table and simulated one after another.
34
The input, which gives the maximum plant gain, is selected as the result of enumeration
method. Both enumeration and optimization results are given in Fig. 6.5.
Optimization result with 8 input variables
Enumeration result with 2 input variables
Operating State
4
1
0
4
1
0
1
2
3
4
5
6
7
8
Time [h]
Fig. 6.5 Results of enumeration and optimization
As seen in Fig. 6.5, both results are identical. This verifies one more time that reducing
input variables to 2, leads to the same optimization result. Moreover, by means of this result,
it can be asserted that optimization for “Test Case 1” gives correct results. It is worth
mentioning that enumeration takes shorter time than that of optimization for this test case. In
10-15 seconds one can simulate all 256 possible input sequences in Dymola. However,
when the number of input variables is fixed to 8 , then the computation time would increase
exponentially and enumeration method would take around more than a week. Therefore,
verification with enumeration is only done with 2 input variables. Nevertheless, one can still
claim that the optimization result of “Test Case 1” is reliable, as it is identical to that of
enumeration. Furthermore it can be argued that decreasing input variables to 2 and
increasing prediction horizon to 24 ℎ𝑜𝑜𝑜𝑜 at “Test Case 1” would not change the optimization
complexity (88 = 224 ) and the optimization of that case can be also assumed as optimal.
6.3. Testing minimum operation time constraint
Test Case 2: This test case is prepared to test whether minimum operation time
constraint is violated or not.
Optimization parameters for Test Case 2 are given in Table 6.2.
35
𝐇𝐩
16
𝐩
Step size
½ ℎ𝑜𝑜𝑜
1 ℎ𝑜𝑜𝑜
𝐭 𝐦𝐦𝐦
𝐧𝐢
2
As seen in Table 6.2, step size and minimum operation time are fixed as different values
in order to allow input changes before minimum operation time is reached.
Q_CHP,th
u_load
c_el
c_heat
400
350
300
Price [ct./kWh]
The prices and the heat load for Test Case 2 are given in Fig. 6.6.
250
200
10
150
8
6
100
4
2
50
0
-2
0
0
1
2
3
4
5
6
7
8
Time [h]
As seen in Fig. 6.6, heat load is chosen greater than the CHP nominal power in order to
make sure that CHP cannot charge the storage tank, when it opeates. Temperature of the
storage tank 𝑇s is below than level 𝑇low , so it is empty in the beginning. Electricity price, as
shown in Fig. 6.6, gets a negative value (-2ct./kWh) for around 20 minutes between the 4th
and 5th hour. It is expected that at this time interval the CHP is better shut down. Because
selling the electricity for negative prices would decrease the gains. As mentioned, the time
step is half an hour. Therefore, it would be smart to shut down the CHP for half an hour when
the electricity prices become negative, and to operate it again in the following half an hour.
However, introducing the minimum operation time of 1 hour makes this solution unfeasible.
Optimization results can be seen in Fig. 6.7. As expected, the CHP is shut down when
the electricity price drops dramatically, as the gain of the boiler becomes greater than the
36
gain of the CHP. In the meantime the boiler covers the heat load, as the storage tank is
already empty. The CHP waits for a minimum operation time before operating again;
however, it could operate even before, as the time step was shorter than the operation time
and electricity prices were already greater than the heat price. If there hadn’t been any
penalty functions, this would have been the best action. Therefore, the results of Test Case 2
have proven that the minimum operation time constraint is not violated and the
implementation is successful.
u_load
u_b
u_CHP
CHP Operation
400
350
300
250
200
150
100
50
0
ON
OFF
0
1
2
3
4
5
6
7
8
Time [h]
6.4. Testing Temperature Constraint
Test Case 3: Temperature of the storage tank should not exceed a certain level. This test
case is prepared to see whether the implementation is successful.
Optimization parameters for Test Case 3 are given in Table 6.3. As seen a daily
optimization is done with step size of 1 hour. Minimum operation time is 1 hour as well; this is
why minimum operation time constraint is not going to be violated during the optimization.
37
Table 6.3 Optimization Parameters for Test Case 3
𝐇𝐩
𝐩
Step size
𝐭 𝐦𝐦𝐦
𝐧𝐢
24
1 ℎ𝑜𝑜𝑜
1 ℎ𝑜𝑜𝑜
2
Q_CHP,th
u_load
c_el
c_heat
100
Price [ct./kWh]
Heat load and the prices are given in Fig. 6.8.
50
0
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
As seen in Fig. 6.8, electricity price is specified greater than the heat price in order to
prioritize the CHP operation. Heat load is chosen too low in comparison to the CHP nominal
thermal power. By this arrangement, it is aimed that when the CHP operates, large amount
of heat is charged to the storage tank and its temperature increases rapidly. Optimization
results are given in Fig. 6.9.
38
u_st
u_CHP
T_high
T_s
10
7.5
5
2.5
0
80
70
60
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Start temperature 𝑇start of the storage tank is set to 60℃ while highest maximum limit
𝑇high is fixed to 80℃. How the heat load is shared between the CHP and the storage tank
illustrated with brown and yellow lines respectively. As seen in Fig. 6.9, the CHP operates 3
hours (1st,9th and 10th hours) until storage temperature reaches to around 79℃ at the end of
the 10th hour. One can question why the CHP is shut down at the end of the 1st hour. It can
be argued that the CHP do not charge between the 2nd and 3th hours, as the storage tank
could be overheated (charging power of the CHP would be higher between the 2nd and 3th
hours). As shown in Fig. 6.9, the storage tank temperature increases rapidly between 8th and
10th hours, as the CHP charges it. When the temperature is close to the high temperature
limit, then the CHP is shut down although the electricity price is greater than the heat price as
seen Fig. 6.8. The storage tank then covers the heat load until it can be charged again
safely. When the temperature drops to around 70℃, then it is charged one more hour and
then it is discharged again.
Optimization results of Test Case 3 have shown that handling the temperature constraint
with the penalty function approach is also successful.
6.5. Selection of Penalty Weights
The optimization results of Test Case 2 and Test Case 3 have proven that the penalty
functions given for the constraints are successful. However, following test case is prepared to
39
Temperature [°C]
12.5
show that sometimes one of these constraints has to be violated. That’s why, the issue of
selecting penalty weights for these constraints should be well considered.
Test Case 4: Two optimizations are done for two sequential time frames. Time step and
minimum operation time are fixed as different values.
Optimization parameters are given in Table 6.4. As seen, a day is optimized in two steps.
Optimization parameters are the same for the two sequential optimizations.
Parameters
𝐇𝐩
𝐩
Step size
𝐭 𝐦𝐦𝐦
𝐧𝐢
1st Optimization
2nd Optimization
½ ℎ𝑜𝑜𝑜
½ ℎ𝑜𝑜𝑜
2
2
24
1 ℎ𝑜𝑜𝑜
24
1 ℎ𝑜𝑜𝑜
Heat load and prices are as the same as given in Fig. 6.8. Basically three assumptions to
realize this test case are as follows:
•
Heat load is chosen lower than the CHP nominal power.
•
Electricity price is greater than the heat price throughout two prediction horizons.
•
Start temperature of storage tank is chosen as 75℃ in order to make sure that the
CHP does not charge in the beginning.
Optimization result of the first half-day is given in Fig. 6.10. As seen, the CHP does not
operate in the beginning. The storage tank covers the heat load until the last half an hour.
Then at the beginning of the last half an hour, the CHP starts operating and covers the heat
load.
40
u_st
u_CHP
CHP Operation
12.5
10
7.5
5
2.5
0
ON
OFF
0
1
2
3
4
5
6
7
8
9
10
11
12
Time [h]
st
Fig. 6.10 Optimization results of 1 half-day (1)
The temperature of the storage tank and charging power of the storage tank are also
Q_CHP,th
u_CHP,St
T_high
T_s
250
200
150
Temperature [°C]
given in Fig. 6.11.
100
50
0
80
75
70
65
0
1
2
3
4
5
6
7
8
9
10
11
12
Time [h]
st
Fig. 6.11 Optimization results of 1 half-day (2)
As seen in Fig. 6.11, the temperature of the storage tank slowly decreases as the storage
tank discharges. The CHP starts operating in the last half an hour, because it does not cause
41
any violation regarding the temperature constraint. As can be seen, in the last half an hour
the temperature increases from 70℃ to around 77℃. As can be anticipated, if the CHP had
operated before the last half an hour, the temperature of the storage would have exceeded
the high temperature level. Because minimum operation time is already 1 hour and 1 hour of
charging would mean, roughly speaking, 15℃ of temperature increase in this particular case.
That’s why the CHP does not operate before the last half an hour in order not to violate
temperature constraint.
Now let us consider the second half-day. At the end of the first prediction horizon, the
temperature of the storage tank reaches to around 77 ℃ and the CHP is at operation mode
“ON”. That means it should maintain its mode of operation at least for another half an hour.
One can easily judge that if CHP operates half an hour more, then the storage tank
temperature exceeds the high temperature limit. On the contrary, if the CHP stops operating
in order not to violate the temperature constraint, then the minimum operation time constraint
is violated. These two cases are illustrated in Fig. 6.12.
Fig. 6.12 Illustration of the constraint violation
As seen in Fig. 6.12, in the beginning of the second optimization dashed and normal lines
are representing two possible actions. Both actions are violating one of the constraints.
The problem is now how to select the penalty weights in a way that one of the constraints
is never allowed to be violated (hard constraint), while the other one might be violated (soft
constraint) in some rare cases like this one. To meet this end, the temperature constraint is
regarded as a soft constraint as it is possible to define the high temperature level arbitrarily.
42
When the temperature of the storage tank exceeds the high temperature level for a short
time, then that would not cause a big safety problem provided that the high temperature level
is already at safe level. One should consider the storage tank capacity and nominal charging
power of the CHP before determining this high temperature limit. Nonetheless, it’s also
possible to do it other way around. One could define a certain temperature limit, which
should not be exceeded at any case. Then the minimum operation time constraint would be
the soft constraint. Hard and soft constraints are easily interchangeable by modifying the cost
function in the model.
It should be carefully reasoned how the penalty weights are chosen to distinguish hard
and soft constraints. In Eq. (4.1) penalty weights for the minimum operation time and the
temperature constraint are denoted by 𝑃1 and 𝑃2 respectively. Arguably, one can claim that
the inequality 𝑃2 < 𝑃1 should be satisfied, if the minimum operation time constraint is
assumed to be the hard constraint. Because, when 𝑃2 < 𝑃1 holds true, then violating the
minimum operation time constraint would drop the total gains with a higher rate than that of
violating the temperature constraint. Then the optimal solution would be violating the
temperature constraint. Moreover, these weights should be also quantitively well determined.
When temperature of the storage tank is above the high temperature limit, then the storage
tank should be discharged, as soon as the minimum operation time constraint is inactive. In
order to achieve this, following relationship between the penalty weights must be realized:
𝐽CHP < 𝑃2 < 𝑃1 .
(𝟔. 𝟏)
That inequality makes sure that the gain of the CHP is always lower than the penalty
being exposed for the temperature constraint. Therefore, when the minimum operation time
constraint is inactive, then it is better to discharge the storage tank, as the penalty function 𝑃2
is already greater than the gain of the CHP. If penalty weights had been chosen lower than
the gain of the CHP, then the storage tank would have been charged forever, which of
course should not be the case.
After having set the weights for penalty functions as in Eq. (6.1), optimization results of
the whole day are obtained as in Fig. 6.13.
43
250
200
150
Temperature [°C]
Q_CHP,th
u_CHP,St
T_s
T_high
100
50
0
85
80
75
70
65
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Fig. 6.13 Optimization results of whole day (1)
As can be seen in Fig. 6.13, the CHP charges the storage tank for an hour (half an hour
in the first half-day + half an hour in the second half-day). Temperature constraint is violated
as it is regarded as a soft constraint and as expected, it falls down after the minimum
operation time constraint is inactive. The storage tank covers the heat load until the end of
the day as can be seen in Fig. 6.14.
u_st
u_CHP
CHP Operation
12.5
10
7.5
5
2.5
0
ON
OFF
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Time [h]
Fig. 6.14 Optimization results of whole day (2)
44
20
21
22
23
24
It is worth noting that, the cost function evaluated by the optimization tool does not give
the actual gains of the system, as the penalty functions might change the value of the cost
function. That’s why the gains of the system are calculated in the model separately in order
to get exact gains of the each single unit.
6.6. Effect of prediction horizon on optimization results
Test Case 5: This test case is prepared to see the effect of prediction horizon on
optimization results. To do that, two following cases are tested:
i.
Optimizing a test day in two steps, each having prediction horizon of 12 hours
ii.
Optimizing the same test day at once with prediction horizon of 24 hours
Parameters for the two cases are given in Table 6.5.
Parameters
𝑯𝒑
𝒑
Step size
𝒕𝐦𝐦𝐦
𝒏𝐢
1st Case
12 ℎ𝑜𝑜𝑜𝑜 + 12 ℎ𝑜𝑜𝑜𝑜
2nd Case
1 ℎ𝑜𝑜𝑜
2
1 ℎ𝑜𝑜𝑜
2
12 + 12
1 ℎ𝑜𝑜𝑜
24
1 ℎ𝑜𝑜𝑜
As seen the step size and the minimum operation time are 1 hour; thus minimum
operation time constraint is not active in this test case.
Following assumptions for the test day are also made:
•
Electricity price is given as a step function. It’s selected as 10 ct./kWh in the first 12
hours and as 20 ct./kWh in the next 12 hours while the heat price is set to 5 ct./kWh.
•
Start temperature of storage tank is chosen as 65℃ in order to make sure that it could
be discharged to cover the demand and its capacity is large enough to cover all the
heat demand throughout the test day.
•
Heat load is chosen lower than the CHP nominal power so that the excess heat
produced can be charged to the storage tank.
Based on these assumptions heat load, CHP nominal thermal power and prices are given
in Fig. 6.15.
45
250
200
150
Price [ct./kWh]
Q_CHP,th
u_load
c_el
c_heat
100
50
0
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
T_high
T_s
c_el
CHP Operation
80
Temperature [°C]
75
70
65
20
10
ON
OFF
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
st
Fig. 6.16 Optimization results of 1 case
As seen in Fig. 6.16, the CHP mostly operates in the first 12 hours and the storage tank
temperature reaches to high temperature level due to excess heat charged by the CHP. After
that time the CHP cannot operate due to the temperature constraint. Knowing that the
46
CHP operation and Electricity Price [ct./kWh]
Optimization results of the 1st case are given in Fig. 6.16
electricity prices are higher in the 2nd half of the day, it would be a better option to operate the
CHP during the next 12 hours. However, when prediction horizon is set to 12 hours, then
optimizer considers no knowledge beyond the 12th hour, although the prices and load
distribution are already available. That’s why, as seen in Fig. 6.16, the operation hours of the
CHP is greater in the first 12 hours. The storage tank is almost full in the second 12 hours,
which leads to a less operation period during that time.
T_high
T_s
c_el
CHP Operation
Temperature [°C]
80
75
70
65
20
10
ON
OFF
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Fig. 6.17 Optimization results of 2
nd
case
As can be seen in Fig. 6.16 the storage tank is discharged in the first 12 hours in order to
create a chargeable space in the storage tank so that the CHP could operate without
violating the temperature constraint in the next 12 hours. As can be deduced the CHP
operation time in the last 12 hours is higher in the 2nd case. The comparison of the CHP
operation time and plant gains are given in Table 6.6.
Table 6.6 Comparison of overall gains
CHP Operation [h]
Overall Gains [€]
st
10
73.67
nd
2 half-day
Whole day
1st half-day
2
12
0
74.37
148.04
20.89
2nd half-day
Whole day
12
12
324.91
345.8
1 half-day
1st Case
2nd Case
47
CHP operation and Electricity Price [ct./kWh]
Optimization results of the 2nd case are given in Fig. 6.17.
As can be seen in Table 6.5, the CHP operation time in 1st half-day and 2nd half day
are different, however the total operation time is identical. As the electricity prices are higher
in the second half-day, overall plant gains are also higher. This particular test case has
shown that prediction horizon should be well determined as it might change the optimization
results dramatically.
48
7. REFERENCE MODEL AND SCENARIO
In order to assess the economic optimization procedure, a comparison between a powerdriven and a heat-driven operation is going to be presented in Section 8. By power-driven
operation it is meant that the power production of the CHP is controlled with respect to the
varying electricity prices. Therefore a power-driven operation aims to make the highest profit
by electricity production, which is the the optimization approach taken for the economic
optimization. On the other hand, a heat-driven operation mainly focuses on covering the heat
demand among the heat production units. In this section a heat-driven DHS model will be
described as a reference model. Then a reference scenario will be created, which is going to
be used for comparison of both operations.
7.1. Heat-driven model
The heat-driven model is a similar to the model described in this thesis. The same units
with the same parameters and variables are included in the model. The constraints described
are also implemented within the heat-driven model. However, the implementation of
constraints is not the same since penalty function method cannot be applied to the heatdriven model. Not all the features and dynamics of the heat-driven model will be explained as
it is only intended to show, in what respect it differs from the power-driven model. The flow
diagram to specify the operating state of the heat-driven model is given in Fig. 7.1. The
notations and operating states are the same as defined before. The main difference of heatdriven model is that determination of its operating states depends only on the heat demand.
Temperature of storage tank is also monitored in order to judge whether there is heat
available in the storage tank or not. For this reason two levels, the same as previously,
𝑇low 𝑎𝑎𝑎 𝑇high , are defined. The storage tank model is implemented exactly the same as in
power-driven model. As long as the temperature of the storage tank 𝑇s is in the
interval�𝑇low , 𝑇high �, then there are 3 operating states, at which the DHS can operate as
shown in Fig. 7.2. When 𝑇s drops below the limit 𝑇low , then the storage tank is regarded as
empty and operating states are chosen among the ones, which storage tank does not supply
heat. Likewise, when 𝑇s exceeds the limit 𝑇high, then operating state 1 and operation state 2,
at which the CHP charges the storage tank, have been excluded. As can be noticed,
operating state 6 (only boiler) and operating state 7 (boiler and storage tank) are not included
within the heat-driven operation. Because considering today’s electricity market, operating
the CHP is generally more profitable than operating the boiler.
49
Fig. 7.1 Decision Tree for selection of the operating states at heat-driven operation
50
Fig. 7.2 Operation state distribution in the tank with respect to temperature levels
7.2. Reference Scenario
The reference scenario is created based on assumptions made in scenario B2023, which
is described by “The Federal Network Agency for Electricity, Gas, Telecommunications, Post
and Railway” 5 as cited in (Elci et al. 2014, para. 12). According to this scenario, power
capacities of photovoltaics (PV) and wind energy will be as shown in Table 7.1.
Table 7.1 Power Capacities (Elci et al. 2014)
Technology
Capacity [GW]
PV
61.3
Wind Onshore
49.3
Wind Offshore
14.1
The load data are based on the measurement data of year 2011, which can be obtained
from the transmission system operators. Then a quadratic relation between the residual load
and electricity prices of year 2011 is obtained by curve fitting as seen in Fig. 7.3.
5
German Translation : Bundesnetzagentur
51
10
5
0
Electricity Price [ct./kWh]
y=-1.485e-09x^2 + 2.489e-04x-3.256
20000
30000
40000
50000
60000
70000
Residaul Load [MW]
Fig. 7.3 Electricity prices and residual load of year 2011
As seen in Fig. 7.4, electricity prices are highly fluctuating and take even negative values
for several times based on the assumptions of scenario B2023, which estimates a higher
share of PV and wind energy plants than that of today’s.
c_el
c_heat
c_fuel
10
Price [ct./kWh]
5
0
-5
-10
Jan
Feb
March
Apr
May
June
July
Aug
Sep
Time of the year
Fig. 7.4 Approximated electricity prices for the future scenario
52
Oct
Nov
Dec
8. RESULTS
At this section the power-driven operation will be compared to the heat-driven operation
based on the reference scenario described before. The main comparison criterion will be the
overall plant gains, since the work itself concerns an economic optimization. In addition to
that, grid interactivity of both operations will be discussed. Lastly, effect of varying the size of
the storage tank on overall costs and grid interactivity will be investigated.
8.1. Comparison with respect to overall plant gains
For comparison a yearly optimization has been carried out. Optimization parameters are
given in Table 8.1.
Table 8.1 Optimization parameters for comparison scenario
𝐇𝐩
𝐩
Step size
𝐭 𝐦𝐦𝐦
𝐧𝐢
24
1 ℎ𝑜𝑜𝑜
1 ℎ𝑜𝑜𝑜
2
As seen in Table 8.1, prediction horizon is 24 hours. Therefore, 365 sequential
optimizations are done to achieve a yearly result. Each optimization process is initialized with
the end state of the previous optimization. A “Python” script, which carries out this
initialization, was available at Fraunhofer ISE. All parameters are selected as the same
values for both heat-driven and power-driven models. Initialization parameters are given in
Table 8.2.
Table 8.2 Parameters values for initialization
Size of Storage Tank (𝐕)
45 m3
CHP thermal efficiency (𝛈𝐂𝐂𝐂,𝐭𝐭 )
0.45
CHP Nominal Thermal Power (𝐐𝐂𝐂𝐂,𝐭𝐭 )
CHP Nominal Electrical Power (𝐏𝐞𝐞 )
CHP electrical Efficiency (𝛈𝐂𝐂𝐂,𝐞𝐞 )
215 kW
198 kW
0.41
Boiler Efficiency (𝛈𝐁𝐁𝐁𝐁𝐁𝐁 )
0.90
Low Temperature Level (𝐓𝐥𝐥𝐥 )
45 ℃
Start Temperature of Storage tank (𝐓𝐬𝐬𝐬𝐬𝐬 )
High Temperature Level (𝐓𝐡𝐡𝐡𝐡 )
53
45℃
80 ℃
For this scenario heat demand of district “Gutleutmatten” considered. It is intended to
show the differences between heat-driven and power-driven operation for the same scenario.
Nominal powers of the CHP and efficiency values are realistic and can be encountered in
real plants.
Results of both operations are given in Table 8.3 and Fig. 8.1.
Table 8.3 Result of the future scenario (1)
Gains
Heat-driven
Power-driven
Storage Gain [€]
9880
4170
- 57 %
CHP Gain [€]
29285
33193
+13 %
Boiler Gain [€]
33186
51843
+ 56 %
Total Gain [€]
72351
89206
+ 23 %
Power Driven
Change
Storage Gain
CHP Gain
Boiler Gain
Heat Driven
0
20000
40000 60000
Gains [€/year]
80000
100000
Fig. 8.1 Results of the future scenario (2)
As can be seen in Table 8.3 and Fig. 8.1, the total gain of the power-driven model is
higher (by 23 %) than that of heat-driven model. It can be also stated that when the CHP
operation is optimized, then the gain of the CHP rises by 13 % itself. Threshold electricity
price value, when the CHP and the boiler gain are the same can be found as
𝐽CHP = 𝐽Boiler ,
(𝟖. 𝟏𝟏)
𝑃el ∙ 𝑐el + 𝑄CHP,th ∙ 𝑐heat − 𝑄fuel,CHP ∙ 𝑐fuel = 𝑢b ∙ 𝑐heat − 𝑄fuel,CHP ∙ 𝑐fuel ,
(𝟖. 𝟏𝟏)
− 𝑄CHP,th ∙ 𝑐heat + 𝑄fuel,CHP ∙ 𝑐fuel + 𝑢b ∙ 𝑐heat − 𝑄fuel,CHP ∙ 𝑐fuel
.
𝑃el
(𝟖. 𝟏𝟏)
𝑐el =
𝑐el =
− 𝑄CHP,th ∙ 𝑐heat + 𝑄fuel,CHP ∙ 𝑐fuel + 𝑢b ∙ 𝑐heat − 𝑄fuel,CHP ∙ 𝑐fuel
,
𝑃el
54
(𝟖. 𝟏𝟏)
When the Eq. (8.1d) is solved for the same amount of heat supply by the CHP and the boiler
(𝑢b = 𝑄CHP,th ), then the threshold level of electricity price can be found as 3.25 𝑐𝑐./𝑘𝑘ℎ. That
means when the electricity price is lower than this level, operating the boiler is more
profitable. Electricity price is generally lower than this threshold price level (can be seen in
Fig. 7.4). That’s why; the power-driven operation makes the highest profit from the boiler.
Fuel consumptions at the boiler and storage tank, operation hours of the CHP and
specific gains of the CHP are given in Table 8.4.
Table 8.4 Comparison of specific gains based on gas consumptions and operation hours
Comparison Factors
Gas Consumption at Boiler[𝒎𝟑 ]
𝟑
Gas Consumption at CHP [𝒎 ]
Operation hours of CHP [h]
Specific Gain of the CHP [€/𝒎𝟑 ]
Specific Gain of the CHP [€/h]
Heat-driven
Power-driven
164788
257435
419930
234189
6893
4566
0.069
0.141
4.24
7.26
As can be seen in Table 8.4, gas consumption at boiler is proportional to the gain of the
boiler, as the heat price is constant. On the other hand, the gas consumption at CHP is
higher at heat-driven model while the gain of the CHP is lower at heat-driven model. In
addition, total operation hours are also higher at heat-driven model, as the CHP operates as
long as there is heat demand without considering any knowledge of prices. It can be also
seen by comparing the specific gains that the power-driven operation outperforms the heatdriven operation.
It can be deduced that in the future, heat-driven operation would not be a smart solution
for meeting residential heat demand considering the financial gains. Systems, which are
capable of supplying heat and electricity concurrently, should consider electricity prices and
develop further strategies to respond to fluctuations in the electricity market.
8.2. Comparison with respect to grid interactivity
First and foremost grid interactivity is a concept corresponding with how good the
electricity production or consumption is scheduled. It concerns whether the electricity is
produced at desired or undesired times with respect to the demand. Residual load 𝐺 is a
good indicator to determine desired and undesired time periods. It is calculated by taking
away the electricity productions of fluctuating renewable energy plants from the total
electricity demand (Shammugam 2014, p. 9):
55
𝐺 = 𝑃load − 𝑃wind − 𝑃PV ,
(𝟖. 𝟐)
where 𝑃load is the electrical power demand; 𝑃wind is the power produced by wind farms and
𝑃PV is the power produced by photovoltaic plants.
In order to quantify the term grid interactivity, Klein et al. has proposed a coefficient called
Absolute Load-Grid Matching Coefficient (𝐿𝐿𝐿𝐿abs ), which is derived as follows (Klein et al.
2014, p. 5):
𝐿𝐿𝐿𝐿abs =
∫ 𝑃el (𝑡) ∙ 𝐺(𝑡) ∙ 𝑑𝑑
,
𝑊el ∙ 𝐺̅
𝑊el = � 𝑃el (𝑡) ∙ 𝑑𝑑 ,
(𝟖. 𝟑𝟑)
(𝟖. 𝟑𝟑)
where 𝑃el is time-resolved electricity production; 𝐺 is time-resolved residual load; 𝑊el is the
total electricity production for an evaluation period and 𝐺̅ is the average residual load during
the evaluation period.
𝐿𝐿𝐿𝐿abs values can be interpreted as follows:
•
𝐿𝐿𝐿𝐿abs > 1 indicates that electricity is produced at favorable times. This kind of
operation known as “positively grid-interactive” or “grid favorable” behavior (Klein et
al. 2014, p. 5).
•
𝐿𝐿𝐿𝐿abs = 1 can be interpreted as “grid-neutral” behavior. By definition, LGMC value
approaches to 1, when electricity is continuously produced during a reference time
(Shammugam 2014, p. 13).
•
𝐿𝐿𝐿𝐿abs < 1 is regarded as “negatively grid-interactive” or “grid-adverse” behavior. It
means that electricity production by conventional power plants takes place when
renewable power plants also produce high amount of electrical power (Klein et al.
2014, p. 5).
In addition to the financial aspects, grid interactivity of both operations is also analyzed
for the same future scenario. Fig. 8.2 shows monthly 𝐿𝐿𝐿𝐿abs values of both operations.
56
Fig. 8.2 Comparison of 𝐋𝐋𝐋𝐋𝐚𝐚𝐚 values
1,6
1,5
LGMC
1,4
1,3
1,2
1,1
1
0,9
1
2
3
4
5
6
7
Months
LGMC_Power Driven
8
9
10
11
12
LGMC_Heat Driven
As seen in Fig. 8.2, heat-driven operation demonstrates a “grid-neutral” behavior, as can
be seen the 𝐿𝐿𝐿𝐿abs value ranges around “1.0”. However, power driven operation results in
better 𝐿𝐿𝐿𝐿abs values throughout the year. Apart from monthly based calculation, 𝐿𝐿𝐿𝐿abs is
also calculated for reference time of one year. It has been found out that power-driven
operation is 33 % more grid interactive than the heat-driven operation.
It can be commented that it is possible to increases the grid interactivity of the system by
optimizing the operation of the CHP based on electricity prices. As electricity prices mainly
depend on the residual load, operating the CHP when electricity prices are high means also
operating it when the residual load is high. This is why power-driven operation shows “gridfavorable” behavior.
8.3. Comparison with respect to size of the storage tank
At this section, effect of the storage size on grid interactivity and financial gains of the
power-driven operation is discussed. To do that, three months at summer season are
considered, as in the winter season storage tanks are generally not used.
The reference scenario includes the same heat demand, residual load and electricity
prices of the future scenario. The effect of varying the size of the storage tank can be seen in
Fig. 8.3.
57
25000
1,24
1,22
Gains [€]
20000
1,2
15000
1,18
10000
1,16
5000
0
LGMC [-]
1,14
5
15
45
60
100
Storage Volume (m^3)
Storage Gain
CHP Gain
150
Boiler Gain
200
1,12
LGMC
Fig. 8.3 Optimization results of the same summer scenario with different storage sizes
Except the storage volume, all other parameters including the optimization horizon (24
hours) stay consistent for comparison of all cases. However, these results would vary
depending on the length of the optimization horizon.
As can be seen in Fig. 8.3, changing the size of the storage tank increases both the
gains and the grid interactivity up to a certain level. In summer season, the heat demand is
generally lower than the CHP thermal nominal power. Increasing the size of the storage tank
enables the CHP to operate longer. Therefore, when the electricity prices are high enough,
then the CHP stores the excess heat at the storage tank without causing any mismatch
between the demand and supply. However, when the size of the storage tank reaches a
saturation volume, which is 150 𝑚3 in this case, then it should be well reasoned whether it
makes sense to have a storage tank with a larger size. It is worth commenting that this size
should be determined by taking into account the average expected heat demand and the
nominal thermal capacity of the CHP.
58
9. CONCLUSION
In this thesis, an economic optimization process has been carried out in order to see the
effect of fluctuating electricity prices on financial gains of the DHS. Two different strategies, a
power-driven operation and a heat-driven operation, which are utilized for meeting residential
area’s heat demand, are compared within a future scenario. It has been concluded that the
power-driven operation demonstrates better results in terms of financial gains and grid
interactivity.
9.1. Interpretation of results
As the main orientation in this thesis is an economic optimization, the results are also in
financial terms. However, it is not intended with this study to give exact system gains of a
DHS in the future. That would vary depending on uncertain power capacities of the
renewable energies, feed in tariffs and energy policies of the countries in the future, which
cannot be explicitly anticipated. However, based on the results obtained in this thesis
following interpretations can be done:
•
Heat-driven operation, which is an accepted way of producing energy at today’s
circumstances, would not be a good solution in the future. When electricity prices
highly fluctuate, as assumed in this thesis, then the fact that the power-driven
operation outperforms the heat-driven operation should not be forgotten.
•
It has been found out that grid interactivity of a power-driven operation is distinctly
higher than that of a heat-driven operation.
•
Storage size, arguably, has an effect on the financial gains and grid interactivity of
the system especially in summer season. However, apart from the operational
gains, other factors such as area requirements, investment costs, and profitability
analysis for the whole operation period etc. should be well considered.
9.2. Limitations of the thesis
Limitations of this thesis can be summarized as follows:
•
Optimization tool, GenOpt, is the only tool utilized to carry out the optimization
process. The fact that only the Particle Swarm Optimization (PSO) algorithm is
available at GenOpt to solve discrete optimization problems restricted the writer of
this thesis to test other tools and algorithms, which could perform better results on
the same optimization problem with respect to execution time and accuracy of the
results.
59
•
Parameters that can cause nonlinearities are simply chosen as constant values.
For instance, temperature dependencies of the properties of water are ignored for
simplicity.
•
It is attempted to validate the model. However uncertainties within the available
measurement data, such as unknown parameters and indefinite storage state
(whether it supplies heat or not), hindered the validation process. It is concluded
that model parameters should be calibrated to catch the real system dynamics.
9.3. Recommendation for further work
It has been argued that solving discrete optimization problems remains a challenging
task. It has been argued that the computational complexity of these problems may depend
exponentially on the number of input variables and the constraints. It has been stated that in
order to mitigate the complexity of such problems one can build “hierarchical or other special
structures” that may simplify the optimization problem (Ramadge and Wonham 1989, p. 83).
That is indeed what has been done in this thesis by reducing the input variables from 8 to 2.
However, in some cases it might not be feasible to build such a hierarchy within the input
variables. For instance, including a heat pump into the DHS would bring one extra input
variable and constraint to the optimization problem described in this thesis. Nevertheless,
considerable effort has been spent on reformulation of discrete optimization problems and
continuous relaxation approaches are playing an important role in solving discrete
optimization problems. Continuous optimization problems are generally easier to be handled,
as the objective function has a gradient and many algorithms that are suitable for continuous
problems can perform faster convergence on the same optimization problem. Therefore
researchers should be encouraged to reformulate the discrete inference problems by
continuous relaxation methods.
60
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