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Optimal operation of a CHP plant for space heating as a peak load regulating plant
Energy 25 (2000) 283–298 www.elsevier.com/locate/energy
Optimal operation of a CHP plant for space heating as a peak load regulating plant Fu Lin a
a,*
, Jiang Yi
b
Department of Power Engineering, Shandong University of Technology, Jinan 250014, PR China b Department of Thermal Energy Engineering, Tsinghua University, Beijing 100084, PR China Received 13 September 1998
Abstract Due to the huge thermal mass of buildings and the district heating (DH) network, room temperatures in buildings do not change much when the heat output from the heat source varies significantly during the day. Therefore, a combined heating and power (CHP) plant can vary its power output during the day to match the load variation of the electrical utility grid which will change the heat output but have little effect on the quality of space heating. So, a CHP Plant for space heating can act optimally as a peak load regulating plant. This paper studies the optimal operation of a CHP plant used as a peak load regulating plant. The system for study consists of a DH network, an electrical utility grid and a CHP plant in which only an extraction unit is considered for simplicity. The thermodynamic characteristics of the heated buildings and the DH network are used to develop the dynamic relationships between the CHP plant heat output and the building room temperatures. The concept of electricity value equivalent is introduced to evaluate of the generated electricity for each interval of the day. An optimal operation model is built with two objectives: that the customers’ space heating requirements are met and that the CHP plant profit is maximized. Then, a new algorithm is proposed for the model and numerical results are presented for a specific case. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction With economic development in China, the demand for electricity has increased rapidly in recent years. However, the power system capacity can not match the increase in electrical demand. In addition, the disparity between the on-peak and off-peak loads of the electric utility grids is increasing. For example, the disparity is 37% of the on-peak load in the Northeast electric utility * Corresponding author. Present address: Department of Thermal Energy Engineering, Tsinghua University, Beijing 100084, PR China. E-mail address: [email protected] (F. Lin)
0360-5442/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 9 ) 0 0 0 6 4 - X
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Nomenclature A coefficient AMRA autoregressive-moving average B coefficient or fuel consumption, t/h C time-of-day price of electricity, yuan/kW.h CHP combined heating and power specific heat at constant pressure, kJ/kg K cp DH district heating EVE electricity value equivalent G coefficient or flow of DH network, t/h (k) kth time interval I number of previous time intervals in analysis or number of time intervals in a day J number of previous time intervals in analysis or number of days for space heating L the number of parts in the divided state vector X(k) for solving the mathematical model. M the number of parts in the divided control variable tg(k) for solving the mathematical model N number of time intervals for space heating p electrical power, kW q heat output, GJ/h T temperature, °C ⌬t number of hours in a time interval, hour X state variable matrix u control variable matrix V price of fuel, yuan/t yuan Chinese money unit, 1 yuan=0.121 US$ Z optimization objective function a,b,g,q,w,j,a,b,c,h coefficients Subscripts s r i j k l m max min ro e t
supply water return water ith of I jth of J kth optimization stage lth value mth value maximum minimum room environment current time interval
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grid [1]. This large disparity can be reduced by shifting some electrical consumption from peak load periods to off-peak load periods. For example, ice-storage systems have been developed to shift part of the air-conditioning load to off-peak. Another solution is to generate more electricity during peak load periods using peak load regulating plants. Most CHP plants in the northern part of China produce heat for space heating and generate power for electrical utility grids. They play an important role in the electrical utility grid. At the end of the year 1997, the total capacity of CHP plants with a single unit capacity of not less than 6 MW reached 2200 MW in China [2]. If the CHP plants generate more electricity at peak electrical load periods and less electricity during off-peak periods, the cost for operating the electrical grid can be reduced significantly because the grid’s thermal efficiency is improved and the capacity of the other peak load regulating plants in the grid can be reduced. With a traditional operating strategy, the heat output for space heating changes little during the day and the electrical output is determined by the heat output. As a rule, CHP plants usually operate as base load units in an electric utility grid. However, because of the large thermal mass of the buildings and the DH network, room temperatures do not change much when the heat output varies during the day. Therefore, CHP plants can be regulated optimally during the daily cycle to match the electrical utility grid load by allowing a variable heat output with no decrease in the space heating quality. Many studies in the area of optimal operation of CHP plants have been published [3–13]. Optimization of load allocations of boilers and generators was studied by Chen and Hong [3]. The economic dispatch of multi- CHP plants was studied in Refs. [4] and [5]. The optimal operation of a CHP plant with a heat storage tank was also investigated [6–11]. Zhao et al. [12] studied the optimum operation of a CHP type district heating system considering the heat storage and the time delay in the distribution network. However, there are few studies on the optimal operation of CHP plants considering the large thermal mass of both the DH network and the buildings for space heating, which will be studied in this paper. The system for study here consists of a DH network, an electrical utility grid and a CHP plant with only an extraction steam unit for simplicity. Analysis of the thermodynamic characteristics of the heated buildings and the DH network leads to a dynamic relationship between the CHP plant heat output and the building room temperatures. The concept of electricity value equivalent is introduced to quantify the value of the generated electricity for each interval during the day, so that the operation is optimized by achieving the two objectives of meeting the customer’s space heating requirements and maximizing the profit of the CHP plant. 2. Analysis of thermal conditions for the space heating system 2.1. Background The space heating system for this study consists of a DH network and heated buildings. Most operating strategies for a DH system are based on the thermodynamic characteristics of the space heating system at steady state. For example, operating experience shows that the supply water temperature and the return water temperature of the Beijing DH circulation network are determined by the two following empirical equations [14]:
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Ts⫽Tro⫹5.63[0.64(Tro⫺Te)]0.74⫹0.354(Tro⫺Te)
(1)
Tr⫽Tro⫹5.63[0.64(Tro⫺Te)]0.74⫺0.354(Tro⫺Te)
(2)
and where the room temperature Tro is normally 18°C. In another operating strategy, a simple linear relationship is used to relate the heating load and the environmental temperature. For a water circulation system with constant flow rate, the difference between the supply and return water temperatures of a circulation network can be modeled by the following equation [15]: Ts⫺Tr⫽A⫹BTe,
(3)
where the coefficients A and B are determined by geographical location, the network configuration and the building type. In Eqs. (1)–(3), Te is the daily average environmental temperature while Ts and Tr are the daily average supply and return water temperatures, respectively. However, these equations can not reflect the dynamic characteristics of the space heating system. Normally, both heat output and electricity output of the CHP plant vary little during the day in China. As a result, the CHP plants can not operate at optimal conditions. However, room temperatures of heated buildings are not changed significantly if the heat output for space heating is changed for a few hours during the day. Fig. 1 reflects the effect of the DH system heat output on the room temperature of heated buildings. The difference between the supply and return temperatures decreases from 56.6°C to 10.5°C and then rises to 53.6°C over a five hour period during the day, but the room temperature of the heated buildings does not change much, which implies that the space heating system has a large thermal mass. If an equation is developed to relate the building room temperature and the heat output produced by the CHP plant at time intervals of a few hours instead of a day, the heat output can be varied over a large range while the room temperature variation is maintained within an acceptable range. The operation of a CHP plant can then be optimized as a peak load regulating plant.
Fig. 1. Effect of differences between Ts and Tr on Tro.
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Fig. 2. Space heating system variables.
2.2. Dynamic thermal model As shown in Fig. 2 for a space heating system with a water circulation network of constant flow rate, the input variables are the supply water temperature, Ts, and the environmental temperature, Te, while the output variables are the return water temperature, Tr, and the building room temperature, Tro. The relationship between the output variables and the input variables of the system is expressed by the AMRA time series model as:
冘
冘
冘
冘
I
Tr,t⫽
冘
I
aiTr,t−i⫹
i⫽1
I
biTs,t−i⫹
i⫽0
giTe,t−i
(4)
i⫽0
and J
Tro,t⫽
冘
J
qjTro,t−j ⫹
j⫽1
J
jiTs,t−j ⫹
j⫽0
wjTe,t−j ,
(5)
j⫽0
where the orders I and J are the number of previous time intervals which may influence the current temperature. All the coefficients in the two equations are determined by characteristics of the space heating system and can be calculated from practical operating data. In this paper, the operating data are taken from the Shenhai DH system with data averaged over four hour time intervals. The generalized least square method was used to determine the coefficients. The standard deviations of Tr,t and Tro,t in Eqs. (4) and (5) for various values of I and J are shown in Table 1 for data over an entire space heating season. The best value of I in Eq. (4) is 2 because the standard deviation changes little for I greater than 2. In the same way, the best value of J in Eq. (5) is 1. Due to the large thermal mass of the space heating system, the building room temperatures are influenced little by the supply water temperature of the DH network at the current time interval t, so Eq. (5) can be rewritten for J equal to 1 as: Tro,t⫽q1Tro,t−1⫹j1Ts,t−1⫹w1Te,t−1.
(6)
Table 1 Standard deviations of Tr,t and Tro,t for various values of I and J I or J
0
1
2
3
4
5
6
7
Eq. (4) Eq. (5)
12.02 6.887
3.975 0.0197
3.544 0.0174
3.431 0.0156
3.367 0.0139
3.307 0.0122
3.276 0.0105
3.264 0.0089
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Table 2 Coefficients of the three models for the space heating system Eq. (4)
0 1 2 standard deviation
ai – 0.5721 0.0607 3.544
Eq. (5) bi 0.2112 ⫺0.0243 ⫺0.0104
gi 0.3317 ⫺0.3169 0.1741
qi – 0.9809 – 0.0174
Eq. (6) ji 0.0012 0.0037 –
wi 0.0016 0.0093 –
qi – 0.9796 – 0.0175
ji – 0.0051 –
wi – 0.0138 –
Fig. 3. (a) Calculation and measured return water temperatures. (b) Calculation and measured room temperatures.
The data in Table 2 show that the difference between the standard deviation in Eq. (6) and that in Eq. (5) is very small. Therefore, the simplified Eq. (6) can be used to relate the building room temperature and the supply water temperature of the DH network. The room temperatures calculated by Eq. (6) and the supply water temperatures of the DH network calculated by Eq. (4) are compared with actual operating data for 20 days during a space heating season in Fig. 3a and b. The coefficients j1 and w1 are much smaller than q1 which shows that the room temperature at the current time interval t is primarily a function of the room temperature at the previous time interval t⫺1, while the supply water temperature has only a small effect on the room temperature. The room temperature at the current time interval can be considered to be the accumulation of
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the effects of the heat output of the CHP plant at the previous few time intervals. Therefore, the room temperature at the current time will remain constant even if the heat output during the previous few hours is changed intentionally. The room temperature can be maintained within an acceptable range while optimizing the CHP plant operating strategy as a peak load regulating plant by varying the heat output and electrical power generation during different time intervals. 3. Optimization of the CHP plant operation The optimal operating strategy for the CHP plant for space heating was determined by including not only the CHP plant, but also the electric utility grid and the space heating system as well. 3.1. Electricity value equivalent Only one CHP plant is considered, which is a very small part of the total generating capacity of the electric utility grid. It is impractical to optimize the operation of the entire system with many power plants. Values of the electricity generated by the CHP plant at various time intervals are used to optimize the CHP plant operation. This method is known as the theory of electricity value equivalent (EVE) in the literature [16]. EVE is a new methodology that represents the value of generated electricity on an hourly basis. The value of electrical power generation is evaluated using modern economic analytic modeling. The time-of-day price of the electricity on an hourly basis is calculated using EVE theory for the electric utility grid studied in this paper. The prices shown in Fig. 4 [16] for a typical day are used to optimize the operation of the CHP plant as an example. 3.2. Mathematical model For simplification, the CHP plant is composed of only a coal-fired boiler and an extraction steam turbine. The DH network is a water circulation network with constant flow rate. The electricity generated by the CHP plant is sent to an electric utility grid.
Fig. 4. Time-of-day price of electricity during a typical day.
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The optimal operation for part of a space heating season requires determination of the heat output and the electrical power at each time interval. Mathematically, the optimization objective is to maximize the difference between the value of electricity generated by the plant and the cost of the fuel consumed by the plant which satisfies the space heating demands. This is expressed by the following equation:
冘冘 J
max
I
Z⫽
(C(i)p(j,i)⫺VB(j,i))⌬t.
(7)
j⫽0 i⫽0
Eq. (7) considers J days of a space heating season with each day divided into I time intervals, so the hours of each time interval ⌬t are equal to 24/I. The fuel consumption of the CHP plant at the kth time interval B(k) is a function of the heat output q(k) and the power p(k) at this time interval. Current practice is that the function is a generalization of a standard quadratic equation that is quadratic in both the heat and the electrical output. The coupling between the electrical and the heat productions is included through a coupled-term that is the product of q(k) and p(k). The fuel consumption function is therefore [4]: B(k)⫽a1⫹a2p(k)⫹a3p(k)2⫹a4q(k)⫹a5q(k)2⫹a6p(k)q(k).
(8)
Fig. 5 shows the heat-power operating region of an extraction unit. This region is enclosed by the boundary curve ABCDE. Along curve AB, the unit runs with maximum electrical power output while the maximum fuel is consumed along curve BC. Curve CD represents the maximum heat extraction, while curve DE represents the minimum fuel consumption. Along curve EA, the unit runs with no heat extraction. These linear curves are described by: alp(k)⫹blq(k)ⱖcl l⫽1⫺5,
(9)
where al, bl and cl are the constant coefficients of the inequalities which are determined by the operating characteristics of the extraction unit. The relationship between the CHP plant and the space heating system is given by: q(k)⫽Gcp(Ts(k)⫺Tr(k))
Fig. 5. Heat-power operating region for an extraction unit.
(10)
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For a space heating system, some variables are not only restricted by Eqs. (4) and (6), but also by: TrⱕTs(k)ⱕTs,max
(11)
Tro,minⱕTro(k)ⱕTro,max.
(12)
and
In this paper, Ts,max is 120°C while Tro,max and Tro,min are 20°C and 16°C, respectively, which are the acceptable room temperature extremes in China. The mathematical model for optimizing the operation of a CHP plant consists of Eqs. (4, 6–12). 3.3. Solution of the mathematical model Since the model includes two discrete time differential equations, i.e. Eqs. (4) and (6), the output variables shown in Fig. 2 at the current time interval are strongly related to the input variables at previous time intervals. The mathematical model is a dynamic non-linear optimization problem with discrete time intervals. The dynamic optimization problem for CHP plant operation has been solved in many different ways, including the non-linear method [10], mixed-integer linear programming [7] and dynamic programming [6,11]. Analysis of the long-term optimal operation over one space heating season is a large-scale optimization problem, so the general optimization methods, i.e. Newton-based methods, are not practical because of the relatively long calculational time. If the time series Eqs. (4) and (6) are changed to state–space equations and the state variables in the equations are divided into discrete values, the dynamic programming approach can be used to divide the large-scale problem into small-scale problems with sequential relations, which can be solved rather simply. Therefore, a dynamic programming method is used [17] combined with the penalty function method to solve the optimization problem. The output variables must first be related to the input variables of the space heating system shown in Fig. 2 by state–space equations. Eqs. (4) and (5) are rewritten as:
冘
冘
I
Tr(k)⫽
I
aiTr(k⫺i)⫹
i⫽1
biu(k⫺i)
(13)
i⫽0
and
冘 J
Tro(k)⫽
j⫽1
冘 J
qjTro(k⫺j)⫹
cju(k⫺j),
j⫽0
where u(k)⫽[Ts(k) Te(k)]T bi⫽[bi gi] cj⫽[jj wj] Now choose the state variables as:
(14)
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x1(k)=Tr(k)−h0u(k)
xI+1(k)=Tn(k)−hI+1u(k)
x2(k)=x1(k+1)−h1u(k)
xI+2(k)=xI+1(k+1)−hI+2u(k)
%%%%%
%%%%%
xI(k)=xI−1(k+1)−hI−1u(k) xI+J(k)=xI+J−1(k+1)−hI+Ju(k) where the coefficients h0⫺hI+J are described as:
冦
冦
h0=b0
hI+1=c0
h1=b1+a1h0
hI+2=c1+q1hI+1
%%%%%
%%%%%
hI=bI+a1hI−1+%+aI−1h1+aIh0 hI+J+1=cJ+q1hI+J+%+qI+Jh1+qIh0
So Eqs. (4) and (5) are changed to the state equation as: X(k⫹1)⫽AX(k)⫹B쐌u(k),
(15)
where X(t)=[x1(k) x2(k)…xI+J(k)]T, B=[h1 h2…hI+J]T
冋 册
A⫽
A1 0
0 A2
冤
0 1
0
% 0
0 0
1
% 0
A1⫽ ⯗ ⯗
⯗
⯗
⯗
% 0
1
0 0
aI aI−1 % a2 a1
冥 冤
0 1
0
% 0
0 0
1
% 0
A2⫽ ⯗ ⯗
⯗
⯗
⯗
% 0
1
0 0
I×I
qJ qJ−1 % q2 q1
冥
J×J
The output equations are: Tr(k)⫽x1(k)⫹h0u(k)
(16)
Tro(k)⫽xI+1(k)⫹hI+1u(k).
(17)
and
With the state equation and the output equations replacing Eqs. (4) and (5), the control variables in the mathematical model are then the element Ts(k) in the 2×1 matrix u(k) and p(k). Another element Te(k) in the matrix can be given by a forecast weather model [18]. The other control variables are the electrical power p(k), which varies independently over the different time intervals, and the supply water temperatures Ts(k) at the different time intervals which are interrelated by the state equation. Therefore, the solution of the optimization model is carried out in two steps. In the first step, p(k) is optimized by a common optimization method, i.e. the penalty function method. In the second step, Ts(k) is optimized by a dynamic programming method. In the dynamic programming method, the original optimization problem is transformed to a
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multistage optimization problem. By introducing a function Gk(X(k),u(k),p(k)) as the optimization objective at the kth time interval, Eq. (7) can be changed to:
冘 N
Z⫽ max
Gk(X(k),u(k),p(k)),
(18)
k⫽0
where N=I×J. The optimization objective from the kth to the Nth time interval is expressed as
再冘 N
ZN−k(X(k),X(N))⫽max
u(k)u(N) p(k)%p(N) k⬘⫽k
冎
Gk⬘(X(k⬘),u(k⬘),p(k⬘)) .
(19)
The following recurrence equation is obtained on the basis of the principle of optimality: ZN−k(X(k),X(N))⫽max
{Gk(X(k),u(k),p(k))⫹ZN−k−1(X(k⫹1),X(N))}.
(20)
u(k) p(k)
The analytical solution can not be obtained by the direct application of Eq. (20) due to the complexity of the model. Thus, the following numerical solution is required to solve the problem. At the kth time interval, the state vector X(k) and the control variable Ts(k) are divided into a discrete set of Xl(k) and Tsm(k), within the region limited by Eqs. (8)–(12), where l varies from 1 to L and m from 1 to M. For a specified pair Xl(k) and Tsm(k), the optimized control variable pl,m(k) is obtained using the penalty function method [17] taking the following equation as the optimization objective. Z⬘⫽ max Gk(Xl(k),um(k),pl,m(k)).
(21)
p1,m(k)
Then, for the state vector Xl(k), the optimized control variables Tsm⬘(k) and pl,m⬘(k) are obtained by solving the recurrence relation Eq. (20). Each Xl(k) with l from 1 to L and the relevant optimized control variables Tsm⬘(k) and pl,m⬘(k) are then stored. The procedure is then completed for every time interval from the Nth to the 0th time interval. Finally, the state Eq. (15) is solved step by step from the 0th to the Nth time interval to obtain each state vector X(k) and the relevant optimized control variables Ts(k) and p(k) for k=0,1,…N. Fig. 6 shows the algorithm used to determine the optimal operation.
4. Case study The CHP plant considered here has an extraction steam turbine. The turbine is a CC12–50/10/3 type with rated power output of 12 MW, inlet pressure of 4.9 MPa and condensing pressure of 0.005 MPa. Low-pressure extraction from the turbine for space heating is done at the rated pressure of 0.3 MPa. High-pressure extraction flow is zero during the space heating season. The turbine isentropic efficiency is 0.79. The boiler steam flow rate to the turbine varies from 50% to 100%
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Fig. 6.
Flow chart for optimal operation.
of the rated output. If Te(k) is lower than ⫺15°C, a peak boiler bears the extra space heating load. The Shenhai space heating system in Shenyang is selected with a DH network water flow of 500 t/h and network losses of 5%. The coefficients of Eqs. (4) and (5) which express the relationship between the output variables and the input variables of the space heating system are the same as shown in Table 2. The calculation considers an entire space heating season, i.e. November 16 to March 31 of the next year, with a day divided into six time intervals. So J=138 and I=6 in the optimization objective Eq. (7). Since the orders in Eqs. (4) and (6) are 2 and 1 respectively, the state variable is X(k)=[x1(k) x2(k) x3(k)]T and the coefficients of state Eq. (15) are: A1⫽
冋 册 0 1
a2 a1
A2⫽q1,
while Eq. (17) is simplified as: Tro(k)=x3(k). More electricity should be generated with less heat output during the time intervals of high time-of-day prices while less electricity should be produced with more heat output during the time intervals of low time-of-day prices. The optimization result confirmed this expectation. During several days of peak space heating load with very low environmental temperatures, the unit operates at or near point C in Fig. 5 with the largest extraction. With this condition, the heat output of the unit reaches the maximum, i.e. 34.95 MW while the electrical power is 6.60 MW.
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Fig. 7. (a) Variation of p and q during a week with maximum heat load. (b) Variation of p and q during a week with medium heat load. (c) Variation of p and q during a week with minimum heat load.
As a result, the CHP plant can not operate as a peak load regulating plant in the three days from the 1st to 9th time intervals in Fig. 7a because the unit runs only at point C with constant electrical output. The unit begins to increase its electrical output when Te increases. First, the electrical power during the time interval of the highest time-of-day price increases while the heat output decreases, as shown in Fig. 7a. As Te gradually increases, the operating state of the unit moves along curve ABC of Fig. 5 from C toward A where the electrical power is the maximum and the
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Fig. 8. Variation of Tro during half of a space heating season.
heat output is zero. Then, as the space heating demand decreases further, the operating states during other time intervals of high time-of-day prices tend towards the production of more electricity. The unit always operates at or near point C during the time intervals of low time-of-day prices in Fig. 7a–c. Fig. 7c shows the variation of the electrical power and heat output of the CHP plant during the period when space heating demand is very low. An important concern with peak load regulating plants is the operational safety. Since the unit runs during the whole season on curve ABC with the inlet steam into the turbine always kept at or near the maximum, the CHP can run safely with this operating strategy. For the optimized operation of the CHP plant, the variation of the room temperature Tro of the heated buildings during the later half of a space heating season is shown on Fig. 8. Tro changes during a day within a range of less than 0.5°C, while it varies over the whole space heating season within the 16.0–17.2°C range which is acceptable in China. Fig. 9 shows the variation of the supply water temperature, Ts, and the return water temperature, Tr, in a typical week when the CHP plant operates with its heat output and electrical power varying as given in Fig. 7b. The highest supply water temperature of the DH network is 103.2°C and the lowest is only 28.4°C. The return water temperature varies within the 45.9–26.8°C range. The optimal operation of the CHP plant is compared in Table 3 with a reference operating strategy which has constant inlet steam flow into the turbine, while the extraction provides enough heat output to maintain constant building room temperature of 16°C. The variation of the heat and electrical production is given over the six time intervals of a typical day in Table 3. The difference between heat and electrical production during the time intervals of high time-of-day prices and that of low prices with the optimal operation is much larger than that with the reference mode of operation. The total electricity value is defined as the sum of the electrical production
Fig. 9. Variation of Ts and Tr during a week with medium heat load.
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Table 3 Comparison of optimal and reference operations Optimal operation Time interval C (yuan/kWh) 0:00苲4:00 0.283 4:00苲8:00 0.335 8:00苲12:00 0.770 12:00苲16:00 0.701 16:00苲20:00 0.566 20:00苲24:00 0.413 Total Total Electricity Value (×104 yuan) Average Electricity Value (×104 yuan/kWh) TEP (×104 yuan) EP (×104 yuan/kWh)
Q (GJ) 70479.0 70138.9 14472.0 31881.6 55242.4 66599.3 308813.2 1539.6 0.550 617.6 0.221
Reference operation P (MW.h) 3746.3 3762.7 6459.3 5614.1 4483.2 3933.8 27999.0
Q (GJ) 59507.4 61478.1 51804.3 39213.2 43196.8 51556.8 306756.6 1493.4 0.524
P (MW.h) 4268.6 4198.3 4744.3 5403.8 5181.3 4713.8 28510.1
496.6 0.174
multiplied by the time-of-day price at each of the six time intervals. The average electricity value is defined as the ratio of the total electricity value to the electrical power which reflects the value of the electricity per kWh generated by a CHP plant in a space heating season. It can be seen in Table 3 that the average electricity value with the optimal operation is 5% higher than that with the reference mode of operation. The advantage with optimal operation is not very great because the simplified CHP plant for the case study can vary the heat output and the power output within only a relatively small range. Another reason is that the environmental temperature, Te, during the time intervals of low time-of-day price is often low, so the heat output with the reference operation is large during these intervals to keep a constant room temperature. As a result, the power output during the time intervals of low time-of-day price is small and the average electricity value increases accordingly. The usual practical operating mode in China is that the heat output of a CHP plant for space heating changes little during the day, and the power output of the CHP plant also changes very little. Thus, the average value of the electricity produced with the practical operating mode is smaller than that with the reference mode. The total electricity profit (TEP) in Table 3 is defined as the value Z of the optimization objective in Eq. (7). The electricity profit (EP) is defined as the ratio of the total electricity profit to the total amount of power generated by a CHP plant, which reflects the profit of the electricity per kWh generated by the CHP plant in a space heating season. Here, the electricity profit with the optimal operation is 27% higher than that with the reference operation, which shows that the optimal operation of a CHP plant does have an evident advantage. 5. Conclusion Due to the large thermal mass of the heated buildings and the DH water circulation network, operation of a CHP plant for space heating can be optimized as a peak load regulating plant of the electric utility grid.
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This optimal operating strategy for a CHP plant not only reduces the difference between electrical supply and demand in the electric utility grid, but also improves the profit of the CHP plant. This operating strategy does not increase safety problems as often happens with ordinary peak load regulating plants. Of course, relevant policies such as time-of-day electric pricing should be implemented to encourage the CHP plant to use this optimal operating strategy. References [1] Wang X. Peak-dispatch of electricity by air-conditioning with cool-Storage. Energy in China 1996;20(11):5–9. [2] Chen W. Analysis of cogeneration development in China. In: Proceedings of the symposium on cogeneration of 1999. Jinan. China, 8–10 June 1999:1–5. (In Chinese) [3] Chen B, Hong C. Optimum operation for a back-pressure cogeneration system under time-of-use rates. IEEE Trans Power Systems 1996;11(2):1074–82. [4] Rooijers F, van Amerongen R. Static economic dispatch for co-generation systems. IEEE Trans Power Systems 1994;9(3):1392–8. [5] Guo T, Henwood M, van Ooijen M. An algorithm for combined heat and power economic dispatch. IEEE Trans Power Systems 1996;11(4):1778–84. [6] Ito K, Yokoyama R, Shiba T. Optimal operation of a diesel engine cogeneration plant including a heat storage tank. J Engng Gas Turbines Power 1992;114:687–94. [7] Yokoyama R, Ito K. Optimal operational planning of cogeneration systems with thermal storage by the decomposition method. J Energy Res Tech 1995;117:337–42. [8] von Spakovsky M, Curti V, Batato M. The performance optimization of a gas turbine cogeneration/heat pump facility with thermal storage. J Engng Gas Turbines Power 1995;117:2–9. [9] Baughman M, Eisner N, Merrill P. Optimizing combined cogeneration and thermal storage systems: an engineering economic approach. IEEE Trans Power Systems 1989;4(3):974–9. [10] Zhao H, Holst J, Arvaston L. Optimal operation of coproduction with a storage. In: 7th Int. Conference on Thermal Energy Storage. Sapporo, Japan, June 1997:187–92. [11] Raven H, Rygard J. Optimal scheduling of coproduction with a storage. Engng Optimization 1994;22:267–81. [12] Zhao H, Benny B, Raven H. On optimum operation of a CHP type district heating system by mathematical modeling. Euroheat and Power 1995;24(11):618–22. [13] Ito K, Shiba T, Yokoyama R. Optimal operation of a cogeneration plant in combination with electric heat pumps. J Energy Res Tech 1994;116:56–63. [14] Jia J. An approach of solving the parallel and supply-demand thermal unbalance in district heating system. MSc thesis, Tsinghua University, 1992. (In Chinese) [15] Wen Z. A new control system for space heating. District Heating 1988;76(3):21–4. [16] Yen M. Electricity value equivalent theory. Electrical Industry Press, 1997. [17] Wan Y. Compilation of programs of some common optimization methods. Worker Press, 1988. [18] Jiang Y. Random weather model for calculating air-conditioning loads. Refrigeration J 1982;17(2):31–5.
Optimal operation of a CHP plant for space heating as a peak load regulating plant Fu Lin a
a,*
, Jiang Yi
b
Department of Power Engineering, Shandong University of Technology, Jinan 250014, PR China b Department of Thermal Energy Engineering, Tsinghua University, Beijing 100084, PR China Received 13 September 1998
Abstract Due to the huge thermal mass of buildings and the district heating (DH) network, room temperatures in buildings do not change much when the heat output from the heat source varies significantly during the day. Therefore, a combined heating and power (CHP) plant can vary its power output during the day to match the load variation of the electrical utility grid which will change the heat output but have little effect on the quality of space heating. So, a CHP Plant for space heating can act optimally as a peak load regulating plant. This paper studies the optimal operation of a CHP plant used as a peak load regulating plant. The system for study consists of a DH network, an electrical utility grid and a CHP plant in which only an extraction unit is considered for simplicity. The thermodynamic characteristics of the heated buildings and the DH network are used to develop the dynamic relationships between the CHP plant heat output and the building room temperatures. The concept of electricity value equivalent is introduced to evaluate of the generated electricity for each interval of the day. An optimal operation model is built with two objectives: that the customers’ space heating requirements are met and that the CHP plant profit is maximized. Then, a new algorithm is proposed for the model and numerical results are presented for a specific case. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction With economic development in China, the demand for electricity has increased rapidly in recent years. However, the power system capacity can not match the increase in electrical demand. In addition, the disparity between the on-peak and off-peak loads of the electric utility grids is increasing. For example, the disparity is 37% of the on-peak load in the Northeast electric utility * Corresponding author. Present address: Department of Thermal Energy Engineering, Tsinghua University, Beijing 100084, PR China. E-mail address: [email protected] (F. Lin)
0360-5442/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 9 ) 0 0 0 6 4 - X
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Nomenclature A coefficient AMRA autoregressive-moving average B coefficient or fuel consumption, t/h C time-of-day price of electricity, yuan/kW.h CHP combined heating and power specific heat at constant pressure, kJ/kg K cp DH district heating EVE electricity value equivalent G coefficient or flow of DH network, t/h (k) kth time interval I number of previous time intervals in analysis or number of time intervals in a day J number of previous time intervals in analysis or number of days for space heating L the number of parts in the divided state vector X(k) for solving the mathematical model. M the number of parts in the divided control variable tg(k) for solving the mathematical model N number of time intervals for space heating p electrical power, kW q heat output, GJ/h T temperature, °C ⌬t number of hours in a time interval, hour X state variable matrix u control variable matrix V price of fuel, yuan/t yuan Chinese money unit, 1 yuan=0.121 US$ Z optimization objective function a,b,g,q,w,j,a,b,c,h coefficients Subscripts s r i j k l m max min ro e t
supply water return water ith of I jth of J kth optimization stage lth value mth value maximum minimum room environment current time interval
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285
grid [1]. This large disparity can be reduced by shifting some electrical consumption from peak load periods to off-peak load periods. For example, ice-storage systems have been developed to shift part of the air-conditioning load to off-peak. Another solution is to generate more electricity during peak load periods using peak load regulating plants. Most CHP plants in the northern part of China produce heat for space heating and generate power for electrical utility grids. They play an important role in the electrical utility grid. At the end of the year 1997, the total capacity of CHP plants with a single unit capacity of not less than 6 MW reached 2200 MW in China [2]. If the CHP plants generate more electricity at peak electrical load periods and less electricity during off-peak periods, the cost for operating the electrical grid can be reduced significantly because the grid’s thermal efficiency is improved and the capacity of the other peak load regulating plants in the grid can be reduced. With a traditional operating strategy, the heat output for space heating changes little during the day and the electrical output is determined by the heat output. As a rule, CHP plants usually operate as base load units in an electric utility grid. However, because of the large thermal mass of the buildings and the DH network, room temperatures do not change much when the heat output varies during the day. Therefore, CHP plants can be regulated optimally during the daily cycle to match the electrical utility grid load by allowing a variable heat output with no decrease in the space heating quality. Many studies in the area of optimal operation of CHP plants have been published [3–13]. Optimization of load allocations of boilers and generators was studied by Chen and Hong [3]. The economic dispatch of multi- CHP plants was studied in Refs. [4] and [5]. The optimal operation of a CHP plant with a heat storage tank was also investigated [6–11]. Zhao et al. [12] studied the optimum operation of a CHP type district heating system considering the heat storage and the time delay in the distribution network. However, there are few studies on the optimal operation of CHP plants considering the large thermal mass of both the DH network and the buildings for space heating, which will be studied in this paper. The system for study here consists of a DH network, an electrical utility grid and a CHP plant with only an extraction steam unit for simplicity. Analysis of the thermodynamic characteristics of the heated buildings and the DH network leads to a dynamic relationship between the CHP plant heat output and the building room temperatures. The concept of electricity value equivalent is introduced to quantify the value of the generated electricity for each interval during the day, so that the operation is optimized by achieving the two objectives of meeting the customer’s space heating requirements and maximizing the profit of the CHP plant. 2. Analysis of thermal conditions for the space heating system 2.1. Background The space heating system for this study consists of a DH network and heated buildings. Most operating strategies for a DH system are based on the thermodynamic characteristics of the space heating system at steady state. For example, operating experience shows that the supply water temperature and the return water temperature of the Beijing DH circulation network are determined by the two following empirical equations [14]:
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Ts⫽Tro⫹5.63[0.64(Tro⫺Te)]0.74⫹0.354(Tro⫺Te)
(1)
Tr⫽Tro⫹5.63[0.64(Tro⫺Te)]0.74⫺0.354(Tro⫺Te)
(2)
and where the room temperature Tro is normally 18°C. In another operating strategy, a simple linear relationship is used to relate the heating load and the environmental temperature. For a water circulation system with constant flow rate, the difference between the supply and return water temperatures of a circulation network can be modeled by the following equation [15]: Ts⫺Tr⫽A⫹BTe,
(3)
where the coefficients A and B are determined by geographical location, the network configuration and the building type. In Eqs. (1)–(3), Te is the daily average environmental temperature while Ts and Tr are the daily average supply and return water temperatures, respectively. However, these equations can not reflect the dynamic characteristics of the space heating system. Normally, both heat output and electricity output of the CHP plant vary little during the day in China. As a result, the CHP plants can not operate at optimal conditions. However, room temperatures of heated buildings are not changed significantly if the heat output for space heating is changed for a few hours during the day. Fig. 1 reflects the effect of the DH system heat output on the room temperature of heated buildings. The difference between the supply and return temperatures decreases from 56.6°C to 10.5°C and then rises to 53.6°C over a five hour period during the day, but the room temperature of the heated buildings does not change much, which implies that the space heating system has a large thermal mass. If an equation is developed to relate the building room temperature and the heat output produced by the CHP plant at time intervals of a few hours instead of a day, the heat output can be varied over a large range while the room temperature variation is maintained within an acceptable range. The operation of a CHP plant can then be optimized as a peak load regulating plant.
Fig. 1. Effect of differences between Ts and Tr on Tro.
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Fig. 2. Space heating system variables.
2.2. Dynamic thermal model As shown in Fig. 2 for a space heating system with a water circulation network of constant flow rate, the input variables are the supply water temperature, Ts, and the environmental temperature, Te, while the output variables are the return water temperature, Tr, and the building room temperature, Tro. The relationship between the output variables and the input variables of the system is expressed by the AMRA time series model as:
冘
冘
冘
冘
I
Tr,t⫽
冘
I
aiTr,t−i⫹
i⫽1
I
biTs,t−i⫹
i⫽0
giTe,t−i
(4)
i⫽0
and J
Tro,t⫽
冘
J
qjTro,t−j ⫹
j⫽1
J
jiTs,t−j ⫹
j⫽0
wjTe,t−j ,
(5)
j⫽0
where the orders I and J are the number of previous time intervals which may influence the current temperature. All the coefficients in the two equations are determined by characteristics of the space heating system and can be calculated from practical operating data. In this paper, the operating data are taken from the Shenhai DH system with data averaged over four hour time intervals. The generalized least square method was used to determine the coefficients. The standard deviations of Tr,t and Tro,t in Eqs. (4) and (5) for various values of I and J are shown in Table 1 for data over an entire space heating season. The best value of I in Eq. (4) is 2 because the standard deviation changes little for I greater than 2. In the same way, the best value of J in Eq. (5) is 1. Due to the large thermal mass of the space heating system, the building room temperatures are influenced little by the supply water temperature of the DH network at the current time interval t, so Eq. (5) can be rewritten for J equal to 1 as: Tro,t⫽q1Tro,t−1⫹j1Ts,t−1⫹w1Te,t−1.
(6)
Table 1 Standard deviations of Tr,t and Tro,t for various values of I and J I or J
0
1
2
3
4
5
6
7
Eq. (4) Eq. (5)
12.02 6.887
3.975 0.0197
3.544 0.0174
3.431 0.0156
3.367 0.0139
3.307 0.0122
3.276 0.0105
3.264 0.0089
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Table 2 Coefficients of the three models for the space heating system Eq. (4)
0 1 2 standard deviation
ai – 0.5721 0.0607 3.544
Eq. (5) bi 0.2112 ⫺0.0243 ⫺0.0104
gi 0.3317 ⫺0.3169 0.1741
qi – 0.9809 – 0.0174
Eq. (6) ji 0.0012 0.0037 –
wi 0.0016 0.0093 –
qi – 0.9796 – 0.0175
ji – 0.0051 –
wi – 0.0138 –
Fig. 3. (a) Calculation and measured return water temperatures. (b) Calculation and measured room temperatures.
The data in Table 2 show that the difference between the standard deviation in Eq. (6) and that in Eq. (5) is very small. Therefore, the simplified Eq. (6) can be used to relate the building room temperature and the supply water temperature of the DH network. The room temperatures calculated by Eq. (6) and the supply water temperatures of the DH network calculated by Eq. (4) are compared with actual operating data for 20 days during a space heating season in Fig. 3a and b. The coefficients j1 and w1 are much smaller than q1 which shows that the room temperature at the current time interval t is primarily a function of the room temperature at the previous time interval t⫺1, while the supply water temperature has only a small effect on the room temperature. The room temperature at the current time interval can be considered to be the accumulation of
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the effects of the heat output of the CHP plant at the previous few time intervals. Therefore, the room temperature at the current time will remain constant even if the heat output during the previous few hours is changed intentionally. The room temperature can be maintained within an acceptable range while optimizing the CHP plant operating strategy as a peak load regulating plant by varying the heat output and electrical power generation during different time intervals. 3. Optimization of the CHP plant operation The optimal operating strategy for the CHP plant for space heating was determined by including not only the CHP plant, but also the electric utility grid and the space heating system as well. 3.1. Electricity value equivalent Only one CHP plant is considered, which is a very small part of the total generating capacity of the electric utility grid. It is impractical to optimize the operation of the entire system with many power plants. Values of the electricity generated by the CHP plant at various time intervals are used to optimize the CHP plant operation. This method is known as the theory of electricity value equivalent (EVE) in the literature [16]. EVE is a new methodology that represents the value of generated electricity on an hourly basis. The value of electrical power generation is evaluated using modern economic analytic modeling. The time-of-day price of the electricity on an hourly basis is calculated using EVE theory for the electric utility grid studied in this paper. The prices shown in Fig. 4 [16] for a typical day are used to optimize the operation of the CHP plant as an example. 3.2. Mathematical model For simplification, the CHP plant is composed of only a coal-fired boiler and an extraction steam turbine. The DH network is a water circulation network with constant flow rate. The electricity generated by the CHP plant is sent to an electric utility grid.
Fig. 4. Time-of-day price of electricity during a typical day.
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The optimal operation for part of a space heating season requires determination of the heat output and the electrical power at each time interval. Mathematically, the optimization objective is to maximize the difference between the value of electricity generated by the plant and the cost of the fuel consumed by the plant which satisfies the space heating demands. This is expressed by the following equation:
冘冘 J
max
I
Z⫽
(C(i)p(j,i)⫺VB(j,i))⌬t.
(7)
j⫽0 i⫽0
Eq. (7) considers J days of a space heating season with each day divided into I time intervals, so the hours of each time interval ⌬t are equal to 24/I. The fuel consumption of the CHP plant at the kth time interval B(k) is a function of the heat output q(k) and the power p(k) at this time interval. Current practice is that the function is a generalization of a standard quadratic equation that is quadratic in both the heat and the electrical output. The coupling between the electrical and the heat productions is included through a coupled-term that is the product of q(k) and p(k). The fuel consumption function is therefore [4]: B(k)⫽a1⫹a2p(k)⫹a3p(k)2⫹a4q(k)⫹a5q(k)2⫹a6p(k)q(k).
(8)
Fig. 5 shows the heat-power operating region of an extraction unit. This region is enclosed by the boundary curve ABCDE. Along curve AB, the unit runs with maximum electrical power output while the maximum fuel is consumed along curve BC. Curve CD represents the maximum heat extraction, while curve DE represents the minimum fuel consumption. Along curve EA, the unit runs with no heat extraction. These linear curves are described by: alp(k)⫹blq(k)ⱖcl l⫽1⫺5,
(9)
where al, bl and cl are the constant coefficients of the inequalities which are determined by the operating characteristics of the extraction unit. The relationship between the CHP plant and the space heating system is given by: q(k)⫽Gcp(Ts(k)⫺Tr(k))
Fig. 5. Heat-power operating region for an extraction unit.
(10)
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For a space heating system, some variables are not only restricted by Eqs. (4) and (6), but also by: TrⱕTs(k)ⱕTs,max
(11)
Tro,minⱕTro(k)ⱕTro,max.
(12)
and
In this paper, Ts,max is 120°C while Tro,max and Tro,min are 20°C and 16°C, respectively, which are the acceptable room temperature extremes in China. The mathematical model for optimizing the operation of a CHP plant consists of Eqs. (4, 6–12). 3.3. Solution of the mathematical model Since the model includes two discrete time differential equations, i.e. Eqs. (4) and (6), the output variables shown in Fig. 2 at the current time interval are strongly related to the input variables at previous time intervals. The mathematical model is a dynamic non-linear optimization problem with discrete time intervals. The dynamic optimization problem for CHP plant operation has been solved in many different ways, including the non-linear method [10], mixed-integer linear programming [7] and dynamic programming [6,11]. Analysis of the long-term optimal operation over one space heating season is a large-scale optimization problem, so the general optimization methods, i.e. Newton-based methods, are not practical because of the relatively long calculational time. If the time series Eqs. (4) and (6) are changed to state–space equations and the state variables in the equations are divided into discrete values, the dynamic programming approach can be used to divide the large-scale problem into small-scale problems with sequential relations, which can be solved rather simply. Therefore, a dynamic programming method is used [17] combined with the penalty function method to solve the optimization problem. The output variables must first be related to the input variables of the space heating system shown in Fig. 2 by state–space equations. Eqs. (4) and (5) are rewritten as:
冘
冘
I
Tr(k)⫽
I
aiTr(k⫺i)⫹
i⫽1
biu(k⫺i)
(13)
i⫽0
and
冘 J
Tro(k)⫽
j⫽1
冘 J
qjTro(k⫺j)⫹
cju(k⫺j),
j⫽0
where u(k)⫽[Ts(k) Te(k)]T bi⫽[bi gi] cj⫽[jj wj] Now choose the state variables as:
(14)
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x1(k)=Tr(k)−h0u(k)
xI+1(k)=Tn(k)−hI+1u(k)
x2(k)=x1(k+1)−h1u(k)
xI+2(k)=xI+1(k+1)−hI+2u(k)
%%%%%
%%%%%
xI(k)=xI−1(k+1)−hI−1u(k) xI+J(k)=xI+J−1(k+1)−hI+Ju(k) where the coefficients h0⫺hI+J are described as:
冦
冦
h0=b0
hI+1=c0
h1=b1+a1h0
hI+2=c1+q1hI+1
%%%%%
%%%%%
hI=bI+a1hI−1+%+aI−1h1+aIh0 hI+J+1=cJ+q1hI+J+%+qI+Jh1+qIh0
So Eqs. (4) and (5) are changed to the state equation as: X(k⫹1)⫽AX(k)⫹B쐌u(k),
(15)
where X(t)=[x1(k) x2(k)…xI+J(k)]T, B=[h1 h2…hI+J]T
冋 册
A⫽
A1 0
0 A2
冤
0 1
0
% 0
0 0
1
% 0
A1⫽ ⯗ ⯗
⯗
⯗
⯗
% 0
1
0 0
aI aI−1 % a2 a1
冥 冤
0 1
0
% 0
0 0
1
% 0
A2⫽ ⯗ ⯗
⯗
⯗
⯗
% 0
1
0 0
I×I
qJ qJ−1 % q2 q1
冥
J×J
The output equations are: Tr(k)⫽x1(k)⫹h0u(k)
(16)
Tro(k)⫽xI+1(k)⫹hI+1u(k).
(17)
and
With the state equation and the output equations replacing Eqs. (4) and (5), the control variables in the mathematical model are then the element Ts(k) in the 2×1 matrix u(k) and p(k). Another element Te(k) in the matrix can be given by a forecast weather model [18]. The other control variables are the electrical power p(k), which varies independently over the different time intervals, and the supply water temperatures Ts(k) at the different time intervals which are interrelated by the state equation. Therefore, the solution of the optimization model is carried out in two steps. In the first step, p(k) is optimized by a common optimization method, i.e. the penalty function method. In the second step, Ts(k) is optimized by a dynamic programming method. In the dynamic programming method, the original optimization problem is transformed to a
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293
multistage optimization problem. By introducing a function Gk(X(k),u(k),p(k)) as the optimization objective at the kth time interval, Eq. (7) can be changed to:
冘 N
Z⫽ max
Gk(X(k),u(k),p(k)),
(18)
k⫽0
where N=I×J. The optimization objective from the kth to the Nth time interval is expressed as
再冘 N
ZN−k(X(k),X(N))⫽max
u(k)u(N) p(k)%p(N) k⬘⫽k
冎
Gk⬘(X(k⬘),u(k⬘),p(k⬘)) .
(19)
The following recurrence equation is obtained on the basis of the principle of optimality: ZN−k(X(k),X(N))⫽max
{Gk(X(k),u(k),p(k))⫹ZN−k−1(X(k⫹1),X(N))}.
(20)
u(k) p(k)
The analytical solution can not be obtained by the direct application of Eq. (20) due to the complexity of the model. Thus, the following numerical solution is required to solve the problem. At the kth time interval, the state vector X(k) and the control variable Ts(k) are divided into a discrete set of Xl(k) and Tsm(k), within the region limited by Eqs. (8)–(12), where l varies from 1 to L and m from 1 to M. For a specified pair Xl(k) and Tsm(k), the optimized control variable pl,m(k) is obtained using the penalty function method [17] taking the following equation as the optimization objective. Z⬘⫽ max Gk(Xl(k),um(k),pl,m(k)).
(21)
p1,m(k)
Then, for the state vector Xl(k), the optimized control variables Tsm⬘(k) and pl,m⬘(k) are obtained by solving the recurrence relation Eq. (20). Each Xl(k) with l from 1 to L and the relevant optimized control variables Tsm⬘(k) and pl,m⬘(k) are then stored. The procedure is then completed for every time interval from the Nth to the 0th time interval. Finally, the state Eq. (15) is solved step by step from the 0th to the Nth time interval to obtain each state vector X(k) and the relevant optimized control variables Ts(k) and p(k) for k=0,1,…N. Fig. 6 shows the algorithm used to determine the optimal operation.
4. Case study The CHP plant considered here has an extraction steam turbine. The turbine is a CC12–50/10/3 type with rated power output of 12 MW, inlet pressure of 4.9 MPa and condensing pressure of 0.005 MPa. Low-pressure extraction from the turbine for space heating is done at the rated pressure of 0.3 MPa. High-pressure extraction flow is zero during the space heating season. The turbine isentropic efficiency is 0.79. The boiler steam flow rate to the turbine varies from 50% to 100%
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Fig. 6.
Flow chart for optimal operation.
of the rated output. If Te(k) is lower than ⫺15°C, a peak boiler bears the extra space heating load. The Shenhai space heating system in Shenyang is selected with a DH network water flow of 500 t/h and network losses of 5%. The coefficients of Eqs. (4) and (5) which express the relationship between the output variables and the input variables of the space heating system are the same as shown in Table 2. The calculation considers an entire space heating season, i.e. November 16 to March 31 of the next year, with a day divided into six time intervals. So J=138 and I=6 in the optimization objective Eq. (7). Since the orders in Eqs. (4) and (6) are 2 and 1 respectively, the state variable is X(k)=[x1(k) x2(k) x3(k)]T and the coefficients of state Eq. (15) are: A1⫽
冋 册 0 1
a2 a1
A2⫽q1,
while Eq. (17) is simplified as: Tro(k)=x3(k). More electricity should be generated with less heat output during the time intervals of high time-of-day prices while less electricity should be produced with more heat output during the time intervals of low time-of-day prices. The optimization result confirmed this expectation. During several days of peak space heating load with very low environmental temperatures, the unit operates at or near point C in Fig. 5 with the largest extraction. With this condition, the heat output of the unit reaches the maximum, i.e. 34.95 MW while the electrical power is 6.60 MW.
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Fig. 7. (a) Variation of p and q during a week with maximum heat load. (b) Variation of p and q during a week with medium heat load. (c) Variation of p and q during a week with minimum heat load.
As a result, the CHP plant can not operate as a peak load regulating plant in the three days from the 1st to 9th time intervals in Fig. 7a because the unit runs only at point C with constant electrical output. The unit begins to increase its electrical output when Te increases. First, the electrical power during the time interval of the highest time-of-day price increases while the heat output decreases, as shown in Fig. 7a. As Te gradually increases, the operating state of the unit moves along curve ABC of Fig. 5 from C toward A where the electrical power is the maximum and the
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Fig. 8. Variation of Tro during half of a space heating season.
heat output is zero. Then, as the space heating demand decreases further, the operating states during other time intervals of high time-of-day prices tend towards the production of more electricity. The unit always operates at or near point C during the time intervals of low time-of-day prices in Fig. 7a–c. Fig. 7c shows the variation of the electrical power and heat output of the CHP plant during the period when space heating demand is very low. An important concern with peak load regulating plants is the operational safety. Since the unit runs during the whole season on curve ABC with the inlet steam into the turbine always kept at or near the maximum, the CHP can run safely with this operating strategy. For the optimized operation of the CHP plant, the variation of the room temperature Tro of the heated buildings during the later half of a space heating season is shown on Fig. 8. Tro changes during a day within a range of less than 0.5°C, while it varies over the whole space heating season within the 16.0–17.2°C range which is acceptable in China. Fig. 9 shows the variation of the supply water temperature, Ts, and the return water temperature, Tr, in a typical week when the CHP plant operates with its heat output and electrical power varying as given in Fig. 7b. The highest supply water temperature of the DH network is 103.2°C and the lowest is only 28.4°C. The return water temperature varies within the 45.9–26.8°C range. The optimal operation of the CHP plant is compared in Table 3 with a reference operating strategy which has constant inlet steam flow into the turbine, while the extraction provides enough heat output to maintain constant building room temperature of 16°C. The variation of the heat and electrical production is given over the six time intervals of a typical day in Table 3. The difference between heat and electrical production during the time intervals of high time-of-day prices and that of low prices with the optimal operation is much larger than that with the reference mode of operation. The total electricity value is defined as the sum of the electrical production
Fig. 9. Variation of Ts and Tr during a week with medium heat load.
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Table 3 Comparison of optimal and reference operations Optimal operation Time interval C (yuan/kWh) 0:00苲4:00 0.283 4:00苲8:00 0.335 8:00苲12:00 0.770 12:00苲16:00 0.701 16:00苲20:00 0.566 20:00苲24:00 0.413 Total Total Electricity Value (×104 yuan) Average Electricity Value (×104 yuan/kWh) TEP (×104 yuan) EP (×104 yuan/kWh)
Q (GJ) 70479.0 70138.9 14472.0 31881.6 55242.4 66599.3 308813.2 1539.6 0.550 617.6 0.221
Reference operation P (MW.h) 3746.3 3762.7 6459.3 5614.1 4483.2 3933.8 27999.0
Q (GJ) 59507.4 61478.1 51804.3 39213.2 43196.8 51556.8 306756.6 1493.4 0.524
P (MW.h) 4268.6 4198.3 4744.3 5403.8 5181.3 4713.8 28510.1
496.6 0.174
multiplied by the time-of-day price at each of the six time intervals. The average electricity value is defined as the ratio of the total electricity value to the electrical power which reflects the value of the electricity per kWh generated by a CHP plant in a space heating season. It can be seen in Table 3 that the average electricity value with the optimal operation is 5% higher than that with the reference mode of operation. The advantage with optimal operation is not very great because the simplified CHP plant for the case study can vary the heat output and the power output within only a relatively small range. Another reason is that the environmental temperature, Te, during the time intervals of low time-of-day price is often low, so the heat output with the reference operation is large during these intervals to keep a constant room temperature. As a result, the power output during the time intervals of low time-of-day price is small and the average electricity value increases accordingly. The usual practical operating mode in China is that the heat output of a CHP plant for space heating changes little during the day, and the power output of the CHP plant also changes very little. Thus, the average value of the electricity produced with the practical operating mode is smaller than that with the reference mode. The total electricity profit (TEP) in Table 3 is defined as the value Z of the optimization objective in Eq. (7). The electricity profit (EP) is defined as the ratio of the total electricity profit to the total amount of power generated by a CHP plant, which reflects the profit of the electricity per kWh generated by the CHP plant in a space heating season. Here, the electricity profit with the optimal operation is 27% higher than that with the reference operation, which shows that the optimal operation of a CHP plant does have an evident advantage. 5. Conclusion Due to the large thermal mass of the heated buildings and the DH water circulation network, operation of a CHP plant for space heating can be optimized as a peak load regulating plant of the electric utility grid.
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