Based on the life cycle cost (LCC) approach, this paper presents an integral mathematical model and particle swarm optimization (PSO) algorithm for the heating system planning (HSP) problem. The proposed mathematical model minimizes the cost of heating system as the objective for a given life cycle time. For the particularity of HSP problem, the general particle swarm optimization algorithm was improved. An actual case study was calculated to check its feasibility in practical use. The results show that the improved particle swarm optimization (IPSO) algorithm can more preferably solve the HSP problem than PSO algorithm. Moreover, the results also present the potential to provide useful information when making decisions in the practical planning process. Therefore, it is believed that if this approach is applied correctly and in combination with other elements, it can become a powerful and effective optimization tool for HSP problem.
Humanity faces serious energy and environment problems at present. The environment is increasingly threatened. For instance, with the increase of greenhouse gas emissions in the atmosphere the environments have already reached concerning levels in terms of their potential to cause climate change. Air pollution, acid precipitation, and stratospheric ozone depletion are other serious environmental concerns. The severity of climate change impacts shows the increasing trend if significant action is not taken to reduce greenhouse gas emissions [
In China, heating utilities have been developed rapidly, but the energy consumption of production and transport is still too much, which accounts for 21.5% of building energy consumption; building energy consumption accounts for 20.9% of social total energy consumption [
Sustainable development of heating system requires application of planning procedures, which includes optimization of both demand and supply sides of heating. Because the heat source site selection and heating pipe network optimizing plan have an important role in the HSP, there are many scholars concerning this subject and lots of optimization methods have been proposed. The methods of HSP can be classified into three separate categories [
Life cycle cost (LCC) has been applied since the 1960s when the United States’ Department of Defense stimulated the development and application of LCC to enhance its cost effectiveness. Defense systems, such as an aircraft or a special land vehicle, are ideal for LCC analyses since the Department of Defense mainly controls the entire life cycle [
There are two popular swarm inspired methods in computational intelligence areas: ant colony optimization (ACO) and particle swarm optimization (PSO). ACO was inspired by the behaviors of ants and has many successful applications in discrete optimization problems. The particle swarm concept originated as a simulation of simplified social system. The original intent was to graphically simulate the choreography of a bird block or fish school. However, it was found that particle swarm model can be used as an optimizer. A substantial review of the properties of the global optimization problems has been given by Parsopoulos and Vrahatis [
The main objective of this paper is to discuss the usefulness of the PSO algorithm for solving the HSP problem. Therefore, based on the LCC approach, an integral mathematical model is presented and PSO algorithm is introduced and improved for solving the problem. In the end, the results of the case study suggest the effectiveness of improved particle swarm optimization (IPSO) application to the optimal planning method for heating system.
LCC is related to the systems engineering process, because economic considerations are very important in the process of creating systems. Life cycle economic analyses should be done early in the system or product life cycle, because the outcome of the systems engineering process cannot be influenced very much when the design is completed. Thus, LCC involves evaluation of all future costs related to all of the phases in the system life cycle including design, construction and/or production, distribution, operation, maintenance and support, retirement, and material disposal, and so on [
Cost models may range from simple to complex and are essentially predictive in nature. Parameters, such as the system’s physical environment, usage demand, reliability, maintainability, labor, energy, taxes, inflation, and the time value of money, may have a great influence on the life cycle costs [
The main objective of this paper is to discuss the usefulness of the PSO algorithm for owners in making sustainable heating system investment decisions and to improve their decisionbases for municipal administration. Therefore, we apply LCC approach to describe the HSP problem.
Moreover, HSP considered in this study works under the following definition and assumptions.
A heat consuming installation can connect with any heat source but cannot connect with two or more heat sources at the same time.
The indirect connection between heat consuming installation and heat source is not allowed.
A heat source must be connected with more than one heat consuming installation; otherwise, it will be closed.
Any connection between any two heat sources is not allowed.
The location of heat consuming installation is fixed.
A heat source can be sited in a given region.
The elevation difference between heat consuming installation and heat source is ignored.
Heating system planning and optimization can be achieved by changing the number and the heating capacity of heat source and the distance between the heat source and heat consuming installation.
The measure between heat source and heat consuming installation is simplified to the Manhattan (or city block) distance.
There is no functional difference between any two heat sources and their products.
The notations used in the mathematical formulations are given as follows.
Optional heating source
Heating equipment
Heat consuming installation
Heat load distributing segment.
In this study, the problem is summarized into a multisource, multifacility, singlecommodity, multiraw material plant location problem, and a mixed 01 integer planning model has been formulated. The cost model of the heat source and the heattransmission network concerned in the optimization model are considered in this study. The objective function of heating system planning problem is to minimize the total heat production cost. The proposed mathematical model formulation for HSP problem can be found as follows.
Minimize
Because the piecewise function of heat load duration curve is introduced in the process of solving the model, this model can be applied to any form of heating system.
The heating source cost model is aimed to resolve the calculation problem of
The heating network cost model is aimed to resolve the calculation problem of
The PSO is proposed by Kennedy and Eberhart [
Specifically, PSO algorithm maintains a population of particles, each of which represents a potential solution to an optimization problem. The position of the particle denotes a feasible, if not the best, solution to the problem. The optimum progress is required to move the particle position in order to improve the value of objective function. The convergence condition always requires setting up the move iteration number of particle.
The position of particle move rule is shown as follows:
The flow chart of general PSO is shown in Figure
Flow chart of general PSO.
For HSP problem and its model in this paper, the value of LCC depends mostly on the distance between heating source and heat consuming installation, and the number of heating source
The evolution of the solution set begins with an initial solution set in the PSO; initial solution set is composed of initial particles. Each solution location is represented by an
The position coordinate of heating source
In the same way, the velocity for location change of heating source
Thus, the update rule of velocity for each particle is indicated by (
The meanings of parameters are consistent with previous description.
The calculated flow of proposed IPSO is described as follows.
The initial solution for HSP problem is obtained by random initial position of each heat source; a matrix is employed in recording the coordinates and the heat loadbearing information of heat source, and the calculated flow of initial solution is as follows.
Set up the number of heat source
Based on randomly and evenly distributed manner, generate the position coordinates of heat sources, into the matrix.
Call the decoding function; calculate the heat loadbearing and the cost for each heat source, into the matrix.
Calculate the LCC, the fitness value of the initial particle.
In this paper, decoding function will call the matrix for current position and heat load of heat consuming installation, and then according to the matrix for the position of heat source, which is represented by current particle, divide the heating range of each heat source, and calculate the LCC.
Information matrix of heat consuming installation (
Read matrix
Calculate the distance to all heat source from the heat consuming installation
By substituting
Find out the minimum cost, and the heat consuming installation
If
After one generation of particles, a new generation is evolved as follows.
Call the decoding function; calculate the fitness value of the particle swarm.
Update the individual optimal solution
Update the speed vector, by using (
Update the speed vector, by using (
The PSO’s convergence is fast, so it is liable to fall into local optimal solution. In order to improve the optimizing capability, we add modular arithmetic of velocity vector into each iterative operation. If the norm of velocity vector
The flow chart of IPSO is shown as Figure
Flow chart of IPSO.
This is a heating plan for a new area in China covering the area of 3.346 million square meters, and heat load is 167.3 MW in total. Based on the road network, the new area is divided into 29 heating districts (Figure
Site location plan of 29 heating districts.
Heating load of 29 heating districts.
The role of the inertia weight
But so far, the research on the most appropriate values for
By applying PSO and IPSO algorithm, respectively, we solved the HSP problem in this paper. The parameters of PSO and IPSO are summarized in Table
PSO and IPSO parameters.
Variable  Symbol  Value  

PSO  IPSO  
Population size  —  100  100 
Maximum iteration number  —  1000  1000 
Inertia weight 

0.7  0.7 
Acceleration constant 

2  2 

2  2 
In this study, 29 kinds of schemes of heating (from one heat source to twentynine heat sources) were calculated for 10 times through reading initial conditions from the excel file successively, which contains the coordinates and heat load of heat consuming installation, preset maximum number of heat source. The results of LCC and the
Algorithm calculation results comparison.
Heat source  LCC (billion Yuan)  

Optimum value  Average value  
PSO  IPSO 

PSO  IPSO 


1  1.5320  1.5320  0.0000  1.5320  1.5320  0.0000 
2  1.5035  1.5035  0.0000  1.5035  1.5035  0.0000 
3  1.4940  1.4940  0.0000  1.4947  1.4945  0.0002 
4  1.4881  1.4880  0.0001  1.4893  1.4890  0.0003 
5  1.4866  1.4864  0.0002  1.4873  1.4872  0.0001 
6  1.4857  1.4848  0.0009  1.4867  1.4863  0.0004 
7  1.4857  1.4842  0.0015  1.4865  1.4852  0.0013 
8  1.4850  1.4828  0.0022  1.4865  1.4854  0.0011 
9  1.4849  1.4832  0.0017  1.4872  1.4857  0.0015 
10  1.4847  1.4843  0.0004  1.4875  1.4866  0.0009 
11  1.4855  1.4852  0.0003  1.4889  1.4874  0.0015 
12  1.4864  1.4856  0.0008  1.4894  1.4880  0.0014 
13  1.4883  1.4865  0.0018  1.4915  1.4887  0.0028 
14  1.4893  1.4873  0.0020  1.4920  1.4910  0.0010 
15  1.4910  1.4903  0.0007  1.4933  1.4917  0.0016 
16  1.4931  1.4908  0.0023  1.4952  1.4944  0.0008 
17  1.4939  1.4933  0.0006  1.4966  1.4954  0.0012 
18  1.4966  1.4944  0.0022  1.5002  1.4971  0.0031 
19  1.4987  1.4974  0.0013  1.5025  1.4993  0.0032 
20  1.4990  1.4985  0.0005  1.5010  1.5007  0.0003 
21  1.5018  1.5001  0.0017  1.5031  1.5027  0.0004 
22  1.5032  1.5018  0.0014  1.5044  1.5040  0.0004 
23  1.5033  1.5031  0.0002  1.5065  1.5056  0.0009 
24  1.5062  1.5050  0.0012  1.5080  1.5075  0.0005 
25  1.5075  1.5065  0.0010  1.5100  1.5095  0.0005 
26  1.5106  1.5102  0.0004  1.5131  1.5120  0.0011 
27  1.5121  1.5110  0.0011  1.5132  1.5130  0.0002 
28  1.5135  1.5133  0.0002  1.5152  1.5150  0.0002 
29  1.5169  1.5145  0.0024  1.5195  1.5170  0.0025 
Algorithm calculation results comparison (optimum value).
Algorithm calculation results comparison (average value).
From analyzing the results, we can draw the following conclusion about the HSP problem.
The original plan (the heating load of each district is supplied by its small gasfired boiler) is not an economic and reasonable plan for the case, and the LCC is the second highest in 29 schemes, which is only better than the scheme which plans to set up one heat source only.
From one heat source to twentynine heat sources, LCC is monotone decreasing until a minimum value first, then monotone increasing.
Only one minimum value of LCC that appeared throughout the change process, which is 1.4828 billion Yuan, the scheme of which plans to set up 8 heat sources, is the best choice for the case. (The detailed calculation results of this scheme are shown in Table
The detailed results of 8 heat sources scheme.
Heating source  Coordinate  Supply heat load (MW)  Heat consuming installation 

1  (395, 555)  24.38  1, 2, 3, 10, 11, 12 
2  (320, 1050)  22.07  18, 19, 26 
3  (1030, 315)  24.43  4, 5, 6, 13 
4  (760, 1070)  14.92  20, 27 
5  (1745, 555)  24.73  7, 14, 15, 23 
6  (1040, 1090)  21.94  21, 22, 28, 29 
7  (2380, 850)  14.78  17, 24, 25 
8  (2145, 340)  20.05  8, 9, 16 
By observing the algorithms, the following is also concluded.
The optimal solution of IPSO is better than PSO. The optimum LCC which calculated by IPSO is not larger than PSO for all 29 schemes. The maximum
The real minimum LCC was not calculated by PSO. The minimum LCC calculated by PSO is 1.9 million Yuan larger than the minimum LCC calculated by IPSO.
Figure
Algorithm calculation comparison (7 heat sources).
Hence, it can be concluded that the improvement approach is effective, and the proposed method IPSO has better significance in solving the HSP problem and competitive to PSO algorithm.
Section
Algorithm calculation in comparison with different parameters (6 heat sources).
Coordinates of heating sources in comparison with different parameters (6 heat sources).
The parameters of each case in the figure are summarized in Table
The parameters of each case in Figures
Variable  Symbol  Value  

Case 1  Case 2  Case 3  Case 4  Case 5  
Population size  —  100  100  100  100  100 
Maximum iteration number  —  1000  1000  1000  1000  1000 
Inertia weight 

0.7  0.4  0.9  0.7  0.7 
Acceleration constant 

2  2  2  0.2  3.8 

2  2  2  0.2  3.8 
The influence aspect of the algorithm is worth further study, but because of the major goal of the present study, the more details were not presented here and will be discussed in a separate paper.
In this paper, we presented an integral mathematical model for solving the heating system planning (HSP) problem taking into account minimizing the cost of heating system for a given life cycle time.
According to the particularity of HSP problem, the particle swarm optimization (PSO) algorithm was introduced and improved, the new definition and update rule of velocity and position vector were proposed, and the improvement approach about generating a random velocity was adopted to avoid particle swarm into local optimal solution. Then an actual case study was calculated to check its feasibility in practical use. The results show that the IPSO algorithm can more preferably solve the HSP problem than PSO algorithm.
Although there is no more discussion about the influence of computational results by changing the values of algorithm parameters (