Modelling and control of a district heating system

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UPTEC ES08 007

Examensarbete 20 p Mars 2008

Modelling and control of a district heating system

Linn Saarinen

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Modelling and control of a district heating system

Linn Saarinen

The aim of this study was to investigate whether the supply temperature to a district heating system could be decreased if a dynamic model of the system is used to determine the supply temperature control point, and to what extent the decrease of the supply temperature would improve the electricity efficiency of the connected combined heat and power plant. This was done by a case study of the district heating system of Nyköping and its 58 MW heat and 35 MW electricity CHP plant. The district heating network was approximated as a point load with a variable time delay.

Prediction models of the heat load, return temperature and transport time of the system were estimated from operational data of the heat plant. The heat load was modelled with an ARX model using the 24 hour difference of the outdoor

temperature as input signal and the 24 hour difference of the load as output signal. A comparison between using a regular control curve and using the dynamic model for controlling the supply temperature indicated that with the same risk of heat deficit, the dynamic control strategy could increase the electricity production with 390 MWh per year, most of it during the winter months. This would correspond to an increased annual income of about 200 000 SEK for the owner.

Sponsor: Vattenfall Research and Development ISSN: 1650-8300, UPTEC ES08 007

Examinator: Ulla Tengblad Ämnesgranskare: Bengt Carlsson Handledare: Andreas Lennartsson

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Sammanfattning

Fjärrvärmenät finns i de flesta större samhällen i Sverige. I ett eller flera centrala värmeverk förbränns exempelvis skogsflis, avfall eller torv och värmen distribueras via ett isolerat vattenledningsnät till kunder runt om i samhället, som växlar över värme från fjärrvärmenätet till sina egna radiator- och varmvattensystem. I vissa fall har värmeverket också en ångturbin för elproduktion, och kallas då kraftvärmeverk. Värmen från förbränningen i pannan värmer vatten till ånga, som får passera ångturbinen och därefter kyls av fjärrvärmenätets vatten.

Elproduktionen från en ångturbin beror av skillnaden mellan temperaturen på ångan före och efter turbinen. För att få en hög elproduktion vill man alltså ha en låg temperatur efter

turbinen. Å andra sidan ställer värmebehovet på fjärrvärmenätet krav på att temperaturen inte får vara för låg, för ju lägre temperaturen på fjärrvärmevattnet blir, desto mindre värme per kubikmeter vatten transporteras från värmeverket till kunden. Vanligen styrs

framledningstemperaturen (temperaturen på vattnet som skickas ut på fjärrvärmenätet från värmeverket) av utomhustemperaturen. När det är kallt ute behöver ju kunderna mer värme.

Sedan systemet med gröna elcertifikat infördes i Sverige har elproduktion på biobränsleeldade kraftvärmeverk blivit mycket lönsam. Detta har gett incitament till industrin att konvertera värmeverk till kraftvärmeverk och även att höja elproduktionen på befintliga anläggningar.

Problemet om man vill höja elproduktionen på ett kraftvärmeverk är att för en viss mängd producerad el produceras också en större mängd värme. Därför är elproduktionen beroende av efterfrågan på värme. I Nyköping har man till exempel börjat kyla bort värme från

returledningen för att kunna öka elproduktionen på kraftvärmeverket vid tidpunkter då

efterfrågan på värme är låg men elpriserna höga. Att sänka framledningstemperaturen innebär att man ökar andelen producerad el jämfört med producerad värme, och skulle kunna vara en billigare och mer energieffektiv metod att öka elproduktionen. I denna studie har en modell för värmebehovet och dynamiken på fjärrvärmenätet tagits fram för att kunna styra

framledningstemperaturen mer precist. Exempelvis tar modellen hänsyn till att värmebehovet följer ett speciellt dygnsmönster som beror av kundernas beteende – som att mest varmvatten används under morgontimmarna. Med hjälp av driftdata från kraftvärmeverket i Nyköping har en modell skattats som både använder mätningar av utomhustemperaturen och beräkningar av värmeförbrukningen några timmar tillbaka för att förutsäga vilket värmebehovet kommer att bli de närmaste timmarna, och därigenom vilken framledningstemperatur som krävs. Detta innebär att framledningstemperaturen kan sänkas när värmebehovet inte är så stort, och då pressas verkningsgraden på elproduktionen upp. Resultaten av den här studien indikerar att elproduktionen på kraftvärmeverket i Nyköping skulle kunna höjas med omkring 390 MWh per år utan att värmeproduktionen ökas, vilket skulle innebära cirka 200 000 kr i ökade intäkter för ägaren Vattenfall.

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Acknowledgements

I would like to express my gratitude to the people who have given me their time and support during this degree project. First of all, thanks to my supervisor Andreas Lennartsson at Vattenfall and my examiner Bengt Carlsson at Uppsala University, who have given me feedback and good discussions. A special thanks also to Katarina Boman and Jozef Nieznaj at Vattenfall, for their help and answers to my numerous questions. And finally, thanks to all the other people at Vattenfall who have taken an interest in my project and assisted me in

different ways, among them Christer Andersson, Majjid Mohammadi, Anna Helgesson, Rolf Abrahamsson and Peter Herbert.

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1 Introduction

The objective of this master’s degree project is to investigate the possibility to increase the electricity efficiency of the combined heat and power plant Idbäcken in Nyköping by decreasing the supply temperature to the district heating system. The decrease of the supply temperature should be made possible by applying a dynamic control strategy for the supply temperature, using a dynamic model of the district heating system. The reliability of the heat distribution to the customers must not be compromised.

1.1 Motivations

District heating is a well spread technology in Sweden. Most cities have heat plants and infrastructure for district heating, and the customers are of many kinds; households, offices, shops, industry etc. The heat plants use a wide range of fuels. Renewables such as wood chips, pellets and waste materials are common, together with peat, municipal waste and electricity used in heat pumps. Waste heat from industrial processes is also used at locations where such energy is available. Some heat plants have a steam turbine and co-produce heat and electricity. After the system of green certificates for electricity was initiated in Sweden 2003, the electricity production from such combined heat and power (CHP) plants has become quite lucrative. The incitement for building CHP plants rather than heat plants has thereby increased, as well as the incitement for increasing the electricity production in already existing plants.

The electricity production from thermal energy is subject to the laws of thermodynamics.

Since the exergy (or energy quality) of thermal energy is lower than the exergy of electricity, it is impossible to convert heat to electricity without substantial losses. The amount of heat that can be converted to electricity depends on what temperature the waste heat has. In condensing power plants, the steam is cooled by large amounts of water or air to a low temperature. This way the electricity efficiency is increased, but still about 60% of the input heat is wasted. In district heating applications, the plant is cooled by the district heating network, to a temperature around 70-120°C. This means that the waste heat is utilised, but on the other hand the electricity production is only about 30% of the heat input. This is a trade- off dictated by thermodynamics –one gets either more electricity and a lot of almost cool waste heat water that cannot be utilised, or less electricity and waste heat water at a high enough temperature to be utilisable for example for district heating. However, if the district heating can utilise lower temperatures, the electricity production of the heat and power plant can be increased. This was the main motivation of this degree project.

1.2 Methods

The need for heating and hot water varies depending on the weather (determining the heat losses from the buildings) and the behaviour of the people using the heat and hot water. If these variations could be predicted, the supply temperature could follow the heat demand. The delivered heat depends on the temperature of the supply water leaving the heat plant (which is controlled by the heat plant) and the flow of water through the network (which is controlled by the consumers). Each consumer will take a certain amount of heat from the water of the supply line, by sending it through their heat exchangers and exert it to the return line. If the temperature of the supply water is high, a smaller flow is needed to deliver the requested heat.

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7 Also, if the heat exchanger can cool the district heating water to a lower return temperature, a smaller flow is required to deliver the requested heat. When the heat demand is low, it is therefore possible to have a lower supply temperature, and increase the electricity efficiency of the CHP plant.

In this project, operational data from the combined heat and power plant Idbäcken was used to estimate a dynamic model of the district heating network in Nyköping. The repeating daily variations of the heat load was modelled through differentiation of the signals, and the weather dependence of the heat load was modelled with an ARX model. A control strategy using the dynamic model to decrease the supply temperature of the district heating system was evaluated and compared to the present control strategy of the Idbäcken plant.

1.3 This report

In this report, the methods and results of the project are presented. In Chapter 2, theory on district heating system dynamics and the relation between supply temperature and electricity efficiency of a combined heat and power plant connected to a district heating system are presented. Feed forward control and the basic concepts of empirical modelling are described.

A short summary of related research is also made. In Chapter 3, the combined heat and power plant Idbäcken in Nyköping and the connected district heating system are described. A

relation between the electricity efficiency and the supply temperature of this plant and the flow capacity of the network are estimated. Also, some basic features of the load and return temperature of the system are presented.

One static and one dynamic approach to modelling was used during this project. The resulting models are presented in Chapter 4. Attempts were made to include not only the outdoor temperature but also solar irradiation and wind speed as input parameters to simulate the heat demand. In Chapter 5, a control strategy based on the dynamic model presented in Chapter 4 is evaluated and compared to the present control strategy of the Idbäcken plant. The economic benefit of the new control strategy is assessed. Finally, in Chapter 6, the results of the project are discussed, and drawbacks as well as benefits of the new control strategy are suggested together with ideas of possible improvements of the model.

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2 Theory

In this chapter, the theoretical background for this study is presented. First, the basic concepts of district heating, the relation between supply temperature and electricity efficiency of a combined heat and power plant and the dynamics of a district heating network are introduced.

Then the feed forward control method and a bit of theory on empirical modelling are described. Finally, a summary of recent publications on district heating modelling is made.

2.1 District heating

A district heating (DH) system consists of a heat producer, a transmission network of pipes, and local substations in which heat from the DH water is transferred to the radiator circuit and the hot water circuit of the heat consumer. The DH network is called the primary side, and the consumer circuit - radiator and hot water circuits – are called the secondary side. Every

substation is connected both to the supply (feed) pipe and the return pipe of the DH system, as shown in Figure 1 (only the primary side is visible in the figure).

Figure 1. Explanatory sketch of the primary side of a district heating network.

A local control system, aiming to meet the momentary heat demand in the house, controls the flow through the heat exchangers, on both the primary and secondary side. This means that the flow in the supply and return line is not controlled centrally at the heat plant, but is the result of the flows through the substations. To work properly, the heat exchangers need a certain pressure difference between the supply and return line. This pressure difference is created and maintained by a central pump, usually located at the heat plant. When the flow is high, the pressure losses throughout the network will increase, and the pump will have to work harder. The pressure will always be sufficient at substations close to the heat plant, but if the capacity limit of the pump is reached, the pressure in the distant parts of the grid will fall, and the heat exchangers situated there will not be able to work properly – these customers will have cold radiators. This situation must be avoided. Therefore, the temperature of the supply water is varied with the load variations, so that at times when the heat demand is high, more energy is transferred with each cubic metre of supply water. In this way, the flow can be regulated indirectly.

Different DH systems are constructed for different supply temperature levels. The advantage of a high supply temperature system is that the flow will be smaller, and hence the pipes can be dimensioned smaller as well. The advantage of a low supply temperature is that the heat losses from the pipes to the ground will be smaller, and if the heat plant is a CHP plant, the condensation temperature of the turbine will be lower, ensuring a higher electricity

production. In Sweden, most DH systems are low temperature systems [1].

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2.2 Supply temperature and electricity output

The supply temperature to the district heating network affects the electricity efficiency of the turbine in a combined heat and power (CHP) plant, since it affects the temperature at which the steam condenses. The electricity output of a steam turbine depends on the mass flow of steam through the turbine, the efficiency of the turbine and generator, and the pressure- and temperature difference of the steam before and after the turbine [2]:

) (i₀ i_c m

P  (1)

where

P is the electric power output m is the mass flow

is the efficiency

i is the enthalpy per mass unit of the steam before the turbine 0

i is the enthalpy per mass unit of the steam after the turbine, in the condenser c

Enthalpy is a state parameter used in thermodynamics (see for example [3]) defined as pv

u

i (2)

where

u is the inner energy per unit mass (which depends on the temperature and the specific heat capacity)

p is the pressure

v is the volume per unit mass

To increase the electricity generation for a given amount of produced heat, one needs to increase the enthalpy before the turbine or decrease the enthalpy after it. The temperature before the turbine is usually limited by the type of fuel used and the durability of the materials in the combustion chamber and the turbine, and will not be considered in this study. The temperature after the turbine on the other hand, is limited by the required supply temperature to and the return temperature from the district heating grid, since the DH water is the heat sink that cools the steam. The heat transferred from the steam to the DH supply water must have a sufficiently high temperature to ensure that the DH customers get as much heat as they consume. If the supply temperature can be decreased, the electricity exchange will increase [2].

The α-value is a commonly used measure of the electricity efficiency of a heat and power plant. It is defined as

heat el

P

P with _th P^heat P^el

P (3)

where

Pel is the produced electric power Pheat is the produced heat power

P_th is the supplied thermal power to the plant η is the plant efficiency

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2.3 Dynamics of district heating networks

The dynamics of a district heating network include processes of very different timescales.

Changes in pressure and flow travel with the speed of sound, and will reach the whole grid in seconds. Changes in temperature, on the other hand, travel with the flowing water, and will take hours to be carried out. This will result in loading and unloading effects when the power output of the heat plant is larger or smaller than the heat consumption on the grid during different times of the day. In this section the time delays, loading effects and return temperature dynamics of a district heating system will be described. Also, methods for calculation and models of the heat load will be presented.

Since the topic for this study is the temperature and flow variations in the system, analysis of the pressure dynamics is omitted, and a quasi-static view of pressure is used.

2.3.1 Time delays

When the output temperature from the power plant is changed, this change will propagate in the pipes with a speed a bit slower than the speed of the water. The reason for this is that the walls of the pipe will need to adjust from the former water temperature to the latter, and to do that they will take heat from the water. The thicker the walls and the narrower the pipe, the slower the temperature change will travel compared to the water. Also, because of losses to the ground, the water will be colder when reaching far out on the grid than it was when it left the plant [1]. However, if these factors are neglected, the transport time of a temperature change can be calculated according to

t

t t

d Q

V ( ) (4)

where

V is the volume of the district heating network

Q( ) is the volume flow emitted from the plant at the time t is the transport time through the network

2.3.2 Loading/unloading the system

Since the transmission on the DH grid is not momentary, and the heat can be delivered at different temperatures, there is a possibility to load the system with extra energy. If the supply temperature is increased, the temperature on the grid will increase, and hence the heat power.

If the load is constant, the flow will decrease, since a smaller amount of water need to pass the heat exchanger to give the same amount of heat. This process is called loading or packing the network. To avoid high flows the system can be loaded in advance when high loads are expected, so that the need for an increased flow will be smaller [4].

Tests with step response when increasing the supply temperature show that the loading time (corresponding to the transport delay of the system) of a DH network is approximately V/Q0, V being the total volume of supply and return pipes and Q₀ being the initial flow from the

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11 plant. If the load is constant, an increased supply temperature will lead to a decreasing flow.

After the time V/(2Q0), the network is 75 % loaded [1].

2.3.3 Calculation of the heat load

A DH system is a large and complex structure. For detailed analyses, models can be built using software like X-power or pfcsf/pfctf (see section 2.6). There is literature on the subject of methodically reducing the models of the networks, see for example [5] and [6]. However, the detailed models are mainly used for analysing the static behaviour of the system,

addressing questions such as which the lowest supply temperature should be. It has been shown in [1] that for production oriented studies, an extremely simplified model of the network can be sufficient.

The heat power provided to the district heating network by the heat plant depends on the supply and return temperature and the flow of the water. These values can be measured, and the supplied power can be calculated as

)) ( ) ( )(

( )

sup(t c mt T t T t

P _p  _S _R (5)

where

Psup is the power supplied to the district heating system by the heat plant c_p is the specific heat capacity of water

) (t

m is the mass flow of the water

T_S is the supply temperature at the CHP plant TR is the return temperature at the CHP plant

However, the power supplied to the district heating network at a specific time is not the same as the power delivered to the customers (also referred to as the load of the DH network) at that specific time. This is a result of the transport delays of the system. If the DH network is approximated as an equally distributed homogenous load, the heat delivered to the load is, according to [1]:

t

d T d

T t m c t P

t t

t R t

t t

S p

del

) ( )

( ) ( )

(  (6)

where

Pdel(t) is the delivered power (the load)

t is the transport time one way through the supply or return line

Note that while the expression for the supply temperature is close to reality, since the temperature from the CHP plant will propagate through the supply line with the flowing water, the expression for the return temperature is more of an approximation. The return water from a specific load will be mixed with the water from more distant loads earlier in time, and more central loads later on. It is not possible to distinguish the return temperature from loads at a specific time, but the expression above will give a mean value.

Another option is to approximate the DH network with a point load, situated at the load centre of mass of the system. The load can then be calculated as

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12 2 ))

( 2 )

( )(

( )

( t

t t T t T t m c t

P_del _p  _S _R (7)

With this approximation, the load variations will be exaggerated [1]. However, since the equally distributed load model to some extent will smoothen out the variations, the point load approximation may be advantageous for modelling purposes.

2.3.4 Heat load models

The heat producer of a district heating system has to follow the heat demand (the load). The load is time varying and partially stochastic, partially deterministic. It depends on parameters like the outdoor temperature, the time of the day, coldwater temperature, solar irradiation and wind speed. The outdoor temperature and the time of the day are most significant. The coldwater temperature has an impact on the heat needed for hot water preparation, and is season dependent. A high wind speed increases the heat losses from the walls of buildings, and thus increases the heat demand, while high solar irradiation contributes to the heating and decreases the heat demand.

The total load can be seen as a sum of load contributions depending on different parameters:

w ...

sol Tcw day Tout

del P P P P P

P (8)

where

P_Tout is the load contribution depending on the outdoor temperature Pday is the load contribution depending on the time of the day

PTcw is the load contribution depending on the coldwater temperature P_sol is the load contribution depending on the solar irradiation P_w is the load contribution depending on the wind speed

The relation between the outdoor temperature and the load PTout can be assumed to be linear with the boundary condition that P_Toutis zero if the outdoor temperature is higher than a certain value. The contribution to the load from daily variations, Pday, has to be estimated from empirical data. The basic features are that the load is smaller at night than in daytime, and that there is a peak in the morning and usually in the evening. The morning peak comes later in the weekends than in the weekdays. This load part depends on what kind of customers the network supplies.

The load contributions from coldwater temperature, P_Tcw, from solar irradiation, P_sol, and from wind speed, Pw, can all be assumed to be linear. Since these contributions all are rather small and to some extent season dependent, they can be modelled by including season dependence in the model. That way, the transmission losses will also modelled indirectly, since they depend on the temperature in the ground, which is season dependent [1].

2.3.5 Dynamics of the return temperature

Theoretically, heat exchangers give a lower return temperature if the supply temperature is higher. However, it has been shown in [1] that this relation is not valid for DH systems.

Instead, the return temperature depends on the outdoor temperature and the social load. In

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13 cold weather, the return temperature increases with falling outdoor temperature. There is a break point at an outdoor temperature of approximately 7 C, where the return temperature has its lowest values. For higher outdoor temperatures, it increases again (see Figure 12 in

Chapter 3). This can be explained by the fact that when it is cold, the heating need dominates the load. The return temperature from heat exchange to the radiator circuits is higher than the return temperature from tap water preparation, because the water in the radiator circuit is never as cold as the cold water supplying the tap water, and therefore cannot cool the DH water as much. When the outdoor temperature goes up, the tap water preparation becomes a bigger part of the load, and hence decreases the return temperature of the system. During really warm days though, the need for heating diminishes and the total load on the system is small, leading to problems with low flow in the system. To keep the flow up, supply water might be by-passed to the return line. Also, with low supply temperatures, the heat exchangers are less efficient. The result is a higher return temperature. Another explanation for why the return temperature for the whole grid does not follow what heat exchanger theory suggests is that if some heat exchangers in the system are malfunctioning, giving a high return

temperature, they have a big influence on the return temperature of the whole grid.

2.4 Feed forward control

The supply temperature of a district heating system is typically controlled by feed forward control (see Figure 2). The idea with feed forward control is to use information of a

measurable disturbance on the system (the outdoor temperature) to compensate for its effects (the load variation). This means that some model of the relation between the load and the outdoor temperature is used to calculate the control point of the supply temperature. The reason to use feed forward control is the long transport delays in the system. If instead the load would be calculated and feed-backed to the supply temperature calculation, the response would be hours too late. The difficulty with feed forward control is to make a good model of the relation between the disturbance signal and the output of the system. Usually, a control curve where the supply temperature increases linearly with decreasing outdoor temperature under a certain break point is used. The break point corresponds to the lowest supply temperature that will ensure the statutory hot water temperature [7].

Figure 2. Feed forward control. Here, v is the measurable disturbance signal, F is the regulator, u is the input signal, G1 and G2 is the system that should be controlled, H is the transfer function of the

disturbance and y is the output.

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14

2.5 Empirical modelling

To use feed forward control of the supply temperature, a model for calculation of the supply temperature from some input parameters is needed. Usually, simple static models like control curves are used. In this project, a more complex static model and dynamic models estimated from empirical data were used instead.

Empirical modelling, also called system identification or black box modelling, means building mathematical models using data collected from the modelled system, rather than the physical relations of the system (which are used in physical modelling). The advantage of empirical modelling is that complex systems can be modelled without knowledge of the details of the system. Only the relevant input and output signals of the system needs to be known to estimate a black box model.

A dynamic model uses not only the current input signal but also earlier input and output signals to calculate the current output. This means that dynamic behaviour can be described.

For example, the temperature inside a house will not drop immediately if the outdoor temperature drops, but will depend both on the outdoor temperature now and the outdoor temperature the last hours, since heat is stored in the house structure. Also, if the heat on the radiators is increased, the indoor temperature will rise slowly. A static model will only be able to describe the temperature indoors if the outdoor temperature and the heat on the radiators are constant for some time, so that the system stabilizes.

A general linear dynamic black box model can be written )

) ( (

) ) (

) ( (

) ) (

( e t

q D

q t C q u F

q t B

y (9)

where

y(t) is the output signal u(t) is the input signal e(t) is white noise

B(q), F(q), C(q) and D(q) are polynomials of the shift operator q

The B(q) and F(q) polynomials model the input signal’s effect on the output signal, and the polynomials C(q) and D(q) models the noise of the system. The shift operator q is equivalent with the z-transform, and shifts a time series back or forward in time. It is used to describe that the output at a specific time depends not only on the output and noise at that time but on previous values of output, input and noise.

The model (9) is called the Box-Jenkins (BJ) model. In many cases, a simplified version of the BJ model, where one or more of the polynomials are set to be ≡1, can be used. If

C(q)/D(q) ≡1, the model is called the Output-Error-model, because the noise e(t) will be the difference between the real output and the undisturbed output. If the polynomials

F(q)=D(q)=A(q), the model is called the ARMAX model (Auto Regression Moving Average eXogen variable). In the ARMAX model, the input and the noise are subjected to the same dynamics. This should be the case if the noise enters the system early. An ARMAX model with C(q) ≡1 is called an ARX model. In Figure 3, the structure of the ARX model is

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15 displayed. An ARX model with A(q) ≡1 is called a FIR (finite impulse response) model. A FIR model calculates the output from old inputs only [8].

Figure 3. Block diagram of the ARX model.

2.5.1 The ARX model

In this study, the ARX model was used and will therefore be described in more detail. The equation for the ARX model is

) ( ) ( ) ) ( ( ) 1 ) ( (

) ) (

( e t w t v t

q t A q u A

q t B

y (10)

If the structure parameters of the model, describing the number of parameters ai and bi in the polynomials A(q) and B(q) are denoted na and nb, the undisturbed output w(t) is

) ( ...

) 1 ( ) ( ...

) 1 ( )

(t b₁u t b ut nb a₁wt a wt na

w _nb _na (11)

and the disturbance term v(t) is

) ( ) ( ...

) 1 ( )

(t a₁v t a vt na et

v _na (12)

The output y(t) can then be written

) ( ) ( ...

) 1 ( ) ( ...

) 1 (

) ( )) (

) ( ( ...

)) 1 ( ) 1 ( ( ) ( ...

) 1 ( ) 1 ( ) ( ) ( ) (

1 1

t e na t y a t

y a nb t u b t

u b

t e na t v na t w a

t v t

w a nb t u b t

u t b t v t w t y

na nb

na

nb

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The parameters a1, a2, ..., ana and b1, ..., bnb can be chosen so that the model fits the data y(t) as well as possible in a least square sense. This is called model calibration, and can be done with the Matlab System Identification toolbox. The model order, that is the value of the structure parameters na and nb, should be chosen with care. If there are too few parameters, the model will not be able to describe the system dynamic properly. If there are too many parameters, the model will be over fitted, and follow not only the dynamic of the system but also the specific noise of the calibration data series. To avoid over fitting, cross validation can be used.

This means that one data set is used to calibrate the model, and another data set is used to validate the model. The number of parameters should only be increased if it improves the fit to the validation data [8].

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16 2.5.2 Prediction with the ARX model

Once the model is calibrated and validated, it can be used for simulation or prediction of the system. In simulation, the model is presented with new input data, and calculates the output from these input data and the modelled output data. In prediction, the model is presented with both new input data and delayed output data. Predictions of the output are made from the last known real output, old predictions of outputs and new inputs. The time difference between the last known real output and the predicted output is called the prediction horizon.

To make a prediction with the prediction horizon k, a k-step predictor is needed. The model is then written in state-space form. The state x(t) of the system is defined as

) (

...

) 1 (

) (

) 1 (

...

) 1 (

) (

nb t u

t u

na t y

t y

t

x (14)

Written in matrix form the new state x(t+1) will be

) (

0 ...

...

0 1

) (

0 ...

0 1 ...

...

0

) (

0 1 ...

0 0 0

...

0

...

1 ...

...

0 ...

...

0 0 0

...

0

0 ...

...

0 0 1

...

0

...

1 0

0 ...

...

0 0 ...

...

0 1

...

) 1 (

1 1

1 2

1

t e t

u t

x b b b

a a

t x

nb nb na

na

(15)

The system can be described by

) ( ) ( ) (

) ( ) ( ) ( ) 1 (

t e t Hx t y

t Ke t Gu t Fx t

x (16)

where

] ...

...

[ a₁ a_na b₁ b_nb

H (17)

The prediction error ε(t) is

(18)

17 )

ˆ( ) ( )

(t y t Hx t

(18)

The optimal predictor for this case, see for example [9], is given by )

( ) ( ) ˆ( ) 1

ˆ(t Fx t Gu t K t

x (19)

Denote F KH . Then the predicted state after two time steps will be

) ( )

1 ( ) ( )

ˆ( ) 1 ( ) ( ))

( ) ˆ( (

) 1 ( )) ( ) ( ) ˆ( ( 0 ) 1 ( ) 1 ˆ( ) 2 ˆ(

t FKy t

Gu t FGu t

x F t

Gu t FGu t

Ky t x F

t Gu t

Ky t Gu t x F t

Gu t

x F t

x (20)

The predicted state after three time steps will be

) ( )

2 ( ) 1 ( )

( )

ˆ( 0

) 2 ( ) 2 ˆ( ) 3

ˆ(t Fxt Gu t F² x t F²Gut FGut Gut F²Ky t

x (21)

The general k-step predictor can then be written

) ˆ( ) ˆ(

) ( ) ˆ( ( )

( )

1 (

) ˆ( )

ˆ( ¹ ¹ ¹

0 1

k t x H k t y

t Ky k x F t Ky F p k

t Gu F k

x F k t

x ^k ^k

k

p p k

(22)

If the prediction horizon is varying, predictions over different horizons can be done simultaneously using the following matrix expression.

) ... (

) 1 (

...

) 1 (

) (

...

0 0 ...

0 )

ˆ( ...

) ˆ(

...

) 1 ˆ( ˆ

0 1 1

0 0

1

t y K HF

HFK HK

N t u

t u

G HF G

HF

G HF HFG

G HF t

x HF

HF H

N t y

t y Y

N N N

t (23)

With this predictor, measured output signals up until time t and input signals up until time t+k are used to predict the output y(t+k), for 1<k<N. When modelling the heat load of a district heating system, more recent values of the input signals than the output signal are known, since the calculation of the load is delayed by the transport of the return water from the load to the heat plant where it is measured.

If the modelled system has more than one input signal, the same predictor can be used, but with u(t), nb and b1, ..., bn as vectors.

During the calibration and validation process, among other things the fit of the models were compared. Fit is used in the Matlab System Identification Toolbox as a measure of the percent of the measured output y that is explained by the model, and is defined as

) ) ( 1 ˆ

(

*

100 y mean y

y

fit y (24)

where

(19)

18 yˆ is the output predicted by the model

y is the real output.

2.5.3 Time dependent variations

The black box models described in 2.5.1-2.5.2 model the output of a system from an input signal that is correlated to the output. One part of the heat load is linear function of the outdoor temperature, and should be possible to describe by such a model. However, the heat load also has a component that depends on the time of the day and the day of the week, the social load. This part of the load is not correlated to any input signal to the system, and should therefore be modelled in another way.

Since the social load is following a certain repeating pattern, one method is to assume that it is the same as the day before. The input and output signals of the system can then be

differentiated with respect to a 24 hour interval, applying the 24 hour differential operator Δ₂₄ that calculates the difference between the value now and the same time yesterday to the input and output signals, as shown in (25). To model the difference between weekdays and

weekends, the week differential operator Δ₁₆₈can be used. This operator computes the difference between the value now and the value the same time one week ago. If both the day and week differential operator is used, one compares the difference of the difference between now and yesterday, and between one week ago and the day before that [10].

192 168

24 168

24

168 168

24 24

1 ) 1

)(

1 (

) 1

(

) 1

(

q q

q q q

(25)

The model applies the operator Δ24 Δ₁₆₈ to the input parameters, and then predicts the load difference y’, from which the load y can be computed:

192 168

24

192 168

24

) ( )

( )

( ) ( ' ) (

) 1

)(

( ) ( '

q t y q

t y q t y t y t y

q q

q t y t

y (26)

An advantage with this method is that it also takes care of the season dependence of the model. Parameters such as cold water temperature and temperature in the ground, as well as the seasonal variations of solar irradiation and wind speed will not change much from one week to another, and will therefore be sufficiently modelled by the reversed differentiation, and will not interfere with the parametric model.

2.6 Previous research

In the beginning of this degree project, a literature study was carried out to find out about available modelling tools and methods for district heating networks. Of special interest was the use of dynamic models to simulate and control the supply temperature of a district heating system. It was found that many different methods are used to model district heating networks, but generally the models are static or quasi-static. Some authors claim that modelling the dynamics is unnecessary; others simply find it outside the scope of their studies.

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19 For modelling of district heating systems, many different methods are described in the

literature. There is dedicated software such as X-power and pfcsf/pfctf, which can handle detailed network models as well as simplified models. In many cases, a simple model is enough. What kind of model that is needed depends on what focus the analysis has. Larsen, Bøhm and Wigbels compares a Danish and a German method of aggregating networks in [5], which both work well on their test case. With these methods, the number of pipes in their model could be reduced from 44 to 3 or 10 respectively, without increasing the error in return temperature and heat production calculations. The Danish method is compared to the network model software TERMIS in [11]. With both methods, the dynamics at a district heating consumer could be modelled fairly well using data from the heat plant, unless the consumer was located at distant pipelines with many bends and fittings. In [6], yet another network aggregation method is presented, based on simulations with pfcsf/pfctf. At Vattenfall in Uppsala, a detailed network model is simulated with the software X-power [12]. However, the detailed models and aggregation methods are necessary mainly for analyses of the network itself, to answer questions about the heat distribution and pressure levels in the different parts of the grid and what the bottlenecks of the system are. For production analyses, it can be enough to use a gross approximation of the network. In [1], Larsson compares simulations of a detailed network model with approximations of the network as a point load and as an equally distributed load. The results from the different models are quite similar.

Attempts to combine a production model with a distribution model are made by for example [6], [13] and [15]. Kvarnström, Dotzauer and Dahlquist models both production and

distribution with Mixed Integer Programming (MIP), and shows in [6] that the production costs of the district heating system in Stockholm would be lower if the production model used to determine which production unit to use was extended with a distribution model. In their distribution model, the transport delay on the grid and the return temperature are

approximated to be constant. What kind of load model they use is not explicit in the report.

Keppo and Athila present a similar study in [13]. They also use MIP to model the heat production, and combine it with a district heating network model. The distribution model models supply and return temperatures at the secondary side of the substations as linear functions of the outdoor temperature. The temperatures and flow on the primary side are calculated according to heat exchanger theory. The transport delay on the network is omitted.

In both [6] and [13], the heat demand of the consumers is assumed to be a function of only the outdoor temperature. This means that the load variation due to the daily behaviour of the consumers, the social load, is neglected. Larsson describes a strategy to include these load variations in the distribution model in [1]. The heat load is modelled as a sum of a linear function of the outdoor temperature and a higher order function of the time of the day. These functions are also adjusted to the season. Larsson implements this load model with the software pfcsf/pfctf, with which the transport delay of the system can also be modelled.

Larsson’s model is verified on the district heating network in Karlskoga in [14]. The aim in that case was to reduce the flow variations to avoid reaching the distribution limits of the network. This was successful – the maximal flow was reduced with 7% and the minimal flow was increased with 12%. The same modelling method is also used by Johnson and Rossling in [15] and by Hedin in a degree project at Vattenfall in Uppsala [16]. A similar model approach is used in [17]. The outdoor temperature dependent part of the load is in that study modelled with piecewise linear polynomials, and the social load is modelled with one constant value for each hour of the week. This model is compared with the ARMA model used by the software Aiolos, and it is found that the models give approximately the same result, since the more sophisticated ARMA model suffers more from errors in the temperature prognosis. In [18],

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20 sine and cosine terms are used to model the social load and a binary variable is used to

distinguish between weekdays and weekends.

In [19], a time series analysis of outdoor temperature prognoses and heat load variations on a district heating system is compared to using neural networks. Malmström et al show that the temperature prognoses provided by for example SMHI could be improved by time series analyses using data from the heat plant site. Three heat load models are compared: a

SARIMAX model (a dynamic model that uses differential operators), an ARX model using both recent values of the input and output signals and values from around 24 hours earlier, and a neural EBP network model, and the conclusion is that time series analysis models are better suited to predict district heating loads than at least this type of neural network model.

The SARIMAX model structure is also used with good results for prediction of electricity consumption in [20].

Research on increasing the electricity efficiency of a CHP plant is abundant. Decreasing the supply temperature is suggested by for example Jansson in [21] and Axby et al in [22]. It is stated by Axby et al that a lower supply temperature affects the electricity production more than a lower return temperature. In their study, the potential gain from a 5°C lower supply temperature is 4.2 GWh/year for the 170 MW CHP plant in Örebro and 6.2 GWH/year for the 155MW CHP + 300MW heat plant in Västerås. Decreasing the return temperature with 5°C would only give 1.1 or 1.7 GWh/year respectively.

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21

3 Case study

In this chapter, the combined heat and power plant Idbäcken and the district heating network of Nyköping are described. A relation between supply temperature and electricity output is derived, and the flow limit of the district heating network is examined.

3.1 Idbäcken’s CHP-plant

The main production of the district heating plant Idbäcken of Nyköping, which is shown in Figure 4, is handled by the 105 MW boiler (P3), that provides the combined heat and power (CHP) part of the plant with steam. One high-pressure (HT) and one low-pressure (LT) turbine together produce 35 MW electricity, and from the exhaust steam, 58 MW heat is transferred to the district heating system by two condensers. A flue gas condenser (Rgk) of 12 MW is used to preheat the return water. When the demand for heat is higher, two heat boilers (P1 and P2) of each 35 MW can be used. There is also an electric boiler (EÅP) of 14 MW and two substations, Lasarettet (LAS) and Brandstationen (BRA), of totally 75 MW connected to the supply line of the district heating system, and a gas boiler (Gaspanna) of 1 MW connected to the return line. There is a hot water accumulator tank that can hold 400 MWh heat, and also an external cooler (not visible in Figure 4) called “Beriden” which is connected to the return line. When the electricity price is high but the heat demand of the district heating customers is modest, “Beriden” is used to cool away heat from the district heating network. Then the production can be increased, and more electricity can be sold. Typically, “Beriden” is turned on and off irregularly on a daily basis. If the heat demand is low, the first step is to reduce the heat extracted from the flue gas condenser. The second step is to start “Beriden”. The third step is to reduce the combustion of the boiler.

Figure 4. Explanatory sketch of the CHP plant Idbäcken.

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22 The bottom line of the heat production of Idbäcken is determined by the heat load applied by the district heating system. To meet the load, a sufficient supply temperature must be

delivered from the plant. The control point for the supply temperature is derived from the control curve in Figure 5. This control curve was originally linear between the upper and lower saturation temperatures and had its lower break point at the outdoor temperature 10 C, but it has been adjusted from time to time according to experience.

Figure 5. Control curve for the supply temperature of the Idbäcken CHP plant.

3.2 The district heating network of Nyköping

A map of the district heating network of Nyköping can be seen in Figure 6. The heat plant is situated in the left down part of the middle cluster. The pipe system contains of 130 km pipe with a volume of 8000m³, of which half is the supply line and half is the return line. The total water volume of the system is about 18 000m³, since there is also an accumulator tank for heat storage at the heat plant in Idbäcken which is connected to the network. At the 450 biggest sub stations, measurements of flow, differential pressure, supply temperature and return temperature are made.

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23

Figure 6. Map of the district heating network in Nyköping. The green lines are the district heating pipes.

3.3 Electricity output and supply/return temperature

To determine the relation between the -value (the produced electricity divided by the produced heat, see also section 2.2) and the supply and return temperatures respectively, two different approaches were used. First, an experiment was carried out at the plant, where step- wise changes were made in the supply and return temperatures respectively, while other parameters were kept as constant as possible. Secondly, the result from the experiment was verified by analysis of operational data from other time periods with higher temperature levels.

In the step experiment, the electricity production was kept constant while the supply and return temperatures were varied with 5 C each, leading to a change in heat production as well.

The -values for the four different states were calculated. The -value is assumed to be negatively proportional to the supply and return temperatures, if all other parameters are constant:

(25)

24

R R R

S S S

T k T

) (

)

( (27)

Where

Δα is the change of the α-value due to a change in supply or return temperature

kS, kR are proportionality constants

ΔTS, ΔTR are changes in the supply and return temperatures

An estimation of the proportionality constants kS for change of the supply temperature and kR

for change of the return temperature can then be made from the measured values. The measured values of the parameters can be seen in Table 1.

Table 1. Result from the -value experiment.

Supply temperature [°C] Return temperature [°C] Alpha value

74.2 46.9 0.579

74.4 49.9 0.574

79.5 50.7 0.565

79.6 45.1 0.571

This experiment gave the values

0015 . 0

0017 . 0

R S

k k

Since the measured data points in the experiment described above were few and not very well spread, there was a need to validate the result with operational data. However, to find

sequences of operational data where the return temperature is constant while the supply temperature changes is difficult. Generally, the supply temperature is stationary, but the return temperature is subject to frequent variations. Hence another approach than the one above is needed to validate the values of the constants.

The share of produced electricity compared to heat, the -value, depends on the condensation temperature of the steam, as described in chapter 2. Steam leaves the turbine at two different pressure levels, and is cooled by two different heat exchangers (see Figure 7). The district heating return water first cools the low pressure steam and then the steam with a bit higher pressure, leaving the last heat exchanger with the temperature required for the DH supply line.

Depending on the load, different amounts of steam goes to each heat exchanger. This means that some of the steam condensates at a temperature close to the supply temperature, and some condensates at a temperature close to the return temperature. The impact of each of these two temperatures compared to the other depends on the load.

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25

Figure 7. Heat balance for the CHP plant Idbäcken. The steam leaves the low pressure turbine LP160H at two stages and the heat is transferred to the district heating network (coming in from the right) in two heat exchangers, before the steam ends up in the condenser.

To simplify, it was assumed that the mean condensation temperature at any given time is the mean value of the supply and return temperatures. Then data from the operation of the power plant was used to find a linear relation between the -value and the condensation temperature.

Data from reasonably stable operation at 40-50 MW_th during the years 2005/2006 and 2006/2007 was chosen [23]. Mean values of the supply and return temperature and the - value were calculated for the selected time periods, and a linear relation was estimated using the Matlab function polyfit, which is based on the least square method (see Figure 8).

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26

50 55 60 65 70 75 80

0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62

Condensation temperature, centigrades

Alpha value

Estimation of the relation between alpha value and condensation temperature

Operation data 2006/2007 Operation data 2005/2006 Experiment data October 2007 Linear regression

Figure 8. Estimation of the relation between the alpha value and the condensation temperature.

The slope of this linear regression is -0.0041, which is about twice the slope calculated from the experiment described above (k_S and k_R). This is reasonable, since a change in either supply or return temperature will give a change of the condensation temperature that is about half as big.

As can be seen in Figure 8, the data points are spread, and seem to differ between different years. There are a number of reasons for why this relation is difficult to detect from process data. Except for the condensation temperature, the parameters before the turbine affects the α- value, and the division of the steam to the two condensers differ depending on the heat load.

Sometimes, steam is also drawn off to other applications, for example warm keeping of the other boilers or the combustion oil. In the following, it will be assumed that the constant kS = 0.0017 calculated from the step experiment is valid for any change of the supply temperature.

3.4 Maximal flow on the DH network

The flow on the DH network is determined by the flow through the substations. Each

substation has to transfer enough heat from the supply water of the DH grid to the secondary system in the house(s) connected to the substation, to ensure that the heat and hot water demand is met. The flow will therefore depend on the heat demand and the temperature of the supply water. With the goal of keeping the supply temperature as low as possible, it is

necessary to determine the upper limit for the flow (which depends on infrastructure such as pipes, pumps etc).

In Figure 9, the situation when the supply temperature is too low compared to the heat demand is illustrated. A shortage of delivered heat will manifest as a low pressure difference between the supply and return line in the substations in the weakest part of the grid, furthest away from the power plant. That is because the substations closer to the power plant are taking a big amount of water from the supply line. The central pump of the network,

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27 controlled by a certain pressure difference in such a weak point on the grid, will then work harder, so that the pressure on the grid increases. A higher flow on the grid leads to greater pressure losses in the system, and a higher pressure from the heat plant is therefore needed. If the flow increases too much, the pump will not be able to keep up the pressure enough. The pressure difference in the periphery of the grid will fall, and the heat exchangers will not be able to transfer enough heat to the secondary systems. People will complain about cold radiators.

Figure 9. Heat deficit on the district heating network, illustrated by operational data 2006-10-24 22:50 to 2006-10-25 15:30 [23]. Mass flow, pump frequency and pressure difference are normalised to enable visual comparison.

To determine the flow limit for the model of the district heating network, the pump frequency is plotted against the flow of the district heating water, see Figure 10. From this plot one can deduce that a linear trend can fairly well describe the relation between the flow and the pump frequency. It also indicates that the flow preferably should be kept under 2000 t/h, and definitely below 2100 t/h. In Figure 9, the pump frequency hits the roof at the flow 2000 t/h.

However, depending on how fast the flow is changing, the flow that the pump can sustain will differ, so the flow limits are not absolute, which can also be seen from the divergence of the data points in Figure 10.

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28

Figure 10. Pump frequency as a function of mass flow, operational data 2006/2007 [23].

3.5 Load and return temperature characteristics

To get a general understanding of the system, it might be interesting to have a look at the operational data to see how the load and return temperature depends on the outdoor temperature in the Nyköping case.

In Figure 11, the calculated load is plotted against the outdoor temperature. One can see that the load is decreasing linearly with increasing temperature, with a break point at around 15°C, over which the load seems constant in relation to the outdoor temperature. One can also see that there is great variation in the data, for example is the load at 0°C spread from 30 MW to 70 MW. Some of this variation depends on the repeating pattern of the social load, and some of it is of a stochastic nature.

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29

Figure 11. Outdoor temperature dependence of the load, calculated from operational data February to May 2007.

A similar plot of the return temperature is shown in Figure 12. In cold weather, the return temperature is decreasing with increasing outdoor temperature. However, for outdoor temperatures above 10°C, the return temperature is instead increasing. This is also the

expected behaviour according to the line of argument in Chapter 2. Data from year 2003/2004 is chosen in this plot because in later years, the return temperature is manipulated with the external cooling device, which means that the unaffected return temperature from the load is unknown.

Figure 12. Outdoor temperature dependence of the return temperature, operational data September to May 2003/2004.

Modelling and control of a district heating system

Examensarbete 20 p Mars 2008

Modelling and control of a district heating system

Linn Saarinen

Abstract

Modelling and control of a district heating system

Sammanfattning

Acknowledgements

Contents

1 Introduction

1.1 Motivations

1.2 Methods

1.3 This report

2 Theory

2.1 District heating

2.2 Supply temperature and electricity output

2.3 Dynamics of district heating networks

2.4 Feed forward control

2.5 Empirical modelling

2.6 Previous research

3 Case study

3.1 Idbäcken’s CHP-plant

3.2 The district heating network of Nyköping

3.3 Electricity output and supply/return temperature

3.4 Maximal flow on the DH network

3.5 Load and return temperature characteristics