Heat production optimization using bio-inspired algorithms

Engineering Applications of Artificial Intelligence 76 (2018) 185–201

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Heat production optimization using bio-inspired algorithms Marcin Woźniak a, *, Kamil Książek a , Jakub Marciniec b , Dawid Połap a a b

Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland Faculty of Energy and Environmental Engineering, Silesian University of Technology, Konarskiego 18, 44-100 Gliwice, Poland

ARTICLE

INFO

MSC: 90C59 74F05 78M50 76B75 97R40 Keywords: Bio-inspired methods Thermal systems optimization Heat exchangers and pumps Computational intelligence

ABSTRACT Energy efficiency of industrial systems is one of key features for optimal use of resources and the lowest costs of energy for users. In the recent time optimization of heating plants and heat distribution systems becomes an important venue for novel methods and innovative constructions. Various proposals can be seen for more efficient performance of heating systems in changing weather conditions. In this article results of using bio-inspired methods for intensification of the district heating plant to work with maximum efficiency at the lowest costs are presented. The research is focused on developing bioinspired approaches for a mathematical model of a district heating plant in various weather conditions. The research model represents a sample district heating plant, in which circulation of hot water is performed in two heat exchangers supplied by controlled pumps. The system was calibrated with the use of proposed Polar Bear Optimization and the results were compared to one of best known heuristics, Particle Swarm Optimization. An objective function describing the operation of the plant was developed and found applicable for proposed bio-inspired approach. The research results have shown that proposed methodology is efficient for all simulated weather conditions and various boundary conditions. Comparison the obtained results with non-optimal parameters confirms huge profits from applying right settings of the system.

1. Introduction Heating plants and their efficiency in different conditions are influenced by many factors. Various analyses are provided to show which aspects of the weather and which technical elements in heating plants influence the overall evaluation. Effectiveness of a power plant can be maximized by means of modifications in structure, application of more efficient subsystems and thoroughly controlled operating parameters. The research reports many advances in these aspects, both by using new technologies for improvements in construction and by application of intelligent algorithms in search of the best available parameters adjustment. The analysis is done in various countries, and therefore it is possible to compare the results to draw conclusion on the influence of weather conditions. Bauer et al. (2010) discussed aspects of heating plant adjusted for seasonal changes in weather in Germany. Palander (2011) presented an analysis of power generation from natural fuels on the example of one of the Finnish heating plants. In Turanjanin et al. (2009) was discussed possibility to replace traditional fuels by the solar energy used in Belgrade heating plant, and Zago et al. (2011) presented an analysis of the heating plant efficiency in northern Italy. These articles have

shown how the weather conditions can influence on the use of heating systems. Since each of these plants was located in other place, various climate features can be noticed. However, the environmental aspects of thermal technology are similarly important. Proposals of technological developments to heating plants give ideas which elements can take changes for higher conversion rates. Construction characteristics for a district heating plant which can be changed for reduced ash production and more efficient chemical conversion were proposed in Dahl et al. (2009) and Pöykiö et al. (2009). In Çomaklıet al. (2004) was discussed the influence of the distribution network on losses in the energy balance. Beside to technological development it is essential for any thermal plant to operate with highest possible efficiency and lowest costs, regardless of fluctuating heat demand. The statistical study of various aspects that define control conditions of heating systems was presented in Kuosa et al. (2013). The authors concluded some interesting ideas for possible improvements of operation. In Aringhieri and Malucelli (2003) optimal settings for a district heating plant were discussed. Each of heating plants needs a control strategy, which is not only to be adjusted to weather conditions, but also to the technical conditions of components and current demand from users. All these may

*

Corresponding author. E-mail addresses: [email protected] (M. Woźniak), [email protected] (K. Książek), [email protected] (J. Marciniec), [email protected] (D. Połap). https://doi.org/10.1016/j.engappai.2018.09.003 Received 8 April 2018; Received in revised form 13 August 2018; Accepted 8 September 2018 0952-1976/© 2018 Elsevier Ltd. All rights reserved.

M. Woźniak et al.

Engineering Applications of Artificial Intelligence 76 (2018) 185–201

Fig. 1. Block flow diagram of the analyzed district heating plant system, in which three controlled pressure pumps supply the returned cold water from the district network to two heat exchangers which warm the water up in two different pressure conditions and forward it back to the network to condition connected buildings.

used to predict features of potential errors and malfunctions. Cheng et al. (2017) described a method based on evolution. The results have shown advantages of nature-inspired algorithms in complex engineering problems. In Hong and Ryu (2017) a simulation method for warship decoy system with the use of a genetic algorithm was described. Dynamic characteristics of electric vehicle in motion were found with the use of heuristic methodology in Woźniak and Połap (2017). In Sangdani et al. (2018) a torque control system for a tracker robot was simulated by a genetic algorithm. Comparative studies on selected heuristics in engineering problems can be also found. In Yıldız and Yıldız (2018) was presented a use of this methods for optimization of vehicle engine connection rod, while in Pholdee et al. (2017) for automotive floor-frame forming. Heuristic algorithms are also very efficient in manufacturing optimization as presented in Yıldız and Yıldız (2017) and miling operations (Yildiz, 2013). Biswas et al. (2018) presented how to use evolutionary algorithm for power flow optimization, and in Tao et al. (2018) similar power optimization problems were solved by using a genetic algorithm. Bio-inspired approaches are also applied for multi-objective optimization in power systems, where optimization is not only constrained due to system composition but vary with the changing demands. In Nguyen and Vo (2017) a hydro-thermal efficiency for lower operation costs of the station was examined by using a cuckoo algorithm to find a balance between operating constraints and lowest emission. While Dubey et al. (2018) presented a comparative overview of recent advances in using bio-inspired methods for managing wind power dispatch. Evolutionary computing is also reported to be efficient

Fig. 2. A schematic organization of a sample district heating system with installed heating plant and connection network.

change and therefore a control strategy must be flexible as proposed in Gustafsson et al. (2010). There are many possible ways to optimize the plant. Recently Computational Intelligence (CI) is reported to succeed in complex optimization problems. Wu et al. (2017) presented risk analysis for complex engineering problems, where CI methods were

Fig. 3. Process flow diagram of analyzed heating plant system. Cold water from distribution network is compressed by three pumps, which supply it to heat exchangers working at 2.0 bars and 6.2 bars respectively. Hot water is returned from the plant to the distribution network and forwarded to users. 186

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201

in computer aided detection (Morra et al., 2018) and multi-objective problems of engineering systems (Russo et al., 2018). Meanwhile in Liao et al. (2018) bio-inspired methods were used to minimize the power consumption for an application of luminance control.

2.2. Analyzed system The analyzed system consists of two heat exchangers and a pumping system of three identical water pumps working in parallel, see Fig. 3. Heat production in water heaters consumes steam at two levels of pressure. Returned water flows into the system (0). Then stream flows through one, two or three identical pumps (1a, 1b, 1c, depending on the system settings) and pump control valves (2a, 2b, 2c). After mixing, water goes to the first heat exchanger (3). A part of the stream (from 10% to 100%) is heated (3a) by using steam at 2 bar (steam saturation temperature: 120 ◦ C). The remaining part of the water flows through the first by-pass (3b). After that, heated (4a) and by-passed (4b) streams are mixed (5). The stream again is split: a part of the water (from 10% to 100%) flows through the second heat exchanger (5a) and is heated by condensing steam at 6.2 bar absolute pressure (steam saturation temperature is higher: 160 ◦ C). The part of water omitting the second exchanger (5b) flows through the second by-pass control valve. The heated (6a) and the by-passed water (6b) is mixed and sent out to the heating network. The aim of this study is to find the optimal feasible configuration of operating parameters for pumps, control valves and heat exchangers. The presented system of pumps and heat exchangers allows for a wide-range manipulation, which is advantageous. It is capable of ensuring hot water for several thousands of houses (a large estate).

1.1. Related works Bio-inspired methods are strategies sourced in nature, which map behavior of animals into computer algorithms. Some of these were applied for optimization of heating lines. A configuration of a district heating network with the use of a genetic algorithm was presented in Li and Svendsen (2013). In Peimankar et al. (2017) bio-inspired methods were reported to be used for optimization of power transformers, where some of the operational characteristics were adjusted for optimal control of the system. In Rey and Zmeureanu (2017) a solar thermal combisystem was intensified with the use of Particle Swarm Optimization. The authors reported that this heuristic is efficient for adjusting solar thermal systems, however in contrary to our approach, a heuristic method was used to optimize a two-storey house heated with solar energy. Bio-inspired methods are also reported to co-work with neural networks for efficient diagnosis of thermal plants. Talaat et al. (2018) presented a model of diagnostic system for gas turbine in a power plant. Geothermal aspects of Organic Rankine Cycle at thermal power plant in Glewe in Germany were analyzed for temperatures about 40– 50 ◦ C in the structure. The authors of Habka and Ajib (2013) pointed out which elements of the system are co-related for higher efficiency of the temperature conversion. In Fig. 1 can be seen a sample presentation of processes that are performed in a district heating plant which is used for conditioning of the temperature in flats in winter season. The schematic idea of this installation is presented in Fig. 2. For this purpose bio-inspired approach have been used. A new method of optimization based on simulation of polar bears hunting habits was proposed. For this approach a devoted cost function which describes operation of the heating plant have been developed. In the research was used our method to minimize the objective function to adjust the system for the minimal total unitary cost of heat production with satisfying boundary conditions. The results of our methodology are compared to one of the most popular bioinspired methods. It is possible to see that Polar Bear Optimization gives better results and shows high efficiency in optimization. The novelty of our research is not only in the proposed methodology with devoted fitness model but also in the use of bio-inspired methods to adjust the district heating plant system. The research was done for four different temperature conditions, which represent demand for efficiency in harsh winter, typical winter weather, mild winter, and out-of-season sanitary hot. In all these conditions the proposed methodology gave good results.

2.3. Pump operation Pumps are used to move the water and raise its pressure. The pressure must be increased due to hydraulic pressure losses in district heating water loop. Pressure rise (𝛥𝑝) of a particular pump is related to its volumetric capacity (𝑉̇ ). This nonlinear relation is called pump characteristic (Lobanoff and Ross, 2013). For practical use, pump characteristic is usually approximated by a square function 𝛥𝑝 = 𝐴𝑉̇ 2 + 𝐵 𝑉̇ + 𝐶.

(1)

where 𝐴, 𝐵 and 𝐶 are empirical constants of pump characteristics. Volumetric flow is equal to mass flow 𝑚̇ divided by fluid density 𝜌 𝑚̇ 𝑉̇ = . 𝜌

(2)

Therefore by substituting 𝑉̇ for ( 𝛥𝑝 = 𝐴

𝑚̇ 𝜌

)2

( +𝐵

𝑚̇ 𝜌

𝑚̇ 𝜌

it is possible to obtain

) + 𝐶.

(3)

The hydraulic resistance of water flowing through hydraulic system elements (i.e. pipelines, heat exchangers) can be predicted based on a set of known reference state parameters (𝛿𝑝0 , 𝑚̇ 0 ) (Shah and Sekulic, 2003) ( )2 𝑚̇ 𝛿𝑝 = 𝛿𝑝0 ⋅ , (4) 𝑚̇ 0

2. The model of a district heating plant 2.1. Combined heat and power plants Heat is very often produced together with electricity. Such solution, also referred to as cogeneration, is very efficient from thermodynamic point of view. Generation of electric energy and heat in the same process consumes considerably less energy of fuels than separate production. Majority of heat and power plants use steam condensation to heat up district heating water. Steam produced in coal- or gas-fired boiler runs a steam turbine and consequently, an electrical generator. Meanwhile, some portion of the steam is extracted from the turbine at various levels of pressure. This extraction steam is utilized either in the heat regeneration system (to improve cycle efficiency), or in dedicated water heaters (to produce the heat). Moreover, heat is a product of significantly lower market price than electricity. Extraction of steam for heat production implies lower generation of electricity. The resulting negative economic effect is hereinafter referred to as steam cost. Also, a part of an electric output of the plant is consumed in pump drivers.

where 𝛿𝑝 is pressure drop in the system for mass flow 𝑚, ̇ 𝛿𝑝0 is pressure drop for the reference state (constant value) and 𝑚̇ 0 is mass flow at the reference state (constant value). In a steady-state operation, pump pressure rise is equal to the total pressure drop of all elements (pipeline, valves, nozzles, orifices, heat exchangers etc.) 𝛥𝑝 =

∑

𝛿𝑝𝑖 .

(5)

𝑖

This condition corresponds to the intersection of pump characteristic curve and system hydraulic resistance curve, as presented in Fig. 4. The pump driver electric power can be calculated as 𝑁𝑒𝑙 = 187

𝑉̇ ⋅ 𝛥𝑝 , 𝜂𝑖𝑃 ⋅ 𝜂𝑒𝑚

(6)

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where 𝑄̇ is heat duty (kW), 𝑐𝑝 is isobaric specific heat ( kgkJK ), 𝑇𝑖𝑛 , 𝑇𝑜𝑢𝑡 are water inlet and outlet temperatures, respectively. On the other hand, the heat duty of exchanger is limited by streams temperatures, heat capacities and complex heat transfer phenomena. For this particular type of heat exchanger (condenser), the heat duty can be calculated based on inlet and outlet temperature differences between water and steam (Shah and Sekulic, 2003). Let 𝑇𝑠 be the condensing steam saturation temperature. It can be stated that [ ] ( ) 𝑈 ⋅𝐴 𝑇𝑠 − 𝑇𝑜𝑢𝑡 = 𝑇𝑠 − 𝑇𝑖𝑛 ⋅ 𝑒𝑥𝑝 − , (9) 𝑚̇ ⋅ 𝑐𝑝 where 𝑈 is overall heat transfer coefficient ( mW 2 K ), 𝐴 is heat transfer area (𝑚2 ). The heat transfer coefficient 𝑈 for a condenser depends on mass flow of heated water in a similar manner as the pressure drop (Eq. (4)) ( )𝑘 𝑚̇ 𝑈 = 𝑈0 , (10) 𝑚̇ 0 Fig. 4. Pump and hydraulic system curves intersection.

where 𝑈0 is overall heat transfer coefficient (for a reference state). Because the type of fluid flow inside tubes is turbulent, the empirical power factor 𝑘 = 0.8. In analyzed case, only a part of main water stream flows through the heat exchangers, see Fig. 3. The remaining part is by-passed, throttled at a valve and then mixed with water from the outlet of heat exchangers. The ratio of heated water flow to the total flow is a split factor 𝑅. For 𝑖th stream split it is defined as 𝑅𝑖 =

(11)

where 𝑚̇ ℎ is water mass flow through the exchanger and 𝑚̇ 𝑏 is water mass flow through the by-pass. Heat demand of a district heating system depends on outdoor conditions such as: air temperature and wind speed. Hence the heat production changes over time within a wide range of heat duties and required forward water temperatures. It is essential to find the least costly configuration for every possible operating point. With the use of presented model, several off-design operating states were analyzed in search of global minimum of heat production cost for each case.

Fig. 5. Shell and tube condenser in schematic representation. 3

where 𝑁𝑒𝑙 is electric power (kW), 𝑉̇ is volumetric flow ( ms ), 𝛥𝑝 is pressure rise (kPa), 𝜂𝑖𝑃 is pump adiabatic efficiency and 𝜂𝑒𝑚 is pump driver electrical and mechanical efficiency. The adiabatic efficiency 𝜂𝑖𝑃 depends on pump operating point. It is a function of pump capacity and rotational speed. For constant speed, a square function approximation is usually performed ( )2 ( ) 𝑚̇ 𝑚̇ 𝜂𝑖𝑃 = 𝐷 +𝐸 + 𝐹, (7) 𝜌 𝜌

2.5. Simplifying assumptions and accuracy of simulation For the purpose of this study following assumptions were applied 1. Water stream is equally distributed among pumps in operation. 2. Water density (𝜌) is assumed constant (independent on temperature and pressure). 3. Specific heat of water (𝑐𝑝 ) is constant (independent on temperature and pressure). 4. Pressure drop in heat exchangers does not affect heat transfer effectiveness. 5. Heat losses to the environment are neglected. 6. Return water parameters are constant, independent of system operation.

where 𝐷, 𝐸 and 𝐹 are empirical constants. The pump efficiency function usually reaches its maximum at nominal operating parameters. 2.4. Heat exchangers Heat exchangers are devices used to transfer heat from one medium to another. In the presented case, a steam-to-water heat exchanger is considered. Condensing steam is separated from water by a thin metallic wall. Steam-powered water heaters are usually shell-and-tube type condensers, see Fig. 5. Water flows through a bundle of tubes inside a sealed shell. Condensation of steam takes place on the outer surface of tubes. As a result, steam turns into condensate and drips downwards, while water is heated up inside the tube bundle. Steam maintains its temperature at a constant level, dependent on pressure (Cengel, 2014). If the heat capacity of water is assumed constant, the heat gained by the water stream in a water heater can be expressed in terms of an energy balance ( ) 𝑄̇ = 𝑚̇ ⋅ 𝑐𝑝 𝑇𝑜𝑢𝑡 − 𝑇𝑖𝑛 ,

𝑚̇ ℎ 𝑚̇ ℎ = , 𝑚̇ 𝑡𝑜𝑡 𝑚̇ 𝑏 + 𝑚̇ ℎ

Models based on these assumptions are expected to provide results of sufficient accuracy for this level of analysis. Application of more rigorous heat transfer and fluid dynamics models is possible. However, this might result in excessively long computation time with no serious change of the final results. 3. Bio-inspired methods Two bio-inspired methods have been selected for the studies. Polar Bear Optimization (Połap and Woźniak, 2017) is one of the recent algorithms. It was inspired by the way in which polar bears hunt.

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Particle Swarm Optimization (Kennedy and Eberhart, 1995) is a wellknown method which is based on a behavior of an intelligent swarm. Comparison of these two methods will show which type of heuristics may bring better results in this type of engineering problems. 3.1. Polar bear optimization Polar Bear Optimization (PBO), one of the newest bio-inspired algorithms, is based on hunting strategy of polar bears in the Arctic. These mammals have to hunt and survive in harsh polar conditions. They have developed an efficient way of hunting for seals. The domain for optimization is Arctic territory. Search points are called polar bears and the global optimum is a location of a victim. One can isolate three following stages of this algorithm. The first one is connected with tracking of seals in large areas. Polar bears can drift on ice floes and look out for an opportunity to hunt. This phase is called exploration of the domain: bears do not exactly know where they can expect promising results so they just roam the land in search. When they find an approximate localization of a seal they circle around and try to catch. These steps are associated with exploitation of the domain. In the algorithm it is realized by trifolium equation. The last stage consists of a mechanism to control the size of a population. During following iterations weak individuals are removed but not always are replaced with new ones. This strategy helps to avoid unnecessary calculations for weak agents.

Fig. 6. Chart of a single trifolium leaf depending on the parameter 𝑎.

where 𝑥𝑖𝑗 is 𝑗th coordinate of 𝑖th agent, 𝑈 [0, 1] is a pseudo-random number from the range [0, 1] (in the case of 𝑈 [0, 𝜔𝑗 ] it is interval [0, 𝜔𝑗 ]). First equation from Eq. (15) is responsible for calculating Euclidean distance between coordinates of current and the fittest point. The second one is a kind of random walk in a promising direction. If a new location is worse than the previous one, the bear returns to that position. Consecutive 20% of fittest bears (from 21% to 40%) execute additional local search around current optimum in accordance with

3.1.1. Proposed PBO application The algorithm is starting from random generation of positions for a population consisting of 75% ⋅ 𝑛 points called polar bears 𝑋𝑖

=

{𝑥𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 ..., 𝑥𝑖𝑚 },

𝑥𝑖𝑗 = 𝑥𝑏𝑒𝑠𝑡 + 𝑈 [−𝛾𝑘 , 𝛾𝑘 ], 𝑗

where 𝛾𝑘 is a given parameter in 𝑘th iteration. During subsequent steps 𝛾 is narrowing (exploitation of domain). In the last stage a mechanism to control the size of the population is used. At the beginning there were created only 75% ⋅ 𝑛 individuals in the population. This value is changing at the end of each iteration. Initially, it is necessary to calculate a difference between 𝑛 and present number of bears (it is denoted by 𝑡). Then one should determine pseudorandom value 𝜅, 𝜅 ∈ [0, 1]. If the value of 𝜅 is less than 0.25, there is run death mechanism. Now it is calculated an auxiliary value called 𝑙𝑖𝑚𝑖𝑡 according to

(12)

𝑥𝑖𝑗

where is 𝑗th coordinate of 𝑖th point, 𝑚 is total number of coordinates. After dispersion around the whole domain bears look for potential victims in the neighborhood. This behavior is modeled by using trifolium equation (13)

𝑟 = 4𝑎 ⋅ cos(𝜑) ⋅ sin(𝜑),

(16)

𝜋 ]. 2

Two described parameters: 𝑎 and 𝜑 where 𝑎 ∈ [0, 0.3] and 𝜑 ∈ [0, are selected randomly for each point. Fig. 6 demonstrates the shape of single trifolium leaf for various parameters 𝑎. Then every agent reaches new position according to the following system of equations ) ( 𝑗−1 ∑ ⎧ 𝑘 𝑘 sin(𝜑𝑞 ) + cos(𝜑𝑗 ) , ⎪𝑥𝑗 = 𝑥𝑗 ± 𝑟 ⋅ ⎪ 𝑞=1 𝑚 ⎪ ∑ (14) ⎨𝑥𝑘 = 𝑥𝑘 ± 𝑟 ⋅ sin(𝜑𝑞 ), 𝑚 ⎪ 𝑚 𝑞=1 ⎪ ⎪ ⎩𝑗 ∈ {1, … , 𝑚}.

𝑙𝑖𝑚𝑖𝑡 = 𝑠𝑖𝑧𝑒 − ⌊0.5 ⋅ 𝑛⌋ − 1,

(17)

where 𝑠𝑖𝑧𝑒 is the total number of bears in the population. It is assumed that 𝑠𝑖𝑧𝑒 cannot be less than 50% ⋅ 𝑛. The last step is based on random choosing an integer (called 𝑝𝑑𝑒𝑎𝑡ℎ ) from the set: {1, 2, … , 𝑙𝑖𝑚𝑖𝑡}, where 𝑝𝑑𝑒𝑎𝑡ℎ weakest bears are removed from population. If the value of 𝜅 is greater than 0.75, it follows a reproduction phase. One should determine pseudo-random integer (called 𝑝𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 ) from the set: {1, 2, … , 𝑟}, where 𝑟 is difference between 𝑛 and 𝑠𝑖𝑧𝑒 for which 𝑝𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 new individuals are created according to 𝑋 𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 =

where 𝑥𝑘𝑗 is 𝑗th coordinate of 𝑘th point, 𝜑𝑞 is generated randomly from the interval [0, 2𝜋]. It should be explained the meaning of sign ±. The bear initially checks if a new position is more promising than the last one. If this presumption is confirmed (lower value of objective function), he will move in that direction (‘‘+’’). Otherwise, he returns to starting position and checks place on the other side (‘‘−’’). After assessing the value of a new location he approves or rejects the new area using the objective function. Bears can drift on ice floes for a long time. Thanks to this, they can explore new unknown territories. In this article a devoted version of PBO is presented. There is a new condition that 20% points with the lowest value of objective function can drift in a direction indicated by currently the best bear according to following equations { 𝜔𝑗 = 𝑥𝑏𝑒𝑠𝑡 − 𝑥𝑖𝑗 , 𝑗 (15) 𝑥𝑖𝑗 = 𝑥𝑖𝑗 + 𝑠𝑖𝑔𝑛(𝜔𝑗 ) ⋅ 𝑈 [0, 1] ⋅ 𝑈 [0, 𝜔𝑗 ],

𝑋 𝑏𝑒𝑠𝑡 + 𝑋 𝑖 , 2

(18)

where 𝑋 𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 is a new polar bear, 𝑋 𝑏𝑒𝑠𝑡 is the best bear in the population and 𝑋 𝑖 is a randomly chosen bear out of 20% the fittest ones (except the best). These operations help to save the time of calculations and correspond to the processes occurring in nature. The process is presented in Algorithm 1.

3.2. Particle swarm optimization The general idea of Particle Swarm Optimization (PSO) is based on the way in which bird flocks or fish schools move. In this algorithm, the population is called a swarm and each point is a particle. Each particle during consecutive iterations makes a compromise between her own best position in the history, the fittest position of the swarm and a random search. 189

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201

Algorithm 1 Pseudocode of Polar Bear Optimization Algorithm Input: number of bears: n, number of iterations: T, narrowing parameter: 𝛾, narrowing force parameter: 𝛼, 𝛼 ∈ (0, 1), objective function: 𝛷 from Eq. (28) Output: coordinates of the optimum with value of objective function 𝛷 Initialization: Create and evaluate random initial population of 𝑆 = ⌊0.75 ⋅ 𝑛⌋ bears. The main phase: 𝑘=1 while 𝑘 < 𝑇 do for 𝑖 = 1 to 𝑆 do Calculate trifolium leaf radius according to Eq. (13). Generate random values of 𝜙, 𝜙 ∈ [0, 2𝜋]. Calculate new position of the bear according to Eq. (14) with sign ‘‘+’’. if New location is worse than the previous one then Calculate new position of the bear by using Eq. (14) with sign ‘‘−’’. if New position is worse than starting location then Turn back to the starting position. else Accept new location of the bear. end if else Accept new location of the bear. end if end for Sort population of bears in order of objective function (from the fittest to the weakest one). for 𝑖 = 1 to ⌊0.2 ⋅ 𝑆⌋ do Determine new position of the bear as stated in (15). if New location is worse than the previous one. then Turn back into the starting position. end if end for for 𝑖 = ⌊0.2 ⋅ 𝑆⌋ + 1 to ⌊0.4 ⋅ 𝑆⌋ do Intensive local search by using (16). end for Generate pseudo-random value 𝜅, 𝜅 ∈ [0, 1]. if 𝜅 < 0.25 and 𝑆 > 0.5 ⋅ 𝑛 then Generate number of bears to removing (integer from 1 to 𝑙𝑖𝑚𝑖𝑡, where 𝑙𝑖𝑚𝑖𝑡 = 𝑆 − ⌊0.5 ⋅ 𝑛⌋ − 1). else if 𝜅 > 0.75 and 𝑘 < 𝑇 − 1 then Generate number of bears to adding (integer from 1 to 𝑛 − 𝑆) and create new agents according to (18). end if 𝛾 ∶= 𝛼 ⋅ 𝛾. 𝑘 ∶= 𝑘 + 1 end while Choose the bear with the smallest value of objective function (𝑋 𝑏𝑒𝑠𝑡 ).

Algorithm 2 Pseudocode of Particle Swarm Optimization Algorithm Input: number of particles: n, number of iterations: T, parameters: 𝛼, 𝛽, 𝛾, objective function: 𝛷 from Eq. (28) Output: coordinates of the optimum with value of objective function 𝛷 Initialization: Create random initial population of 𝑛 particles. Generate an initial velocity of all particles in the swarm. Evaluate points and choose currently the fittest one: 𝑝𝑏𝑒𝑠𝑡 . The main phase: 𝑘=1 while 𝑘 < 𝑇 do for 𝑖 = 1 to 𝑛 do Calculate new values of velocity vector of the particle: 𝑉 𝑖𝑘 = 𝛼 ⋅ ( ) ( ) 𝑈 [0, 1]⋅𝑉 𝑖𝑘−1 +𝛽⋅𝑈 [0, 1]⋅ 𝑃 𝑖𝑘−1 − 𝑋 𝑖𝑘−1 +𝛾⋅𝑈 [0, 1]⋅ 𝑃 𝑏𝑒𝑠𝑡 − 𝑋 𝑖𝑘−1 . 𝑘−1 Determine new position of the particle: 𝑋 𝑖𝑘 = 𝑋 𝑖𝑘−1 + 𝑉 𝑖𝑘 . if New location is the fittest in history of this particle then 𝑃 𝑖𝑘 = 𝑋 𝑖𝑘 end if end for Find currently the best particle in the swarm: 𝑋 𝑏𝑒𝑠𝑡 . if 𝑋 𝑏𝑒𝑠𝑡 is fittest than 𝑝𝑏𝑒𝑠𝑡 then 𝑝𝑏𝑒𝑠𝑡 = 𝑋 𝑏𝑒𝑠𝑡 end if 𝑘 ∶= 𝑘 + 1 end while Choose the particle with the smallest value of objective function (𝑝𝑏𝑒𝑠𝑡 ).

Table 1 Constant values of the system. Description

Value

Unit

Return water temperature Return water pressure Water specific heat

70 7.00 4.19

◦C

Overall heat transfer coefficient (reference state) Heat exchange area Steam saturation temperature Pressure drop (reference state)

=

𝑖 + 𝑉 𝑘,

W m2 K

3200 421.1 120.0 0.25

m2 ◦ C bar

Overall heat transfer coefficient (reference state) Heat exchange area Steam saturation temperature Pressure drop (reference state)

3200 421.1 160.0 0.25

m2 ◦C bar

Heating network pressure drop (reference state) Pump mechanical efficiency Pump driver efficiency Water density Electricity cost

5.80 92 98 978.04 250 59.95 16 3.84 23 5.52

Heat exchanger 2

Steam 2.0 cost

𝑖 𝑋𝑘

kJ kg K

Heat exchanger 1

3.2.1. Proposed PSO application The first step is creating 𝑛 particles where each point has 𝑚 coordinates, similarly as in Eq. (12). Initially, one should assign a random position for created points, evaluate them and choose temporary optimum. Each particle is given a velocity (at the beginning it could be for instance 0). During main phase of the algorithm particles move according to 𝑖 𝑋 𝑘+1

bar

Steam 6.2 cost

W m2 K

bar % % kg m3

PLN/MWh EURO/MWh PLN/GJ EURO/GJ PLN/GJ EURO/GJ

memory. Points can come back to past position which was remembered as the best so far. The second component is associated with the best position of the swarm. Knowledge about the favorable location is avail-

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able for all particles. These two parts of velocity pattern are insufficient

𝑖

where 𝑋 𝑘+1 is a vector of coordinates of 𝑖th point in the 𝑘 + 1 iteration,

since points can get stuck in local minima. The third component is often

𝑉 𝑘 is a velocity vector of 𝑖th particle in the 𝑘 iteration. The most important part of this method is velocity equation which has three components. One of them – as in nature – is particle historic

called a random movement factor of particles in the domain. Due to

𝑖

these facts, points realize assumptions of an intelligent swarm. Therefore 190

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201 Table 2 Requirements set for optimization in each case. Description

Case 1 Harsh winter

Case 2 Normal winter

Case 3 Mild winter

Case 4 Out-of-season

Temperature outside (◦ C) Requested heat duty (kW) Requested forward water temperature (◦ C)

under −10 65 000 145

from −10 to 0 45 000 125

from 0 to 10 25 000 130

above 10 10 000 110

Table 3 Average results of 50 measurements for peak-level heat demand (e.g. frosty winter): requested heat duty: 65 000 kW, requested water temperature: 145 ◦ C.

Table 6 Best results after 50 measurements for normal heat demand requested heat duty: 45 000 kW, requested water temperature: 125 ◦ C.

Description

PBO

PSO

Description

PBO

PSO

Averaged cost (PLN/GJ) Standard deviation

20.1241 0.89166

20.3476 0.7229

Averaged cost (EURO/GJ) Standard deviation

4.8256 0.213812

4.87918 0.173345

Averaged objective function Standard deviation

19.99500 0.66666

20.36570 0.72240

Averaged water temperature Standard deviation

144.817 0.798243

144.923 0.6405

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty Water flow through first heat exchanger Water flow through second heat exchanger Total water flow Number of pumps in operation

18.2716 4.38137 124.917 45 002.8 1 0.344084 195.578 3

18.8895 4.52954 124.962 45 003.4 0.869985 0.422041 195.42 3

Averaged heat duty Standard deviation

65 000.8 8.82914

64 999.4 3.29126

Averaged water flow through first heat exchanger Standard deviation

0.836215 0.250687

0.771352 0.203759

Description

PBO

PSO

Averaged water flow through second heat exchanger Standard deviation

0.901886 0.0483909

0.920602 0.0469895

Averaged cost (PLN/GJ) Standard deviation

18.5543 0.362552

19.829 0.897217

Averaged total water flow Standard deviation

207.375 2.23813

207.067 1.76579

Averaged cost (EURO/GJ) Standard deviation

4.44916 0.0869369

4.75483 0.215145

Averaged number of pumps in operation

2.96

3

Averaged objective function Standard deviation

18.596 0.36955

19.8547 0.90298

Averaged water temperature Standard deviation

129.628 0.446455

130.135 0.406841

Averaged heat duty Standard deviation

24 998.6 3.38156

24 999.7 3.08414

Averaged water flow through first heat exchanger Standard deviation

0.985374 0.0776098

0.710952 0.188744

Averaged water flow through second heat exchanger Standard deviation

0.417937 0.0378969

0.559318 0.0840898

Averaged total water flow Standard deviation

100.064 0.747547

99.224 0.670144

Averaged number of pumps in operation

2.42

2

Table 7 Average results of 50 measurements for mild winter: requested heat duty: 25 000 kW, requested water temperature: 130◦ C.

Table 4 Best results after 50 measurements for peak-level heat demand (e.g. frosty winter) requested heat duty: 65 000 kW, requested water temperature: 145 ◦ C. Description

PBO

PSO

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty Water flow through first heat exchanger Water flow through second heat exchanger Total water flow Number of pumps in operation

19.5512 4.68821 145.047 64998 1 0.869939 206.704 3

19.5548 4.68906 145.13 64997.1 1 0.872038 206.475 3

Table 8 Best results after 50 measurements for mild heat demand requested heat duty: 25 000 kW, requested water temperature: 130 ◦ C.

Table 5 Average results of 50 measurements for normal heat demand: requested heat duty: 45 000 kW, requested water temperature: 125 ◦ C.

Description

PBO

PSO

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty Water flow through first heat exchanger Water flow through second heat exchanger Total water flow Number of pumps in operation

18.4704 4.42903 129.957 24 999.9 1 0.418747 99.514 2

18.4662 4.42803 129.9 24 995 1 0.417383 99.5892 2

Description

PBO

PSO

Averaged cost (PLN/GJ) Standard deviation

18.9023 1.06802

20.1497 1.07681

Averaged cost (EURO/GJ) Standard deviation

4.53261 0.256102

4.83171 0.258209

Averaged objective function Standard deviation

18.92390 1.09454

20.18140 1.07387

Averaged water temperature Standard deviation

124.959 0.825372

125.329 0.617545

Averaged heat duty Standard deviation

45 001.2 11.5715

45 000.4 3.16246

Averaged water flow through first heat exchanger Standard deviation

0.86819 0.219408

0.615803 0.217152

Averaged water flow through second heat exchanger Standard deviation

0.410588 0.103681

0.540463 0.0970038

( 𝑖 ) 𝑖 𝑖 𝑖 𝑉 𝑘+1 = 𝛼 ⋅ 𝑈 [0, 1] ⋅ 𝑉 𝑘 + 𝛽 ⋅ 𝑈 [0, 1] ⋅ 𝑃 𝑘 − 𝑋 𝑘 ( 𝑏𝑒𝑠𝑡 ) 𝑖 + 𝛾 ⋅ 𝑈 [0, 1] ⋅ 𝑃 𝑘 − 𝑋 𝑘 ,

Averaged total water flow Standard deviation

195.465 2.94126

194.135 2.18382

where 𝛼, 𝛽 and 𝛾 are coefficients: 𝛼 is an impact of randomization

Averaged number of pumps in operation

2.88

3

the formula on the velocity of the particle is as follows

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component, 𝛽 is an impact of the best point in the memory of the particle 𝑖 and 𝛾 is an impact of the fittest point in the swarm, 𝑉 𝑘+1 is a velocity of 𝑖th particle in the 𝑘+1 iteration, 𝑈 [0, 1] is a pseudo-random number from 𝑖 𝑖 the range [0, 1], 𝑃 𝑘 is the best position of 𝑖th point up to 𝑘th iteration, 𝑋 𝑘 191

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201

Table 9 Average results of 50 measurements; requested heat duty: 10 000 kW, requested water temperature: 110 ◦ C.

After calculations, the fittest positions of each particle and the best position of the swarm are updated. Afterwards, these steps are repeated. Proposed composition of the algorithm is presented in Algorithm 2.

Description

PBO

PSO

Averaged cost (PLN/GJ) Standard deviation

17.6445 0.900703

18.8773 1.30547

3.3. Designed objective function

Averaged cost (EURO/GJ) Standard deviation

4.231 0.215981

4.52661 0.31304

Averaged objective function Standard deviation

17.6086 0.626989

19.4734 1.42702

Averaged water temperature Standard deviation

110.281 0.343889

110.031 0.0960546

Averaged heat duty Standard deviation

10 001.2 8.53578

10 002 3.21591

Averaged water flow through first heat exchanger Standard deviation

0.76822 0.117131

0.595967 0.174652

Averaged water flow through second heat exchanger Standard deviation

0.113106 0.0623819

0.215197 0.100129

Averaged total water flow Standard deviation

59.261 0.49609

59.6312 0.143889

Averaged number of pumps in operation

1.04

1

Choosing the right objective function is a matter of fundamental significance for efficient optimization of technical systems. It is necessary to satisfy the boundary conditions for which one have to minimize the total utility cost per unit of energy. The objective function has to assess the quality of a solution given by proposed optimization algorithms. In our case, an optimal work of the heating plant system relies on high efficiency, fulfillment of boundary conditions (temperature, pump operation range, heat duty and pump control valves pressure drop) and the lowest possible operation costs. Heuristics have to find a compromise between mentioned factors and choose the most appropriate parameters. Each solution found by the algorithm is evaluated, so during following iterations the method can learn which propositions are the closest to the global optimum. In our case, the proposed objective function 𝛷 is developed as a composition of operating costs 𝐶𝑡𝑜𝑡𝑎𝑙 and four boundary conditions related to these values. For the optimization it is assumed that the lower the value of 𝛷, the more appropriate the solution is. In the system visible in Fig. 3 operation costs are minimized according to

Table 10 Best results after 50 measurements; requested heat duty: 10 000 kW, requested water temperature: 110 ◦ C. Description

PBO

PSO

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty Water flow through first heat exchanger Water flow through second heat exchanger Total water flow Number of pumps in operation

17.4663 4.18828 110.023 10 003.7 0.785735 0.1 59.6538 1

17.4696 4.18905 109.993 9998.86 0.784727 0.100161 59.67 1

𝐶𝑡𝑜𝑡𝑎𝑙 =

Case 1 Harsh winter

Case 2 Normal winter

Case 3 Mild winter

Case 4 Out-of-season

1400 0.025667

1609 1.51 ⋅ 10−6

2213.5 ≈0

2066 ≈0

0.66667 8.82 ⋅ 10−10

0.84 ≈0

0.85417 ≈0

] [ 𝐺𝐽 , 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 6.2 = 𝑄̇ 2 ⋅ 0.0036 ℎ 𝑄̇ 2 is heat duty for second heat exchanger.

Mann–Whitney U Statistic 𝑝-value

0.36170 0.003975

𝑏𝑒𝑠𝑡

is a location of 𝑖th particle in 𝑘th iteration and 𝑃 𝑘

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Additionally to Eq. (21), can be defined other boundary conditions responsible for maintaining right water temperature, heat duty, nonnegative pressure drop at pump control valves and allowable pump operation range. Denote by 𝑃𝑖 penalty function relevances to actual heat duty (𝑄̇ = 𝑄̇ 1 + 𝑄̇ 2 ) as

Kolmogorov–Smirnov Statistic 𝑝-value

(21)

where 𝐶𝑡𝑜𝑡𝑎𝑙 is total operating cost of the system per 1 GJ of heat produced, 𝐶𝐸𝐿 means electricity cost, 𝐸𝐸𝐿 expresses electricity consumed per hour. Symbol 𝑠2.0 is a steam 2.0 cost and 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 2.0 means consumption of steam 2.0 per hour [ ] 𝐺𝐽 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 2.0 = 𝑄̇ 1 ⋅ 0.0036 , (22) ℎ 𝑄̇ 1 is heat duty for the first heat exchanger. Analogically 𝑠6.2 expresses steam 6.2 cost and 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 6.2 signifies consumption of steam 6.2 per hour

Table 11 Results of comparing heuristic algorithms by the use of statistical tests for significance level 0.05. Statistical test

𝐶𝐸𝐿 ⋅ 𝐸𝐸𝐿 + 𝑠2.0 ⋅ 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 2.0 + 𝑠6.2 ⋅ 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 6.2 , 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 2.0 + 𝑄̇ 𝑠𝑡𝑒𝑎𝑚 6.2

⎧1 ⎪ ⋅ 𝑒𝑥𝑝(𝑄 − 𝑄𝑠𝑒𝑡 ) 𝑃1 = ⎨ 5 ⎪0 ⎩

is the best position

of the swarm up to 𝑘th iteration.

|𝑄 − 𝑄𝑠𝑒𝑡 | > 5, | |

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𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Table 12 Comparison of all test cases between best results from PBO and an exemplary not optimized solution. Case 1

Case 2

Case 3

Case 4

Description

PBO

Unoptimized parameters

PBO

Unoptimized parameters

PBO

Unoptimized parameters

PBO

Unoptimized parameters

Cost (PLN/GJ) Cost (EURO/GJ) Cost (PLN/h) Cost (EURO/h) Cost (EURO/month) Water temperature Heat duty 𝑅1 𝑅2 Total water flow Number of pumps

19.5512 4.68821 4574.84 1097.01 789 844 145.047 64 998 1 0.8699 206.704 3

21.1016 5.05998 4937.76 1184.03 852 503 145.005 64 999.7 0.5575 0.9697 206.826 3

18.2716 4.38137 2960.19 709.828 511 076 124.917 45 002.8 1 0.3441 195.578 3

20.2336 4.85184 3277.92 786.017 565 932 125.04 45 001 0.5929 0.5545 195.131 3

18.4704 4.42903 1662.33 398.611 287 000 129.957 24 999.9 1 0.4187 99.514 2

20.3167 4.87177 1828.49 438.457 315 689 129.981 25 000 0.6048 0.6072 99.4747 2

17.4663 4.18828 629.02 150.834 108 600 110.023 10 003.7 0.7857 0.1 59.6538 1

19.3407 4.63773 696.264 166.958 120 210 110.03 59.6206 0.5320 0.2608 59.6206 1

𝑅1 describes water flow through first heat exchanger, 𝑅2 through the second one. 192

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Table 13 Comparison of all test cases between heuristic methods and nonlinear interior point method. Case 1

Case 2

Description

PBO

PSO

Interior point method

PBO

PSO

Interior point method

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty 𝑅1 𝑅2 Total water flow Number of pumps

19.5512 4.68821 145.047 64 998 1 0.869939 206.704 3

19.5548 4.68906 145.13 64 997.1 1 0.872038 206.475 3

19.5492 4.68773 145.0 65 000.1 1 0.868731 206.842 3

18.2716 4.38137 124.917 45 002.8 1 0.344084 195.578 3

18.8895 4.52954 124.962 45 003.4 0.869985 0.422041 195.42 3

18.2783 4.38298 125.0 45 000 1 0.346013 195.27 3

Case 3

Case 4

Description

PBO

PSO

Interior point method

PBO

PSO

Interior point method

Cost (PLN/GJ) Cost (EURO/GJ) Water temperature Heat duty 𝑅1 𝑅2 Total water flow Number of pumps

18.4704 4.42903 129.957 24 999.9 1 0.418747 99.514 2

18.4662 4.42803 129.9 24 995 1 0.417383 99.5892 2

18.4735 4.42979 130.0 25 000 1 0.419766 99.4431 2

17.4663 4.18828 110.023 10 003.7 0.785735 0.1 59.6538 1

17.4696 4.18905 109.993 9998.86 0.784727 0.100161 59.67 1

17.4676 4.18858 110.0 10 000 0.78514 0.1 59.6659 1

𝑅1 describes water flow through first heat exchanger, 𝑅2 through the second one.

Fig. 7. Comparison of objective function value for averaged simulation results.

Fig. 9. Comparison of water temperature for averaged simulation results.

Fig. 8. Comparison of heating cost in EUR for averaged simulation results.

Fig. 10. Comparison of total water flow in the heating system for averaged simulation results.

193

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Fig. 11. Comparison of water flow on first heat exchanger for averaged simulation results.

Fig. 13. Comparison of steam 2.0 bar pressure consumption in heating process for best simulation results.

Fig. 12. Comparison of water flow on second heat exchanger for averaged simulation results.

Fig. 14. Comparison of steam 6.2 bar pressure consumption in heating process for best simulation results.

where ||𝑄 − 𝑄𝑠𝑒𝑡 || is the absolute value of a difference between requested and actual heat duty. The second one, and the most important penalty relevant is connected with temperature values ( ) |𝑇𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑇𝑠𝑒𝑡 | > 1, ⎧50 ⋅ 𝑒𝑥𝑝 20 ⋅ (𝑇𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑇𝑠𝑒𝑡 ) | | ( ) ⎪ |𝑇𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑇𝑠𝑒𝑡 | ∈ [ 1 , 1], (25) 𝑃2 = ⎨3 ⋅ 𝑒𝑥𝑝 𝑇𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑇𝑠𝑒𝑡 | | 10 ⎪ ⎩0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. |𝑇𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑇𝑠𝑒𝑡 | is the absolute value of a difference between requested | | and actual forward water temperature. Right value of temperature is essential for the usefulness of results given by applied algorithms. Third boundary condition concerns pump control valves pressure drop (𝛿𝑝𝑣𝑎𝑙𝑢𝑒𝑠 ) which cannot be negative: { 5 10 𝛿𝑝𝑣𝑎𝑙𝑢𝑒𝑠 < 0, 𝑃3 = (26) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. Moreover, each pump has an allowable operation range (𝑉̇ ). In presented system it is assumed that this value must be within the range [120, 300], otherwise pumps will not be working correctly. ⎧ ⎪1020 𝑃4 = ⎨ ⎪0 ⎩

[ 3] [ 3] m m 𝑉̇ < 120 𝑜𝑟 𝑉̇ > 300 , h h 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Fig. 15. Comparison of electricity consumption for best simulation results.

Boundaries of penalty functions were chosen experimentally to ensure

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high quality of solutions. Finally, the pattern for the objective function 194

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Fig. 16. The characteristics of the simulation process by using PBO. From left to right, in each row, number of bears during subsequent iterations, values of heated water flow split factors 𝑅1 and 𝑅2 , values of 𝑚0 , number of pumps in operation and the objective function 𝛷 can be seen. First row presents simulation process for Case 1 (harsh winter), second row for Case 2 (normal winter, third for Case 3 (mild winter) and the last one for Case 4 (out-of-season use).

flow, which will be set in the range: [20, 250]( 𝑘𝑔 ). Values from Table 1 𝑠 will be constant for all examined working conditions. The outcome of calculations are four values: part of the water stream which is flowing through heat exchanger no. 1 (split ratio 𝑅1 ), part of the water stream which is flowing through heat exchanger no. 2 (split ratio 𝑅2 ), total water flow (𝑚̇ 0 ) and the number of pumps in operation. All searches were made by using 100 individuals (bears in the case of PBO and particles in the case of PSO) and 200 iterations. For each of 4 settings of the system, there were made 50 attempts and obtained values have been averaged. Experiments were conducted by using following 𝑅 𝑚 parameters of Polar Bear Optimization: 𝛾1 1,2 = 0.25, 𝛾1 0 = 10, 𝛼 = 0.92

𝛷 will be the following 𝛷 = 𝐶𝑡𝑜𝑡𝑎𝑙 + 𝑃1 + 𝑃2 + 𝑃3 + 𝑃4 .

(28)

4. Experimental results There was performed optimization of the district heating plant from Fig. 3 in four cases representing changing weather conditions: harsh winter, typical winter weather, mild winter, and out-of-season period, when only sanitary hot water is required. Demands from the system in changing weather conditions are presented in Table 2. For each case bio-inspired methods were used to set the system for highest efficiency at lowest costs.

𝑅

(𝛾1 1,2 is in charge of width of domain for 𝑅1 and 𝑅2 in local search, 𝑚 𝛾1 0

is connected with total water flow). In the case of Particle Swarm Optimization: 𝛼 = 𝛽 = 𝛾 = 1 were adopted. Because of the fact that one parameter is an integer (number of pumps in operation), at the beginning algorithms assign randomly this value for each point. Modifications of values involve only water split ratios and total water flow. When the number of iterations is equal to 0.5 ⋅ 𝑇 (𝑇 is a total number of iterations) all agents obtain such amount of operated pumps like the best point at this moment.

4.1. Configuration of bio-inspired methods Algorithms need some additional information. One has to set limits 1 , 1]. for water split ratio for two heat exchangers: it will be the range [ 10 This ratio cannot be greater than 1. Another is restriction for total water 195

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Fig. 17. The characteristics of the simulation process by using PSO. From left to right, in each row, values of heated water flow split factors 𝑅1 and 𝑅2 during subsequent iterations, values of 𝑚0 , number of pumps in operation and the objective function 𝛷 can be seen. First row presents simulation process for Case 1 (harsh winter), second row for Case 2 (normal winter, third for Case 3 (mild winter) and the last one for Case 4 (out-of-season use).

This represents medium-level heat demand, occurring during typical winter weather conditions. Again more efficient was the Polar Bear Optimization, but this time the difference between results was bigger (see Table 5 for averaged results and Table 6 for the best results). The most efficient set of parameters coming from PBO brings savings 0.12 EURO/GJ in comparison to settings from PSO, see Table 6. The water temperature was only less by 0.83 K than it was expected. It is worth to see that definitely different stream split ratios can produce similar final results so these tasks are very hard to optimize. Parameters of the system for Case 2 are available in Table 15.

4.2. Case 1 - harsh winter First considered case concerned requested heat duty equal to 65 000 kW and water temperature equal to 145 ◦ C. This corresponds to peaklevel heat demand (e.g. frosty winter). It is possible to see that Polar Bear Optimization achieved slightly better results than Particle Swarm Optimization (see Table 3 for averaged results and Table 4 for the best results). Averaged cost (in EURO/GJ) was ≈ 4.83 in case of PBO and ≈ 4.88 in the matter of PSO. Except for boundary conditions, water temperature requirement is essential − it is more important than accurate values of heat duty. In practice, forward water has to be hot enough to reach the heat consumers at sufficient level of temperature, despite the heat losses from pipelines. As a result of optimization it is possible to see that bio-inspired methods suggest to use 3 pumps. Parameters of the system for the Case 1 are available in Table 14.

4.4. Case 3 - mild winter In the third considered case (requested water temperature: 130 ◦ C, requested heat duty: 25 000 kW) representing mild winter conditions Polar Bear Optimization again beats Particle Swarm Optimization. Averaged results are presented in Table 7. In Table 8 can be seen the best results. Averaged total cost of heating is equal to ≈ 4.45 EURO/GJ according to data from PBO and ≈ 4.75 EURO/GJ for optimization from PSO. Their

4.3. Case 2 - typical winter weather The second test required parameters of the system, in which water temperature should be equal to 125 ◦ C and heat duty 45 000 kW. 196

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201 Table 14 Results for Case 1 : Parameters for efficient heat plant work at harsh winter conditions presented in Table 4. Description

Value (PBO)

Value (PSO)

Unit

Return water temperature Pump required pressure rise Pump adiabatic power Pump driver electrical power Adiabatic efficiency of pump

70 8.82406 253.614 84.5877 81.5113

70 8.82907 253.333 84.5254 81.5274

◦C

70 206.704 100 0 0 3285.53 50 10.1187 109.881 − 109.881 34 540.8 0.267041

70 206.475 100 0 0 3282.62 50 10.1151 109.885 − 109.885 34 505.6 0.26645

Inlet temperature Water flow through heat exchanger no. 2 Water flow through heat exchanger no. 2 Water flow through by-pass no. 2 Water flow through by-pass no. 2 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 2) Water outlet temperature (of by-pass no. 2) Mixed stream temperature Heat exchanger no. 2 duty Pressure drop (actual)

109.881 179.82 86.9939 26.8841 13.0061 2938.98 50.1187 9.69497 150.305 109.881 145.047 30 457.1 0.202095

109.885 180.054 87.2038 26.421 12.7962 2942.04 50.1151 9.69842 150.302 109.885 145.13 30 491.4 0.202622

K K ◦ C ◦ C ◦ C kW bar

Actual heat duty Requested heat duty Actual forward water temperature Requested forward water temperature Heating network pressure drop (actual)

64 997.9 65 000 145.047 145 6.19535

64 997 65 000 145.13 145 6.18163

kW kW ◦ C ◦ C bar

Steam 2.0 bar consumption

124.347

124.22

Steam 6.2 bar consumption

109.645

109.769

Electricity consumption

0.253763

0.253576

GJ h GJ h MWh h

bar m3 h

kW %

Heat exchanger no. 1 Inlet temperature Water flow through heat exchanger no. 1 Water flow through heat exchanger no. 1 Water flow through by-pass no. 1 Water flow through by-pass no. 1 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 1) Water outlet temperature (of by-pass no. 1) Mixed stream temperature Heat exchanger no. 1 duty Pressure drop (actual)

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar

Heat exchanger no. 2

precision is adequate for the most important parameter, i.e. water temperature is equal to 129.957 ◦ C (PBO) or 129.9 ◦ C (PSO). Bio-inspired methods suggest to use only 2 pumps in these weather conditions. Requested heat duty has been reached within the range of tolerance. Parameters of the system for Case 3 are available in Table 16.

◦

C

kg s

% kg s

% W m2 K

comparison of objective function values from Eq. (28). PBO was better, returning results of about 5% to 8% lower than PSO for Cases 2–4, while for Case 1 the results are almost the same. Similar situation is visible for heating costs in the system of heat exchangers. By comparing the results in Fig. 8, it can be observed that costs for system by the use of PBO are 2% to 5% lower than PSO. Proposed bio-inspired methods are efficient in this task. Further analysis of the results shows that proposed bio-inspired approach is also precise to keep the boundary limitations of the heating plant system. Analyzing results in Fig. 9, it can be deducted that both methods maintain boundary temperature of heating water in the system, which is an important clue to evaluate optimization methods for precision in engineering problems. Both algorithms maintain comparable total water flow in the system, as visible in Fig. 10. On the other hand, from Figs. 11 and 12 it follows that PBO lets more water to the first heat exchanger, while for the second one PSO simulates higher water flow. In Figs. 13 and 14 can be compared suggested by the methods steam flow in each heat exchanger. The highest values are visible for Case 1 representing high demand in harsh winter conditions, while the lowest demand is visible for Case 4 representing only a sanitary use in out-ofseason time. Additionally both bio-inspired methods correctly allowed more steam at lower pressure (2.0 bar) into first exchanger. The second heat exchanger serves as a water temperature controller. This makes the heating plant more efficient, what is visible in optimized heating costs. In Fig. 15 can be observed that both PBO and PSO set the system at low

4.5. Case 4 — out-of-season sanitary hot Last considered setting is concerned with the lowest heat demand, which is reflected by the water temperature - 110 ◦ C - and the corresponding heat duty value 10 000 kW. This is a case for sanitary hot water heating (as in summer, when no heating is needed). Averaged results are presented in Table 9. In Table 10 can be seen the best results. For Polar Bear Optimization averaged cost was almost 0.30 EURO/GJ less than for PSO. For both methods, the best results were almost the same in this case, see Table 10. Difference between the requested and the actual water temperature was only 0.007 K (for PSO) and 0.023 K (for PBO). One pump in operation was an adequate choice selected by heuristic methods. Parameters of the system for normal winter conditions are available in Table 17. 4.6. Discussion on efficiency of proposed bio-inspired methods In Figs. 7–15 comparison charts for optimization results from PBO (blue bars) and PSO (yellow bars) can be seen. Fig. 7 presents a 197

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201 Table 15 Results for Case 2 : Parameters for efficient heat plant work at normal winter conditions presented in Table 6. Description

Value (PBO)

Value (PSO)

Unit

Return water temperature Pump required pressure rise Pump adiabatic power Pump driver electrical power Adiabatic efficiency of pump

70 9.05795 239.963 81.6133 82.0535

70 9.06113 239.769 81.5718 82.0577

◦C

70 195.578 100 0 0 3143.27 50 9.94038 110.06 − 110.06 32 827.7 0.239067

70 170.012 86.9985 25.4075 13.0015 2810.03 50 9.49443 110.506 70 105.239 28 854.2 0.180651

Inlet temperature Water flow through heat exchanger no. 2 Water flow through heat exchanger no. 2 Water flow through by-pass no. 2 Water flow through by-pass no. 2 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 2) Water outlet temperature (of by-pass no. 2) Mixed stream temperature Heat exchanger no. 2 duty Pressure drop (actual)

110.06 67.2953 34.4084 128.283 65.5916 1338.79 49.9404 6.76113 153.239 110.06 124.917 12 175.1 0.0283041

105.239 82.4753 42.2041 112.945 57.7959 1575.38 54.7608 8.02892 151.971 105.239 124.962 16 149.2 0.0425135

Actual heat duty Requested heat duty Actual forward water temperature Requested forward water temperature Heating network pressure drop (actual) Steam 2.0 bar consumption

45 002.9 45 000 124.917 125 5.54636 118.18

45 003.4 45 000 124.962 125 5.5374 103.875

Steam 6.2 bar consumption

43.8305

58.1371

Electricity consumption

0.24484

0.244715

bar m3 h

kW %

Heat exchanger no. 1 Inlet temperature Water flow through heat exchanger no. 1 Water flow through heat exchanger no. 1 Water flow through by-pass no. 1 Water flow through by-pass no. 1 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 1) Water outlet temperature (of by-pass no. 1) Mixed stream temperature Heat exchanger no. 1 duty Pressure drop (actual)

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar

Heat exchanger no. 2 ◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar kW kW ◦ C ◦ C bar GJ h GJ h MWh h

simulation, slightly adjusting them during further calculations. Values of water flow split factors 𝑅1 and 𝑅2 for PSO show higher fluctuations in optimization process. In Case 1, 2 and 4 PSO was optimizing them during every iteration by achieving additional improvement. In Case 3 changes are visible up to 20 iterations and after that the values are almost constant. Value of 𝑚0 was intensively optimized in Case 1, while for other cases only up to 20 iteration. PSO in first 10 iterations decided on the number of used pumps and kept this decision in each case until the end of simulation. In case of objective function 𝛷 the biggest fluctuations are visible up to 50 iterations for all cases. In Case 1, 2 and 4 further improvements can be seen, however without so spectacular fluctuations, but in Case 3 after 50 iterations the objective function value was almost constant. By comparing PBO to PSO it is possible to say that PBO was achieving better results in two or three phases during simulation process, while PSO was optimizing them in almost all following iterations. However, by the basis on Figs. 7–15, it can be claimed, that final results were slightly better for PBO. Therefore this method can be evaluated as a more efficient one in our experiments.

level of electricity consumption, which is an environmentally friendly setup. Graphic presentation of variability in optimized parameters during subsequent iterations is presented in Fig. 16 for PBO and in Fig. 17 for PSO. Number of individuals which are used for search in PBO is changing between 50 and 100. Values of heated water flow split factors 𝑅1 and 𝑅2 in following iterations of PBO are visible in second and third column, respectively. For Case 1, the biggest changes take place in 0–50 and 100–150 iterations. Above 150 iterations PBO was not changing these values much. In Case 2 the situation is different, since the biggest changes in 𝑅1 and 𝑅2 are visible up to 150 iteration and above it corrections were marginal. In Case 3 these values were optimized up to 100 iterations and after similarly as in Case 2 only marginally corrected. In Case 4 the most significant corrections were done between 0–50 and 100–120 iterations with almost no improvements in other iterations. Value of 𝑚0 was mostly optimized up to 20 iterations and after that there are lower fluctuations in Case 1 and 3, and almost no correction in Case 2 and 4. The biggest changes in the number of used pumps are visible in Case 3, where PBO was suggesting all possible combinations of pumps. PBO in first 10 iterations decided to propose the use of 2 pumps and kept this decision until the end of simulation. In the last column objective function changes can be seen. After analyzing this charts it is possible to say that in all cases PBO started the optimization from very large value of the objective function. Despite this fact, PBO very quickly achieved stable values and kept them until the end of

4.7. Statistical tests For comparison of the efficiency of two tested algorithms, there were carried out two statistical tests: Mann–Whitney U and Kolmogorov– Smirnov. The Mann–Whitney U is a non-parametric test, which checks if median value of the differences between observations from two 198

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201 Table 16 Results for Case 3 : Parameters for efficient heat plant work at mild winter conditions presented in Table 8. Description

Value (PBO)

Value (PSO)

Unit

Return water temperature Pump required pressure rise Pump adiabatic power Pump driver electrical power Adiabatic efficiency of pump

70 9.82297 183.147 70.0582 79.1166

70 9.82152 183.286 70.0854 79.1339

◦C

70 99.514 100 0 0 1830.77 50 7.86855 112.131 − 112.131 17 567.3 0.061894

70 99.5892 100 0 0 1831.87 50 7.87075 112.129 − 112.129 17 579.6 0.0619876

Inlet temperature Water flow through heat exchanger no. 2 Water flow through heat exchanger no. 2 Water flow through by-pass no. 2 Water flow through by-pass no. 2 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 2) Water outlet temperature (of by-pass no. 2) Mixed stream temperature Heat exchanger no. 2 duty Pressure drop (actual)

112.131 41.6712 41.8747 57.8428 58.1253 912.42 47.8686 5.29972 154.7 112.131 129.957 7432.62 0.010853

112.129 41.5668 41.7383 58.0224 58.2617 910.591 47.8707 5.29411 154.706 112.129 129.9 7415.36 0.0107988

Actual heat duty Requested heat duty Actual forward water temperature Requested forward water temperature Heating network pressure drop (actual) Steam 2.0 bar consumption

24 999.9 25 000 129.957 130 1.43594 63.2422

24 995 25 000 129.9 130 1.43811 63.2867

Steam 6.2 bar consumption

26.7574

26.6953

Electricity consumption

0.140116

0.140171

bar m3 h

kW %

Heat exchanger no. 1 Inlet temperature Water flow through heat exchanger no. 1 Water flow through heat exchanger no. 1 Water flow through by-pass no. 1 Water flow through by-pass no. 1 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 1) Water outlet temperature (of by-pass no. 1) Mixed stream temperature Heat exchanger no. 1 duty Pressure drop (actual)

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar

Heat exchanger no. 2

algorithms is equal to zero (𝐻0 ). The Kolmogorov–Smirnov test is used for checking the equality of distributions based on data from two heuristics (𝐻0 : two distributions are equal). The results of described statistical tests are presented in Table 11. In each case it turned out that the null hypothesis was rejected since p-values were very small. The most similar results for both methods were obtained in Case 1 but still 𝐻0 was rejected. It is possible to say that there is a statistically significant difference between two algorithms. On the other hand, results from PBO more often exceed the boundary conditions than the competitor. One can have more trust to PSO, furthermore the distribution of results from PSO is close to normal distribution. Therefore in a sense of these results both methods are comparably efficient in this task. These results confirmed that proposed application of using bio-inspired methods to solve complex engineering problems is efficient. Both methods set the system for the most efficient work in weather conditions.

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar kW kW ◦ C ◦ C bar GJ h GJ h MWh h

EURO in Case 3 and 11 610 EURO in Case 4). For these example results, profits are: 7.35% in Case 1, 9.69% in Case 2, 9,90% in Case 3 and 9.66% in Case 4. We can estimate that the proposed optimization gave the highest savings in regular winter conditions, which is reasonable since in harsh winter we all rather expect our flats to be warm instead of saving money. On the other hand the savings for harsh winter were over 7% in comparison to unoptimized system what shows that the proposed method is also efficient in these conditions. It is very important, that the boundary conditions are fulfilled in two compared solutions (optimized and unoptimized) in each test case. However, the costs of energy production are significantly different. These experiments confirmed that it is worth to set a big heating plant in an optimized way. Thanks to this, it is possible to save money during heating season.

4.8. Importance of optimization

4.9. Discussion on numerical method in comparison to the proposed heuristics

Results shown in Table 12 help to evaluate the importance of the proposed optimization. In this table we compared the best solutions coming from the proposed PBO and exemplary non optimized parameters of the heating plant system. Although the differences between costs per hours seem to be not as great, the summary waste of money without the optimization is visible in monthly costs. The right choice of parameters can bring a dozen thousand euros of savings (62 659 EURO in Case 1, 54 856 EURO in Case 2, 28 689

For further analysis the results from PBO and PSO were also compared to the numerical method called Nonlinear Interior Point. This is an algorithm designated for nonlinear optimization with constraints. The data presented in Table 13 indicate that numerical methods also reach good scores. It is worth trying to improve parameters, because even a slight correction ensures long-term profits (it was proved on the basis of the data from Table 12). Therefore, searching for the new methods, like the proposed heuristic approaches, which can set the system in as 199

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Engineering Applications of Artificial Intelligence 76 (2018) 185–201 Table 17 Results for Case 4 : Parameters for efficient heat plant work at mild winter conditions presented in Table 10. Description

Value (PBO)

Value (PSO)

Unit

Return water temperature Pump required pressure rise Pump adiabatic power Pump driver electrical power Adiabatic efficiency of pump

70 9.37113 219.576 77.3467 81.963

70 9.37028 219.635 77.3589 81.9649

◦C

70 46.8721 78.5735 12.7817 21.4265 1002.44 50 5.8263 114.174 70 104.709 8675.45 0.0137312

70 46.8247 78.4727 12.8453 21.5273 1001.63 50 5.82376 114.176 70 104.666 8667.17 0.0137034

Inlet temperature Water flow through heat exchanger no. 2 Water flow through heat exchanger no. 2 Water flow through by-pass no. 2 Water flow through by-pass no. 2 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 2) Water outlet temperature (of by-pass no. 2) Mixed stream temperature Heat exchanger no. 2 duty Pressure drop (actual)

104.709 5.96538 10 53.6884 90 192.68 55.2912 2.1513 157.849 104.709 110.023 1328.23 0.000222411

104.666 5.97661 10.0161 53.6934 89.9839 192.97 55.3337 2.15558 157.844 104.666 109.993 1331.69 0.000223249

Actual heat duty Requested heat duty Actual forward water temperature Requested forward water temperature Heating network pressure drop (actual) Steam 2.0 bar consumption

10 003.7 10 000 110.023 110 0.515993 31.2316

9998.86 10 000 109.993 110 0.516274 31.2018

bar m3 h

kW %

Heat exchanger no. 1 Inlet temperature Water flow through heat exchanger no. 1 Water flow through heat exchanger no. 1 Water flow through by-pass no. 1 Water flow through by-pass no. 1 Overall heat transfer coefficient (actual) Initial temperature difference Terminal temperature difference Water outlet temperature (of heat exchanger no. 1) Water outlet temperature (of by-pass no. 1) Mixed stream temperature Heat exchanger no. 1 duty Pressure drop (actual)

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar

Heat exchanger no. 2

Steam 6.2 bar consumption

4.78162

4.79407

Electricity consumption

0.0773467

0.0773589

more efficient in various working conditions is justified. Examples of such algorithms are the proposed Polar Bear Optimization or Particle Swarm Optimization. Detailed comparison of numerical and heuristic methods in other complex engineering problems will be a subject of our further work.

◦

C

kg s

% kg s

% W m2 K

K K ◦ C ◦ C ◦ C kW bar kW kW ◦ C ◦ C bar GJ h GJ h MWh h

4.10.1. Summary Using this system for long period at unfavorable settings is certain to bring high and unnecessary costs. It is worth to set it in the most efficient way. As shown above, bio-inspired algorithms can be a very helpful tool in adjusting the system. As presented in Fig. 8 optimized system is producing the heat with cost not higher than 5 euro per unit. Proposed methods can be adapted to different requirements (weather conditions, boundary values, etc.) and effectively approximate the optimal solution for each of co-working elements. It proves that application of bioinspired algorithms in solving engineering problems is promising and economically viable.

4.10. Conclusions Obtained results confirm that bio-inspired algorithms can be an effective tool for solving engineering problems. Although the objective function is based on boundary conditions, the presented methods reach good results and can properly set presented system of heat exchangers and pumps in a wide range of requirements. One can claim that Polar Bear Optimization achieved slightly better scores than Particle Swarm Optimization, however also PSO has showed a high value for optimization. Sometimes lower values of cost function are connected with obtaining water temperatures slightly lower than requested. One should also take into account different number of calculation steps per iteration, which was higher in case of PBO algorithm. On the other hand, in this algorithm a certain number of bears is removed from the population (see column one for all cases in Fig. 16) so it reduces the number of redundant calculations. It can be therefore concluded that both presented bio-inspired methods are efficient and can properly adjust the heating system.

5. Final remarks The presented methodology can be widely used in power and chemical plants. Achieving the desired performance with the lowest possible operating costs is extremely important in civil power plants. Heuristic algorithms are able to ensure an efficient work of large energetic systems which can generate the heat for several thousands of houses (an estate). In real-life situations, there are plenty of constraints essential for proper operation and maintenance of industrial equipment. They are all to be taken into account so as to avoid untimely damage and costly repairs. Bio-inspired algorithms are applicable tools for finding optimal settings of the system by fulfilling boundary conditions. Most importantly, an optimized system configuration can be practically used. 200

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For further research, it is possible to extend the presented model by additional process equipment and test some more complicated systems. One can also compare the efficiency to other algorithms or use artificial neural networks.

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