OPTIMIZATION OF DISTRICT HEATING SYSTEM DESIGN AND OPERATION Author: Andor Jasper

Budapest University of Technology and Economics Faculty of Mechanical Engineering

Department of Building Services and Process Engineering Pattantyús-Ábrahám Géza Doctoral School of Mechanical Engineering

Sub-programme of Building Services and Process Engineering

Thesis booklet of the PhD dissertation titled

OPTIMIZATION OF DISTRICT HEATING SYSTEM DESIGN AND OPERATION

Author:

Andor Jasper

qualified building services engineer

Subject head:

Dr. László Garbai professor emeritus

Budapest 2016

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1. T

Public utility networks play highly important roles in the life of settlements, with the most complex technology represented by district heating systems. District heating supply and services constitute an important branch of energetics both in Europe and in Hungary. District heating systems are highly asset-intensive; they are costly in terms of investment and operation; at the same time, they play a major role in energetics because they provide premises and opportunities for the cogeneration of thermal and electric energy, as well as an ideal option for the utilization of renewables and for compliance with EU energy policy directives. In Hungary, 650,000 apartments – mostly built by prefabricated building methods – are supplied with district heating.

The economic optimization of the establishment and operation of district heating systems can make district heating substantially cheaper. Optimization of establishment involves determination of the track and optimal pipe diameters, while in the course of the optimization of operations, the optimal volumetric flow and temperature temperature differential are determined in function of external temperatures.

The local control of the heating systems of pre-fabricated block buildings constructed in the course of recent decades was practically unfeasible. “This technical feature is a planned state, and is a result of the fact that the individual control of varying heating levels was not considered to be allowable by professional specialists. Therefore the application of single pipe heating systems seemed to be a sufficient and even desirable technical solution, where there is only a minor scope for intervention in heating control. The establishment of single pipe heating systems without control options corresponded to the other endeavour of the era to construct as many apartments as possible, and as inexpensively as possible. Energy seemed to be cheap, and apartment users only paid for about 1/3 of actual heating costs. Differences between actual costs and consumer charges were settled towards service provider companies in the form of state subsidies.” This is how the situation arose that by the time of the completion of the programme, out of the 650 thousand apartments, 380 thousand had a single pipe heating system, including 140 thousand of the so-called flow-through type, where there was no option whatsoever for intervention in heating control. s a result of reconstruction works, the number of flow-through systems has been reduced to nearly zero by now.

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In the various phases of district heating supply system construction works, technology mostly reflected the level of technical standard of the given age. In the 1990s, as a consequence of an enormous rate of development in measurement technology and informatics, such technical developments came to be demanded in district heat supply systems as well.

In view of European environment protection objectives and new climate policy perceptions, the importance of district heating services has been revaluated even compared to the previous state of affairs. District heating services have exited from the value system of direct economic assessment. District heating supply systems provide the broadest opportunity for the extensive implementation of renewables – including biomass and geothermal energy in particular – at the lowest investment development cost, as well as for the extension of application.

“The current consumer pool and situation of district heating is a “historical category”;

in respect of pre-fabricated buildings, it has no alternative, its redemption is technically unreal, it is against common sense not only at an economic level, but also in view of environmental requirements.” [1]

Both experience and theoretical investigations in the course of the past two decades show that in the renovation of district heat supply systems, the economy and further acceptance of district heating systems can be improved, to a great extent, by applying optimization methods, on the one hand, and by the enforcement of aspects of energy efficiency and cost efficiency in the operation of existing systems, on the other hand.

In view of and considering the above, my dissertation discusses the complex optimization of the establishment and the operation of the so-called hot water district heating systems. The district heating systems examined satisfy hot water demands for heating and sanitary purposes. In the following, the terms district heating and district heating supply and provision will be used in an identical meaning and sense.

2. P

,

In my dissertation, I first discuss the complex system-based optimization of existing district heating supply systems. This means that the system is considered as given:

known data include the technical and mechanical data of the system, connections,

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network geometry, dimensions and operational data, as well as characteristic historical data on operations in function of external meteorological conditions, the heat supplied and the heat demands satisfied. Consequently, the statistics of operational data are assumed to be known. Measured consumer heat demands are treated as substantial data. It is important to tackle the issue where and how accurately the heat provided and consumed were measured, together with the degree of uncertainty of such data.

The decision-making model is as follows:

 Methodology to determine heat demands (heating and sanitary hot water) by probability theory descriptions.

 The target function of operation, representing a minimization of the costs of the heat generated and supplied in order to satisfy consumer demands at a given reliability level. Target function elements include the cost of heat generation, the cost of hot water circulation, the cost of heat loss, and potentially the cost of heat center operation.

 In the minimization of the target function, our fundamental variables for decision making are as follows: primary forward hot water temperature, secondary return water temperature, volumetric flow of the circulated hot water, and – if it can be influenced – the volumetric flow of the secondary circulated warm water.

 The target function is suitable for taking into account each consumer heat center separately, in a decentralized manner, in line with network (graph) topology.

Briefly, the cost functions are as follows:

For the primary system

K1= 𝐾(𝑡𝑒, 𝑡𝑣, 𝑉̇) → 𝑚𝑖𝑛!.

Jointly for the primary and secondary systems

K_1,2= 𝐾(𝑡′_𝑒, 𝑡′_𝑣, 𝑡′′_𝑒, 𝑡′′_𝑣, 𝑉̇₁, 𝑉̇₂) → 𝑚𝑖𝑛!.

In my dissertation, I first present the input-output model and the input-output analysis of the district heating system. I define the so-called basic task and the inverse task.

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The hydraulic analysis of systems, the basic task and the inverse task are indispensable for the system-based optimization of operation.

After reviewing and solving the basic task and the inverse task, I provide a methodology and an algorithm for the system-based optimization of operation, in the framework of which I specify the optimal forward and return, primary and secondary water temperatures and the volumetric flows of hot and warm water to be circulated in line with the actual heat demand. This is a task of stochastic decision making, in which I first need to analyze the uncertainties of heat demand, which I perform on a probability theory basis. I determine the density and distribution functions of heat demands, their expected values and standard deviation, their errors and confidence bands, as associated with external meteorological conditions. Such probability theory discussion of heat demands is followed by the statement of the target function of the system, decision making variables and conditional equations. I examine the possibilities of stating a model of concentrated parameters and of distributed parameters, respectively, their interconnections, as well as the quality and errors of the results yielded.

In the second part of my dissertation, I discuss the partial optimization of system establishment. In the framework thereof, the task envisaged is to determine the fundamental nominal – dimensioning and design – parameters of a district heating system in the phase of planning and establishment by system-based cost minimum optimization controlled by an economic target function.

In complex optimization, the basic dimensioning and design parameters are as follows:

 dimensioning and meteorological status, temperature, external and internal heat gain, as well as their uncertainties in a probability theory approach,

 track,

 optimal conduit diameters,

 optimal pump size and nominal operating point,

 pressure upkeep.

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Partial optimization does not involve the task of track and pressure upkeep dimensioning.

Track optimization and pressure upkeep selection and technical solution thereof does not fall within the scope of mathematical modelling and cannot be algorithmized. I do not discuss these two problems in my dissertation.

In the event that the capacities of mechanical equipment in consumer heat centers – primarily heat exchangers – are given, then the design values of primary hot water temperature will only be slightly variable, resulting in little room for changing hot water temperature values. If this is not so, then the optimal design values of primary hot water temperature can be determined by counting the partial optimum set rather than by a strictly taken mathematical algorithm.

I would like to point out that in my dissertation, the primary emphasis is on methodology and optimization theory issues: I do not discuss, for instance, already known issues of heat loss calculations and pressure loss calculations, nor the details of stating costs.

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3. T

3.1 Heat power convection coefficient

The heat power convection coefficient – which is, in a certain sense, analogous with the heat convection coefficient applied to complex wall structures – makes it possible to calculate varying operational states by extremely simple calculations, involving the heat power possible to be introduced in the heated space, subject to various input parameters. This way, the space temperature can be calculated [2], [3].

In heat power transfer equations, differences between the highest and lowest temperatures measurable in the system represents the driving force – to the model of conduction laws.

Subject to a variety of input variables, the values sought for include the heat balance generated, the heat power possible to be transferred, the internal space temperature produced, as well as the primary and secondary water temperature exiting from heat exchangers.

A solution can be yielded by two methods. The problem is represented by the logarithmic temperature difference ∆𝑡_𝑘 of the heat exchanger, in which some of the variables are present in an implicit form. The equation system can only be solved in an iterative manner. As we know, the logarithmic temperature difference can be properly approximated in many cases by the arithmetic mean, making it simpler to solve the equation.

Linearization of the heat exchanger equation yields a simple explicit solution, the error of which is negligible in case of manual calculations. The arithmetic mean temperature difference is used instead of logarithmic mean temperature ∆𝑡𝑘 in the solution.

Thesis 1

In the changing operational conditions of district heat supply systems, subject to known external meteorological parameters, and in case of various primary and secondary circulations and forward water temperatures, the heat power convection coefficient derived from the method for determining the heat power possible to be transferred through the system will be:

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𝒌 = 𝟏

𝟏

𝟐𝒎̇ _𝟏𝒄𝟏+ 𝟏

(𝒌𝑨)_{𝑭𝑯𝑪𝑺}+ 𝟏

(𝒌𝑨)_𝒓𝒂𝒅+ 𝟏 (𝒌𝑨)_{𝒉𝒐𝒎𝒆}

.

With Bošnjaković coefficient:

𝒌 = 𝟏

(𝒌𝑨)𝟏𝒓𝒂𝒅+ 𝟏

(𝒌𝑨)𝒍𝒂𝒌á𝒔+ 𝟏

𝒎̇_𝟏∙ 𝒄 ∙ 𝝓 − 𝟏 𝟐 ∙ 𝒎̇_𝟐∙ 𝒄

.

Also taking into consideration the temperature dependence of the heat transmission coefficient of the radiator:

𝒌 = 𝟏

(𝒌𝑨)𝟏𝒍𝒂𝒌á𝒔+ 𝟏

𝒎̇_𝟏∙ 𝒄 ∙ 𝝓 + (𝟏 𝒌𝑨)

𝟏

𝟏+𝑴∙ 𝑸^{𝟏+𝑴−𝟏𝟏} − 𝟏 𝟐 ∙ 𝒎̇_𝟐∙ 𝒄

.

The primary forward hot water temperature required to maintain the prescribed room temperature at a discretional external temperature can be specified easily by the heat power convection coefficient, or any fourth parameter by fixing the other three.

Related publications: (18)

3.2 Probability features of sanitary hot water (SHW) demands

Professional literature published abroad so far has not discussed the probability description of sanitary hot water and heating demands [4],[5],[6],[7],[8],[9],[10]. The DIN 4708 standard, still applied in Germany today, provides a method for calculating sanitary hot water demands on a deterministic basis [11]. In Hungary, this topic was looked into as early as the 1970s, and the probability theory approach was introduced.

Measurement experiences have shown that the joint examination of several consumers reveals regularities that may provide assistance in estimating the real values of future consumption. Typically, there is a morning peak between about 6.00 and 8.00 a.m., when people get up and go to work or school; and there is an evening peak between about 6.00 and 10.00 p.m., when people have a bath or a shower at the end of the day.

Consumptions during these two peak periods are characteristically higher than

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demands measurable during the daytime; however, there are discrepancies as to which period represents the benchmark.

Due to the incidental nature of consumptions, a probability model must be created to describe this phenomenon, and the phenomenon should be examined within this model.

The figure to characterize peak period SHW consumption is the total quantity of water used during the given period. It can be used to determine the average per minute SHW quantity utilized in a given peak period if the duration of the peak period is known.

The peak period average consumption figures thus yielded can be compared to each other within a building and if expressed as specific to the number of apartments as average per unit of consumption, the entire data set can be managed together as a whole. In 2004-2005, FŐTÁV Zrt. measured the consumption of 5,080 apartments in 56 buildings, for an average duration of 19.7 days. I processed this data set yielded by such series of measurements. The empirical distribution of consumption figures can be determined by sorting this mass of data containing the average of 1,104 measurement days, yielded by expressing consumption as average per unit. I previously presumed that in a peak period, consumption intensity can be described as a stationary process, and the entire peak period can be characterized by a single probability variable. I assumed that the empirical distribution of peak period consumption average values will approximate a distribution possible to be described mathematically. Based on the density function, I presumed that distribution was to follow the Weibull distribution;

therefore I performed the 𝜒²test, which verified this conjecture.

Thesis 2

I elaborated a probability theory model based on the theory of stochastic processes in order to describe the probability distribution and sorted duration diagram of peak period consumption intensity figures. I designed this model based on measurements conducted by FŐTÁV in 2004-2005. I established that the process of peak period consumption intensities can be approximated by a stochastic process. The expected average value of peak period consumption intensity can be approximated by Weibull distribution as follows:

𝒇(𝒙) =𝒌 𝝀∙ (𝒙

𝝀)^𝒌−𝟏∙ 𝒆^{−(𝒙𝝀)𝒌},

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𝑭(𝒙) = 𝟏 − 𝒆^{−(𝒙𝝀)𝒌}.

I established that peak period consumption distribution is of normal distribution, and its duration diagram can be derived by normal distribution in function of the number of apartments as follows:

𝑭(𝒙) = 𝟏𝟐𝟎 𝟏

𝝈√𝟐𝝅 ∫ 𝒆^{−(𝒙−𝒎)}

𝟐 𝟐𝝈^𝟐 𝒅𝒙

𝒕

−∞

.

I established that linear regression can be established between peak consumption average values per unit – described by the Weibull distribution and referring to a single apartment – and average standard distributions, that is,

𝒚 = 𝟑. 𝟑𝟎𝟐𝟒𝒙 + 𝟎. 𝟒𝟎𝟓𝟖.

Upon the examination of daily sanitary hot water consumption figures, I established that consumption intensity in the peak period can be described as a stationary process:

the entire peak period can be characterized by a single probability variable. I presumed that the empirical distribution of peak period average consumption values will approximate a Weibull distribution. I performed an 𝜒² test which proved my conjecture. The data thus yielded can be used for producing peak period consumption intensity distribution figures, which approximate normal distribution. I applied an 𝜒² test for normality examination.

3.3 Description of the duration diagram of daily mean temperatures occurring in the heating season by a probability theory method

A highly important – and mostly only graphically displayed – tool of building services design and operation is the so-called duration diagram to indicate the frequency of occurrence and continuance of external temperatures. Duration diagrams have been applied in design and operation for several decades. Practical importance thereof lies

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in the fact that it can be used to identify that particular external temperature range which is very low and short in duration, for which the capacity of heat generation and transport facilities is designed, while the area below the curve is in proportion to the heat generated and consumed. While their practical importance is beyond doubt, duration diagrams have never been described by exact mathematical means, and no attempt whatsoever to produce a probability theory model has ever been made.

Simulation models have been developed for the determination of the unforeseeable effects [12],[13],[14],[15],[16],[17]. No probability theory description has yet been considered.

I completed the Ryan-Joiner test [18] to demonstrate that the probability distribution of daily mean temperatures really follows a normal distribution. As a result of the test, correlation coefficient 𝑟^∗= 0.997. As it is much higher than the criterion value 𝑟^∗= 0.96, our assumption has been proved.

Thesis 2

I defined a probability variable applied for heat demand estimates for heating in district heating provision, subject to various meteorological parameters, and describing the annual duration diagram of daily mean temperatures. I demonstrated that its course follows a normal distribution. I provided a method for determining distribution circumstances (parameters), the expected value, and standard deviation. I presented the method to identify the confidence band associated with the prescribed reliability level. I used the above to develop a new method for degree day calculations. The integral curve thereof can be used for calculations of the heat demand for heating or the energy resource demand presented during the course of a specific period. The formula to describe the duration diagram for Budapest in the heating season from 15 October to 15 April:

𝑷(𝒕𝒌≤ 𝝃) = 𝟏

𝝈√𝟐𝝅 ∫ 𝒆^{−(𝒙−𝒎)}

𝟐 𝟐𝝈^𝟐 𝒅𝒙

𝝃

−∞

= 𝟏

𝟓. 𝟒𝟒√𝟐𝝅 ∫ 𝒆^{−(𝒙−𝟒.𝟐𝟖)}

𝟐 𝟐∙𝟓.𝟒𝟒^𝟐 𝒅𝒙

𝝃

−∞

.

The area under the duration curve: 𝑻𝒕𝒂𝒓𝒕𝒂𝒎= 𝑻𝟏+ 𝑻𝟐

~ 12 ~ 𝑻𝟏= 𝟏𝟖𝟑 𝟏

𝝈√𝟐𝝅 ∫ ∫ 𝒆^{−(𝒙−𝒎)}

𝟐 𝟐𝝈^𝟐 𝒅𝒙𝒅𝒕

𝒕

−∞

𝒕_𝒌

−∞

,

𝑻_𝟐= 𝟏𝟖𝟑(𝟐𝟎 − 𝒕_𝒉) 𝟏

𝝈√𝟐𝝅 ∫ 𝒆^{−(𝒙−𝒎)}

𝟐 𝟐𝝈^𝟐 𝒅𝒙

𝒕_𝒌

−∞

.

Related publications: (4), (5), (8), (9), (10), (12).

3.4 Breakdown of a looped system into a radial network of two feed points and optimization of operation

Key issues of network operation include the hydraulic examination of pipe networks, as well as the adjustment of a hydraulic optimum, an optimum pump operating point.

Some pipe networks are of radial, others are of looped topography. Looped networks are more complicated to operate: the evolving flow pattern is usually unstable due to changes in consumer demands [19],[20]. Linear optimization has been based on linear optimization by Krope [21],[22],[23],[24],[25]. Panty of simulations have been developed to create the flow pattern [27],[28],[29],[30].

It is a widely held view that looped networks are more advantageous hydraulically than radial networks. In order to provide an overview of this issue, I present a hydraulic analysis method for district heating networks containing a single loop.

Target functions are drawn up for such networks and the flow pattern is specified by minimizing such target functions. One of the target functions formulates the minimization of flow work. As we know, in the mathematical sense this involves the solution of a variation calculation, and the solution is yielded by applying Kirchoff’s second law. The other target function aims at the minimum of the energy fed in. The flow patterns yielded by these two target functions differ from each other; and logically, feed work is lower in the solution for the minimization of feed work than in the solution requiring the fulfillment of Kirchoff’s second law. Through this procedure, Kirchoff’s second law will be impaired. Two pressure values appear at the feed point, which can be technically achieved by two pumps, so the left side and right side branches of the circular conduit are operated by two different pumps.

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The formula developed by me can be used for determining a proportional factor k, showing – at consumer 0 considered to be the hydraulic end point – in what proportion the demand 𝑉̇₀ is satisfied from the left and the right direction, respectively; thereby the pressure conditions and hydraulic end point of a looped network can be calculated more easily, even by a manual method.

Thesis 3

I provided a method for calculating the hydraulic end point of a district heating system including a single loop. I demonstrated that the method can also be applied to networks containing several, but still a small number of loops. In the course of the procedure I demonstrated that an economically higher optimum can be reached by breaking up the determinant loop and applying two smaller pumps of different operating points therein. Comparisons of the radial and looped hydraulic operational modes form part of optimization.

In the course of the procedure a factor k can be used to easily define from which directions and in what proportions the quantity of hot water is received by the hydraulic end point. A further advantage of this procedure is that on-going optimization can be carried out with it on an operating network.

Flow pattern at the minimization of dissipated energy:

𝐤^𝟐(𝑨 − 𝑨^∗)𝑽̇_𝟎^𝟐+ 𝟐𝒌(𝑩 + 𝑨^∗𝑽̇𝟎𝟐+ 𝑩^∗) + (𝑫 − 𝑫^∗− 𝑨^∗𝑽̇𝟎𝟐− 𝟐𝑩^∗).

Flow pattern in the minimization of input power:

𝒌^𝟐∙ (𝑨 − 𝑨^∗)𝟑 ∙ 𝑽̇𝟎𝟑+ +𝒌 ∙ [𝟐𝑨 ∙ 𝑽̇_𝟎^𝟐∙ ∑ 𝑽̇_{𝒊,𝒊−𝟏}

𝒏

𝒊=𝟏

+ 𝟒𝑩 ∙ 𝑽̇_𝟎+ 𝑨^∗(𝟐 ∙ 𝑽̇_𝟎^𝟐∙ ∑ 𝑽̇^∗_{𝒊,𝒊−𝟏}

𝒏∗

𝒊=𝟏

+ 𝟔 ∙ 𝑽̇_𝟎^𝟑) + 𝟒𝑩^∗∙ 𝑽̇_𝟎] +

+ [𝟐𝑩 ∑ 𝑽̇𝒊,𝒊−𝟏 𝒏

𝒊=𝟏

+ 𝑪 ∙ 𝑽̇𝟎− 𝑨^∗(𝟑 ∙ 𝑽̇𝟎𝟑+ 𝟐 ∙ 𝑽̇𝟎𝟐∑ 𝑽̇^∗𝒊,𝒊−𝟏 𝒏∗

𝒊=𝟏

) − 𝑩^∗∙ (𝟒 ∙ 𝑽̇𝟎+ 𝟐 ∑ 𝑽̇^∗𝒊,𝒊−𝟏 𝒏∗

𝒊=𝟏

)

− 𝑪^∗∙ 𝑽̇𝟎] = 𝟎 .

Related publication: (11)

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3.5 Decomposition procedure to optimize district heating system operation Optimization of the operation of district heating systems for given heat demands has been in the focus of scientific and professional interest for two decades. So far, this issue has been examined on simplified models. The system was substituted by a consumer, and the district heating system was modelled by a forward and a return conduit [31],[32],[33]. A simplified target function was applied, consisting of the heat loss determined globally for the system and the cost of total primary hot water circulation. The quantity of the water circulated can be decreased by increasing the temperature of the forward hot water. Heat loss increases in this case, together with a reduction in both the necessity and the cost of circulation. Otherwise, costs will change in the opposite direction.

In my dissertation, I apply a decentralized system model. This enables more accurate calculations, making it possible to examine different types of heat centers separately.

In the taxonomic sense, a district heating system is a diverging branch system.

Given are the consumer demands of the prescribed reliability level, which must be satisfied by the primary system. The question is what should be the forward temperature of the hot water and what volumetric flow should be circulated to transport the heat to each heat center. In case of higher forward temperatures, heat loss increases but the cost of circulation will be reduced, and vice versa.

Given are the geometric features of the system, which can be used to determine hydraulic and thermal resistance factors.

Forward primary hot water temperature is selected to be the state variable of system components. In the first stage of decision making, which is the heat center located in the presumed hydraulic end point of the system, the optimal values of 𝑚̇1′ 𝑎𝑛𝑑 𝑚̇1 are specified in function of forward temperature as a parameter. These can be used to calculate the return water temperatures 𝑡𝑣 𝑎𝑛𝑑 𝑡𝑣1′ , and the resultant return water temperature at the node from their mingling. Optimization was performed by a recursive procedure of dynamic programming, by moving backwards.

In the second stage of decision making, there is no decision factor, therefore there is no decision making situation.

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In the third stage of decision making, only the value 𝑚̇2′ of the volumetric flow of the hot water removed at the branch should be optimized in function of the forward primary hot water temperature 𝑡𝑒.

The procedure is continued recursively, by progressing towards the pump. Upon reaching the pump, the optimal value of the forward primary hot water temperature can be selected by in the last decision making state by determining the optimum 𝑡𝑒 in the last optimal function.

This model is characterized by the fact that the secondary system is provided with the primary program to which the program of the secondary system needs to adapt.

Therefore it needs to be examined whether the heat exchangers in the heat center are suitable for transferring the heat demand requested to the secondary system, given the forward temperature and water flow circulation of the primary system. More specifically, the circulation parameters of the secondary system should be adjusted to this end.

As we have seen, the state variable parameter in the model is the forward primary hot water temperature. By this being fixed, the rate of circulation will be determinate. The heat power convection coefficients stated in Thesis 1 are used for interim calculations.

Thesis 4

I provided a systems theory based decomposition process for the optimization of district heating system operation. The decomposition process is based on the application of dynamic programming. The target function aims to minimize the operational costs of the district heating system. Elements of the target function include the cost of primary circulation and the cost of heat loss in the primary system. Cost elements of secondary systems can also be involved in optimization.

Optimization is performed by taking as a basis known heat demands of a given reliability level. Optimization can be performed for various arbitrary cases of heat demand (meteorological states) during the heating season. The white box model of the decision making model is illustrated by the figure below.

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Figure 3.1. Decentralized model of the district heating system

3.6 Determination of optimal pipe diameters in closed-track radial district heating networks

Determination of optimal pipe diameters in district heating networks means that the task is to select a conduit diameter for each section of the district heating network resulting in a minimum amount of annual network development cost, direct annual cost of operation, and joint cost of hot water circulation and heat loss.

An efficient solution for this problem, effective from the engineering point of view and of a high practical value, is still missing to date.

The problem appeared in the 1960s and 1970s. Endeavors of the time were characterized by a search for so-called analytic solutions. Diameters and costs were taken into consideration with continuous value series. The diameters yielded were rounded to standard figures by a variety of standard procedures. The “distance”

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between the standard diameters thus yielded and the real optimum has been left unrevealed in the literature to date [34],[35],[36].

Later models were characterized by optimum search using discrete diameter series and discrete cost functions over a set of standard diameters. Extreme value search on discrete sets was conducted by authors using diverse counting techniques, such as the Branch and Bound method and dynamic programming.

Theoretically, dynamic programming proved to be a stable and convergent method. in the course of the application of this method, however, the rapidity of convergence and the attainment of the absolute optimum were not properly analyzed. In the course of applying this method, Bellmann’s principle of optimality is used. The state variable to connect the stages of decision making at nodes – the meeting points of the backbone and branches – is the pressure difference between the forward and the return conduit.

By making as dense divisions as possible, the solution can be made even more accurate. Search for a solution can also be perceived as a task of trying to find the optimal route.

In my thesis, I present a procedure to guarantee convergence and finding the absolute optimum based on dynamic programming. I discuss the task from a systems theory and decision making theory objective, and I analyze both the efficiency of the method and error limits.

Thesis 5

I provided a discrete optimization procedure for radial district heating systems to determine the optimal pipe diameter by further developing dynamic programming, and by analyzing the error and uncertainty band of the resultant optimum yielded by joining the so-called independent optimums in order to determine the absolute optimum by a finite number of steps. The newly generated target function is stated as follows for the hydraulically determinant backbone and branches:

𝑲 = [( ∑ 𝑹_𝒊 𝟏 𝒅_𝒊^𝟓𝑽̇_𝒊^𝟐𝑽̇_𝟎

𝒊∈𝛀_𝒗

) 𝒆𝝉 + ∑ 𝑲_𝒊(𝒅𝒊)𝒍𝒊𝒂_𝒊

𝒊∈𝛀_𝒗

] + ∑ 𝑲_𝒊(𝒅𝒊)𝒍𝒊𝒂_𝒊

𝒋∈𝛀_𝒇𝟏

+ ⋯ + ∑ 𝑲_𝒊(𝒅𝒊)𝒍𝒊𝒂_𝒊

𝒋∈𝛀_𝒇𝒎

.

~ 18 ~

The target function is minimized in two steps. In the first step, the aggregate of those cost members of the cost function is minimized which are located along the hydraulically decisive route. Thereby the optimal pump work and the feed pressure difference are yielded. By setting up the pressure pattern, the differential pressure available is yielded for branches. Branches are optimized at a fixed feed pressure by a smaller-scale dynamic programing session.

Related publications: (1), (2), (13).

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4. U

In my dissertation, I discussed the complex optimization of the establishment and operation of district heating systems. I examined the heat convection coefficient of the system for changing operational conditions of district heating systems, in case of various primary and secondary circulations and forward temperatures. In the course of the duration diagram analysis of daily mean temperatures I came to the conclusion that the diagram follows a normal distribution, which I even demonstrated by a normality test. I provided a method for determining the expected value and standard deviation of distribution, and produced a calculation schedule for confidence bands of the prescribed reliability level. I defined heat demand for heating as a probability variable for district heating supply systems. Parameters of the formula to determine heat demand for heating should be considered as probability variables, using which the distribution and parameters of heat demand for heating as a probability variable are generated. At a given reliability level, the error, uncertainty and confidence band of the heat demand for heating can be determined. I provided a discrete optimization procedure by further developing dynamic programming, and by analyzing the error and uncertainty band of the resultant optimum yielded by joining the so-called independent optimums in order to determine the absolute optimum by a finite number of steps. I provided a decomposition procedure on a systems theory basis to optimize the operation of district heating systems. Such decomposition procedure is based on the application of dynamic programming. The target function aims at minimizing the operating costs of the district heating system. Target function elements include the cost of primary circulation and the cost of the heat loss of the primary system. Cost elements of secondary systems may also be included in optimization. Optimization is performed on the basis of known heat demands of a given reliability level.

Optimization can be performed for various arbitrary cases of heat demand (meteorological cases) during the heating season. Comparisons of the radial and looped hydraulic operational modes form part of optimization. I provided a calculation formula for the target functions of the two modes of operation and their optimization, as well as for the comparison of the results yielded.

In the course of my research work, I examined the phases of heat supply systems design and operation failed to be analyzed in sufficient detail, and improved them in a manner also usable for practising engineers.

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5. O

(1) Garbai, László; Jasper, Andor: A matematikai rendszerelmélet feldolgozása és alkalmazása épületgépészeti optimalizációs feladatok megoldására [Adaptation and application of mathematical systems theory to solve building services optimization tasks – In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 59:(3) pp.

3-6. (2011)

(2) Garbai, László; Jasper, Andor: A matematikai rendszerelmélet feldolgozása és alkalmazása épületgépészeti optimalizációs feladatok megoldására 2. rész [Adaptation and application of mathematical systems theory to solve building services optimization tasks, Part 2 – In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 59:(9) pp. 13-17. (2011).

(3) Garbai, László; Jasper, Andor: Épületgépészet, oktatás, tudomány [Building services, education, science – In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 59:(1-2) pp. 4-7. (2011)

(4) Jasper, Andor: A hőigények rendezett gyakorisági diagramja, tartamdiagramja, [Reqular frequency diagram and duration diagram of heat demands – In Hungarian] In: Garbai, László: Távhőellátás, hőszállítás [Dictrict heating supply, heat transport]. 935 p. Budapest: Typotex Kiadó, 2012. pp. 56-67. (ISBN:978- 963-279-739-7)

(5) Garbai, László; Jasper, Andor: A külső hőmérsékletek tartamdiagramjának matematikai leírása [Mathematical description of the duration diagram of external temperatures In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 61:(10) pp. 3-7.

(2012)

(6) Garbai, László; Jasper, Andor: A fűtési energiaszükséglet meghatározására szolgáló diagramok matematikai leírása [Mathematical description of diagrams to determine heat demand for heating – In Hungarian], MAGYAR ENERGETIKA 19:(6) pp. 35-37. (2012)

(7) Jasper, Andor: Gyárlátogatás Viechtachban [Factory visit to Vietach – In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 62:(7-8) p. 33. (2013)

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(8) Garbai, László; Jasper, Andor: Mathematical Description of the Duration Curve of External Temperatures. In: Anikó Szakál (ed.), EXPRES 2013: 5th International Symposium on Exploitation of Renewable Energy Sources.

Subotica, Serbia, 21.03.2013 to 23.03.2013. Budapest: Óbudai Egyetem, 2013.

pp. 22-25. (ISBN:978-86-85409-82-0)

(9) Garbai, László; Jasper, Andor: A fűtési időszak napi középhőmérsékletének statisztikája az 1901-2011. évekre [Statistics of daily mean temperatures in the heating season for the years 1901-2011 – In Hungarian], MAGYAR ENERGETIKA 20:(5) pp. 35-37. (2013)

(10) Garbai László, Jasper Andor: Probability Theory Test of External Temperature’s Duration Curve, In: anon (ed.) International Conference of BUILDING SERVICES AND AMBIENTAL COMFORT. Conference venue and date:

Timisoara, Romania, 11.04.2013 to 12.04.2013. Timisoara: Editura Politehnica, 2013. pp. 370-379.

(11) Garbai, László; Jasper, Andor: Operation of Looped District Heating Networks.

In: Nyers, József (ed.) Internationale symposium "EXPRES 2014" Subotica: 6th International Symposium of Renewable Energy Sources and Effectiveness.

Conference venue and date: Subotica, Serbia, 27.03.2014 to 29.03.2014.

Subotica: Visoka Technicka skola strukovnih studija u Subotici, 2014. pp. 120- 123. (ISBN:978-86-85409-96-7)

(12) Garbai, László; Jasper, Andor; Magyar, Zoltán: Probability theory description of domestic hot water and heating demands, ENERGY AND BUILDINGS 75: pp.

483-492. (2014)

(13) Garbai L, Jasper A, Kontra J: Determination of optimal pipe diameters for radial fixed-track district heating networks. PERIODICA POLYTECHNICA-CIVIL ENGINEERING 58:(4) pp. 319-333. (2014)

(14) Garbai, László; Jasper, Andor: Fűtési hőigények kockázati elvű meghatározása valószínűségelméleti megközelítésben 2. rész [Risk-based determination of heat demands for heating by a probability theory approach, Part 2 – In Hungarian].

MAGYAR ÉPÜLETGÉPÉSZET 64:(12) pp. 10-12. (2015)

(15) Garbai, László; Jasper, Andor: Fűtési hőigények kockázati elvű meghatározása valószínűségelméleti megközelítésben 1. rész [Risk-based determination of heat

~ 22 ~

demands for heating by a probability theory approach, Part 1 – In Hungarian].

MAGYAR ÉPÜLETGÉPÉSZET 64:(11) pp. 11-15. (2015)

(16) Garbai, László; Szánthó, Zoltán; Jasper, Andor: A dabasi szennyvíztisztító rendszer korszerűsítése [Modernization of the Dabas waste water treatment plant – In Hungarian], MAGYAR ÉPÜLETGÉPÉSZET 64:(7-8) pp. 7-10. (2015) (17) Magyar, Zoltán; Garbai, László; Jasper, Andor: Risk-based determination of heat

demand for central and district heating by a probability theory approach, ENERGY AND BUILDINGS 110: pp. 387-395. (2016)

(18) Garbai, László; Jasper, Andor: Analysis of steady states in district heating systems. In: Nyers, József (ed.) Internationale symposium "EXPRES 2016"

Subotica: 8th International Symposium of Renewable Energy Sources and Effectiveness. Conference venue and date: Subotica, Serbia, 31.03.2016. - 02.04.2016. Subotica: Visoka Technicka skola strukovnih studija u Subotici, 2016. pp. 81-83. (ISBN:978-86-85409-96-7)

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6. B

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