Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3968324, 10 pages

http://dx.doi.org/10.1155/2016/3968324

## Dynamic Heat Supply Prediction Using Support Vector Regression Optimized by Particle Swarm Optimization Algorithm

School of Environment Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China

Received 31 December 2015; Revised 30 March 2016; Accepted 11 April 2016

Academic Editor: Antonino Laudani

Copyright © 2016 Meiping Wang and Qi Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We developed an effective intelligent model to predict the dynamic heat supply of heat source. A hybrid forecasting method was proposed based on support vector regression (SVR) model-optimized particle swarm optimization (PSO) algorithms. Due to the interaction of meteorological conditions and the heating parameters of heating system, it is extremely difficult to forecast dynamic heat supply. Firstly, the correlations among heat supply and related influencing factors in the heating system were analyzed through the correlation analysis of statistical theory. Then, the SVR model was employed to forecast dynamic heat supply. In the model, the input variables were selected based on the correlation analysis and three crucial parameters, including the penalties factor, gamma of the kernel RBF, and insensitive loss function, were optimized by PSO algorithms. The optimized SVR model was compared with the basic SVR, optimized genetic algorithm-SVR (GA-SVR), and artificial neural network (ANN) through six groups of experiment data from two heat sources. The results of the correlation coefficient analysis revealed the relationship between the influencing factors and the forecasted heat supply and determined the input variables. The performance of the PSO-SVR model is superior to those of the other three models. The PSO-SVR method is statistically robust and can be applied to practical heating system.

#### 1. Introduction

With the recent development of intelligent heating system in China, every component of the heating system requires intelligentization. In order to ensure heat supply of the users and energy conservation, the users’ heat supply demand should be accurately calculated. However, the dynamic heat supply prediction remains a longstanding challenge because of the nonlinear problems of the huge pipe networks system under the increasing growth tendency of heat users, such as heat decline in conveying process, large time delay, and huge thermal inertia of the heating system itself. The accuracy of the prediction model directly affects user’s comfort and the economical operation of the heating system. Therefore, it is necessary to accurately predict dynamic heat supply of the whole heating system.

Though the tremendous progress has been made, large prediction deviation for dynamic heat supply still exists and leads to the high energy consumption of the whole heating system and the thermal comfort problem of users. Most of existing prediction methods belong to linear models, in which it is assumed that heat supply of the system only varies with the outdoor temperature without considering other influencing factors. The methods are relatively accurate only when they are used to describe the dynamic thermal load of building envelope. However, the dynamic heat supply of the whole heating system includes heat consumption of exterior protected construction and heat loss from the pipeline and different devices to surrounding environment in heating system. The prediction of the dynamic heat supply is a complicated nonlinear problem. When the predicted heat value is too low or too high, it will cause resource waste or insufficient heat supply to users. These factors, including the supply and return water temperature, the supply and return water pressure, and the flow, which are used to reflect internal performance of heating system, should be considered together with outdoor meteorological parameters. For a huge and complex heating system, it is almost impossible to obtain an accurate mathematics model. Therefore, it is necessary to develop a more precise method to predict the dynamic heat supply of heating system. Support vector regression (SVR) is generally used to solve the nonlinear small-sample problem. The model is based on support vector regression and widely used in power system [1, 2] and the prediction accuracy is significantly enhanced.

The SVR prediction model combined with particle swarm optimization (PSO) algorithms was used to predict the heat supply of the heat source for the heating system. Combined with the analysis of influencing factors, the method can achieve reasonable heat supply prediction, thus guaranteeing the normal operation of the heat supply system and energy conservation.

The paper is structured as follows. Some related works are outlined in Section 2. In Section 3, the SVR model and its optimization algorithm are introduced and the flowchart of the proposed method is designed. In Section 4, the influencing factors of heat supply are analyzed and the experiment and comparisons are performed to validate the proposed approach. Our conclusions are summarized in Section 5.

#### 2. Literature Review

##### 2.1. System Performance and Influencing Factors of Dynamic Heat Supply

Before predicting the dynamic heat supply of the heating system, it is necessary to explore the system performance and influencing factors. In general, the heat load is calculated as a steady-state value, which is not consistent with the actual value. Westphal and Lamberts presented a transfer function method to analyze the dynamic heat load of nonresidential buildings based on simplified meteorological data [3]. A model and corresponding computer code, which are based on the accurate high-order numerical solution of the transient energy equation and the hydraulic prediction of pressure and fluid flow rates within the complex pipe network, are developed to simulate the thermal transients in local heating systems [4]. In a parameter estimation procedure [3], the hourly space heating and cooling loads from the monthly energy consumption were restored. The procedure was based on a nonlinear multivariate regression approach. In the literature [5], an assessment method for describing daily heat load variations was described in view of two basic conditions. Firstly, it was independent from the system size. Secondly, external parameters were not analyzed. A simplified building procedure spreadsheet was presented to evaluate energy demand in an early design stage of dwellings [6] and the spreadsheet required only fewer input data to describe the building design within a short time. The two influencing factors of prediction loads in district-heating systems, the outdoor temperature and the social behaviors of consumers, were analyzed [7]. Solar radiation and heat demand increase caused by a higher wind speed were taken into consideration in the prediction of heat supply for a single building [8]. In seasonal heating load calculations [9], it was important to carry out climatologic investigations and develop empirical equations for determining the total duration of daytime temperatures and solar radiation intensities. The above studies were based on climate parameters and the characteristic of heating system and mainly focused on heat load of building envelope. However, only external factors of heat load were considered. With the expansion of city construction and increasing customers in China, dynamic heat supply prediction becomes more complicated because of the nonlinear problem of central heating system. Dynamic heat supply is not only affected by external factors, but also related to internal factors.

##### 2.2. Prediction Methods of Dynamic Heating Load

Nowadays, various predicting techniques have been developed. Due to the complex nonlinear property of central heating system, dynamic heating load prediction needs to be improved through nonlinear models. The existing prediction methods include autoregressive integrated moving average (ARIMA) model [10, 11], linear regression (LR) technique [12], artificial neural network (ANN) model [13, 14], wavelet neural network (WNN) [15], and gray model [16]. These prediction models did not perform well enough because each model considered a few factors or only one relevant factor. These models had some defects, such as the optimization of the weight and threshold of the neural network. In addition, these models had relatively poor robustness. Recently, with the developments of artificial intelligence, some coupled algorithms have obtained considerable development [17–19]. The accuracy and generalization of the algorithms have been improved more than previous traditional methods.

Although many methods for the dynamic heat supply prediction have been presented, the application is limited because of the nonlinear characters of heating systems, such as large thermal inertia, attenuation, and large time delay. It is difficult to accurately forecast dynamic heat supply of central heating system with precise mathematical models and traditional linear methods. The core of SVR in Vapnik [20, 21] is to describe the relationship between overfitting and generalization ability and to control the capacity of machine learning through introducing structure risk minimization. Compared with traditional methods, the method requires fewer samples, avoids the dimension disaster and the local minimum problem, and shows the good adaptability to nonlinear problems. In addition, the significant characteristic of the SVR is to transform a nonlinear problem into a linear problem in a higher-dimensional feature space with kernel functions. The SVR is applicable to predict nonlinear dynamic heat supply of the heating system. The presented method might be just a feasible solution of dynamic load prediction of heating system.

In order to improve the prediction accuracy, a fusion algorithm for heat supply based on SVR and PSO was proposed and some actual data were provided to verify the proposed method.

#### 3. The Proposed Prediction Method

##### 3.1. Support Vector Regression (SVR)

The notions of support vector machine (SVM) for the case of regression are introduced briefly. Given a set of data , where ; ; is the total number of data patterns, a nonlinear mapping function, : , is defined to map the training (input) data into the so-called high dimensional feature space (which may have infinite dimensions), . Then in the high dimensional feature space, there theoretically exists a linear function, , which can be used to formulate the nonlinear relationship between input data and output data. The linear function, namely, SVR function, is expressed aswhere denotes the prediction values; , which maps inputs vectors into a high dimensional feature space, is a nonlinear mapping; coefficient () and () are adjustable and can be estimated by minimizing the following regularized risk function:where and are prescribed parameters. In (2), is called the -insensitive loss function. The loss equals zero if the forecasted value is within the -tube (3). The second term, , indicates the flatness of the function. Therefore, is considered to specify the trade-off between the empirical risk and the model flatness. The parameters and are user-determined. Two positive slack variables and denote the distance from actual values to the corresponding boundary values -tube. Therefore, (2) can be transformed into the quadratic optimization problem with inequality constraints:with the constrains , , , .

The optimal weight of the regression hyperplane is calculated aswhere , are the Lagrangian multipliers obtained by solving a quadratic program.

So the SVR regression function is obtained as (6) according to the duality theory,where is defined as the kernel functions and the value of the kernel equals the inner product of two vectors, and , in the feature space and , respectively. That is to say, . Any function which meets Mercer’s condition [22] can be used as the kernel function.

There are several kernel functions, such as the polynomial kernel with an order of and the constants of and , , and the Gaussian radial basis functions (RBF) with a width of , . However, it is difficult to determine which kind of kernel function is suitable for some specific data patterns [23]. The Gaussian kernel RBF can be easily applied and map original domain into the higher-dimensional space in a nonlinear way. Therefore, it is suitable to deal with the nonlinear relationship problem. The Gaussian kernel RBF is adopted in this study.

The determination of three parameters (including the penalty parameter (), the insensitive loss function , and the kernel parameter ) of the SVR model will largely affect the forecasting accuracy. Thus, it is necessary to develop an efficient approach to simultaneously meet three optimal parameters. Particle swarm optimization (PSO), which was a good method to optimize SVR model parameters, was used to predict the cyanotoxin content [24]. Therefore, in the study, we adopted the optimization model to predict the dynamic heat supply of central heating system.

##### 3.2. Particle Swarm Optimization (PSO) Algorithm

The particle swarm optimization (PSO) is an intelligent optimization algorithm [25, 26]. In PSO, a population of particles or proposed problems gradually evolves towards the optimal solution of the problem after each iteration. Genetic Algorithms [27, 28], Differential Evolution [29], and Ant Colony System [30] are also intelligent optimization algorithms, in which the evolution process of the population involves random factors. Indeed, a new population in the PSO algorithm is obtained by shifting the position of the previous one at each iteration. In its movement, each individual is influenced by its neighbor and its own trajectory.

Supposing a -dimension search space, one population is composed of particles. The state information of the particle can be expressed through two -dimensional vectors: expresses location information of particle and expresses velocity information, which decides the flying direction and distance of a particle.

In the simulation of each particle, two factors should be considered. One is the personal best position, expressed as ; the other one is the global optimum value, expressed as . The algorithm updates the positions and the velocities of the particles following [31]The velocity of each particle at iteration depends on three components:(i)The previous step velocity term, , affected by the constant inertia weight, ;(ii)The cognitive learning term, which is the difference between the existing best particle position (called , local best) and the current particle position ;(iii)The social learning term, which is the difference between the global best positions found in the entire swarm (called , global best) and the current particle position ;where and are random numbers distributed uniformly in the interval ; is used to indicate the understanding of the particle and called the weight coefficient tracking the historical best value of particle itself; is used to indicate the knowledge to whole group and called the weight coefficient tracking the best value of the group. *β* is called constriction factor used to restrain the speed of the updating position, where is the inertial weight coefficient to maintain the original speed. Generally, when is a nonnegative number, the algorithm shows the better global search capability for the large and the better local search capability for the small ,where and represent the maximum and minimum weight factors, respectively; is the current iteration; is the maximum iteration. In (9), when the value gradually decreases in the search process, it meets the demand of the adaptive process for the algorithm from global optimization to local optimization.

##### 3.3. Particle Swarm Optimization (PSO) Optimizing the Parameters of SVR Model (PSO-SVR)

Based on the above SVR modeling theory and PSO optimization algorithm, the proposed algorithm of dynamic heat supply is shown in Figure 1.