Resistance Characteristic Parameters Estimation of Hydraulic Model in Heating Networks Based on Real-Time Operation Data

Abstract

Heating systems are essential municipal infrastructure in winter, especially in severe cold cities of China. The safety and efficiency of heating systems greatly affect building energy efficiency and indoor thermal comfort. Heating networks (HNs), playing the role of transportation, are the key parts of heating systems. In HNs, hydraulic models could be affected by the accuracy of resistance characteristic parameters, which are expressed by pipe friction parameters (PFPs) in this paper. As the uniqueness of the estimation results of PFPs has not been discussed in previous studies, this paper builds an estimation method of PFPs by dividing two types of pipes, substituting variables and establishing a split-step linearization method. Combining with the theory of matrix equations, the decision conditions and solution methods for obtaining the unique estimation results of PFPs are determined. Theoretical analysis and case study results show that estimation values of PFPs can be obtained by utilizing measured data under multiple hydraulic conditions. In the example of DN and the simple actual HN, the average estimation deviation of PFPs is 1.42% and 1.86%, which are accurate enough for actual engineering. Estimation results of PFPs obtained by this method guarantee the accuracy of analysis and regulation in heating systems and improve social energy utilization efficiency.

1. Introduction

With rapid urbanization and residents’ requirements for thermal comfort, the total heating floor area in Northern China reached 10.8 billion m² in 2019. Approximately 5% of the total energy consumption of the country was utilized for space heating [1]. Reducing the energy consumption of heating systems is the key to achieving carbon peak and carbon neutrality in China. Heating networks (HNs) are the carrier of heat medium (hot water) in heating systems, and their major function is to transport and distribute heat medium on demand [2]. An HN needs to make the flow and thermal energy meet the requirements at the same time, that is, to achieve hydraulic balance and thermal balance. Otherwise, uneven heating, low energy efficiency or a high possibility of equipment failure appear easily. In heating systems, hydraulic balance is an essential precondition for thermal balance, and the hydraulic balance depends on the hydraulic model. Thus, obtaining the hydraulic model of HN is the key to realizing the hydraulic and thermal balance.

The hydraulic model has two essential factors, which are the topology and resistance characteristics of an HN. The design documents can provide the topology but not resistance characteristics of an HN. There are three reasons listed as follows:

• Due to the complexity of an HN, it is impossible to ensure that the design and actual resistance characteristics of the HN are completely consistent in detail;
• Due to the initial adjustment of an HN, the resistance characteristics of the HN are uncertain when it is just put into use;
• Due to corrosion or blockage of pipelines (especially the long service ones), the resistance characteristics may be far away from the initial state.

It is recognized as an effective method in that the resistance characteristics are determined by the real-time measured values of operation parameters (such as node heads and outflows). In actual engineering, pipe friction parameters (PFPs) are commonly used to express the resistance characteristics. If the estimation values of PFPs can be obtained, a real-time hydraulic model of the HN can be constructed. Wang et al. [3] concluded that the pump power consumption can be reduced to 85% by using the identified PFPs in the hydraulic model. Moreover, the analysis of hydraulic and thermal conditions based on estimation values of PFPs will greatly improve the energy efficiency of the HN, which will be helpful in achieving carbon peak and carbon neutrality.

In actual HNs, the nodes are usually equipped with pressure sensors. Real-time measured values of node heads can be obtained. HNs, being different from water supply networks, are closed networks and composed of supply networks and return networks. (In this paper, supply networks are taken as an example to analyze, and the analysis process of return networks is the same as that of supply networks.) In the supply networks, the flows of heat sources and heat users can be regarded as node outflows, which can be obtained usually. Based on these conditions, the measured conditions for real-time estimation of PFPs can be determined.

At present, the research on characteristic parameters estimation mainly focuses on water supply networks. Scholars have studied estimation and calibration problems based on least-squares methods of water supply networks for several decades. Nash and Karney [4] calibrated hydraulic models based on the least-squares method, and an objective function was expressed as the difference between measured and calculated values. Reddy et al. [5] combined a least-squares method with Gauss–Newton method. This calibration method was verified in a small-scale water supply network. Savic and Walters [6] researched a simulation and calibration problem of water supply networks. The research result shows that the accuracy of a calculation result by utilizing measured data over a period of time is better than only relying on measured data under one hydraulic condition. Meirelles et al. [7] considered operational flows as a basis of PFPs calibration. Meirelles’s method relieves adverse effects on PFPs identification results obtained by using inadequate data. Shamloo and Haghighi [8] added a genetic algorithm to solve an optimization model by utilizing the sequential quadratic programming method. Better results can be obtained. However, the time consumption of these calculation processes is enormous, especially for PFPs estimation under multiple hydraulic conditions. Optimization methods based on least-squares methods have been widely applied in engineering. The objective is to search for an acceptable result for hydraulic calculation equations. However, ensuring the acceptable solution (PFPs values) is the same as the actual resistance characteristic cannot be guaranteed.

The mainstream of PFPs identification methods is based on least-squares principles. However, analytic methods also play an important role in PFPs identification studies. Lansey and Basnet [9] proposed a method based on the gradient method and non-linear programming technology to estimate unknown parameters of water supply networks. Calculation processes contain parameter estimation, calibration assessment and data collection design. Jun et al. [10] chose measured data under multiple hydraulic conditions and expressed results of non-linear equations. This method can be applied to inadequate data. By using the Taylor series approach, Datta and Sridharan [11] proposed a method to estimate unknown parameters by utilizing measured values of node heads and pipe flows. Kapelan et al. [12] combined Levenberg–Marquard algorithm with genetic algorithms. Relatively reliable results could be obtained with this method. For a water supply network, Liu et al. [13,14] proposed an identification method of PFPs based on Moore–Penrose pseudo-inverse solution. Identification results are accurate enough for engineering when the number of measured sites is relatively small. Further, in order to meet the current demands of digital twin technology and intelligent heating technology, it is necessary to develop a method to obtain real-time PFPs values [15].

Being different from water supply networks, an HN needs to be decomposed into two sub-HNs, which increases calculation quantity. More importantly, in order to meet the requirements of heating demand, the flows of HNs are usually constant and huge. Therefore, accurate estimation of PFPs values is more important for HNs. Wang et al. [3] developed an identification method to obtain hydraulic resistance of a branch HN. However, 500 operating conditions need to be provided if relatively small errors of the identified hydraulic resistances are expected. Liu et al. [16] proposed a method to determine the PFPs of a branch HN uniquely by using measured data of heads and outflows in all the heating substations. However, it is difficult to apply this method directly to real-time PFPs analysis of an HN, including loops. Tol [17] and Zheng et al. [18] applied Newton iteration method in PFPs estimation of a branch HN. In the studies mentioned above, estimation methods have been proposed for branch HNs rather than loop HNs, which are more common in engineering. Equations described in hydraulic models are non-linear ones, also creating a barrier for the HN study.

By utilizing measured data under multiple hydraulic conditions, this paper suggests a new expression and decision condition to find unique estimation results of PFPs. Following the four steps mentioned below, this study presents two examples of HNs, illustrating the ability to obtain PFPs unique estimation values by utilizing measured data under multiple hydraulic conditions. First, build a PFPs calculation equation by using mass and energy conservation equations of an HN, and eliminate dependent variables by the relationship of pipe flows according to the types of the pipes. Second, describe energy conservation equations corresponding to tree pipes by relationships of operational data between different hydraulic conditions. Third, transform matrix equations of a tree into a linear form. Finally, analyze theoretically the probability of obtaining unique estimation values of PFPs, and express the results of PFPs estimation by a solution of the corresponding matrix equation.

2. Methods

Hydraulic calculation equations (mass and energy conservation equations) can be established by using operational data or PFPs of an HN. In the design calculation of HNs, PFPs are viewed as known variables. In the estimation process of PFPs, some measured data such as node heads and outflows are viewed as known variables. Pipes in an HN can be divided into a spanning tree (tree) and a corresponding cotree (cotree). PFPs calculation equations of an HN are first described by partitioned matrices and then transformed into linear algebraic equations. According to theories of a linear algebraic equation and matrix analysis, in order to obtain a unique solution, the number of equations being more than or equal to the number of unknown variables should be ensured. However, in actual engineering, the number of measurement sites is limited. In most cases, the number of unknown variables is more than the number of (independent) equations in PFPs calculation equations. This paper researches the PFPs estimation problem of an HN by increasing the number of hydraulic conditions and establishes a theoretical framework for estimating the PFPs values and obtaining unique values.

2.1. Basic Equations

Two matrix functions are defined before further study. Considering a matrix M of dimension m × n, abs (M) is a matrix of dimension m × n. Every element of the matrix abs (M) is the absolute value of the element of the matrix M at the corresponding position. Considering a vector X of dimension n × 1, a matrix transform function D = D (X) is defined. D is a diagonal matrix of dimension n × n. The ith diagonal element of D satisfies this relationship D (i, i) = X (i), (i = 1, 2, …, n).

Considering an HN with n + 1 nodes and b pipes, hydraulic calculation equations can be expressed in the following matrix form.

AG = Q,

(1)

$$abs\left({\mathit{A}}^T\mathit{P}\right)=abs\left(\Delta \mathit{H}\right)=D^2\left(\mathit{G}\right)\cdot \mathit{S},$$

(2)

where A is a basic incidence matrix, which contains information about which pipes are connected to a particular node. The value of every element in matrix A can be defined as −1 or 1 according to the pipe flow being towards or away from the node; else, if the pipe flow does not connect with the node, the corresponding element is defined as 0. G is a pipe flows vector, Q is a node outflows vector, P is a node heads vector, $\Delta H$ is a head losses vector and S is a PFPs vector.

Generally, some operational data (node heads and pipe flows) can be obtained in the PFPs estimation study. The main aim of this study is to search for unique values of PFPs by using operational data. If all measured data of node heads and outflows under a single hydraulic condition are available, there will be b + n equations and 2b unknown variables in Equations (1) and (2). In an HN, b is greater than n. Thus, the number of unknown variables is greater than the number of algebraic equations. Equations (1) and (2) are actually under-determined. Thus, it is not able to find unique results of PFPs in the situation. To obtain more algebraic equations, operational data of more hydraulic conditions are provided. However, these algebraic equations are non-linear and cannot be solved directly. Thus, a linearization method is needed to transform the non-linear equations into linear ones during studying PFPs estimation.

Analyzing the feasibility of obtaining unique values of PFPs, the paper establishes a split-step linearization method for PFPs estimation by using measured data under multiple hydraulic conditions. Since measured data of node heads are widely used in the detection and calibration of hydraulic networks [19,20], considering the most unfavorable estimation conditions, the estimation processes assume there are no pipe flow sensors. (If there are pipe flow sensors, the theory mentioned in this paper can be applied, obtaining better results.)

Since there are unknown variables S needing to be identified in the energy conservation equation (Equation (2)), the mass conservation equation (Equation (1)) can be used to eliminate dependent unknown variables. Flows in tree pipes, satisfying the mass conservation equation, can be written as [21].

$${\mathit{G}}_t=-{\mathit{A}}_t^{-1}{\mathit{A}}_l{\mathit{G}}_l+{\mathit{A}}_t^{-1}\mathit{Q},$$

(3)

where A_t is composed corresponding to tree pipes, A_l is composed corresponding to cotree pipes, satisfying A = [A_t, A_l]; G_t is composed by tree pipe flows, G_l is composed by cotree pipe flows, satisfying G = [G_t, G_l].

The energy conservation equation can be divided into two types. Head losses of tree and cotree pipes can be written as follows, respectively.

$$abs\left({\mathit{A}}_t^T\mathit{P}\right)=abs\left(\Delta {\mathit{H}}_t\right)=D^2\left({\mathit{G}}_t\right)\cdot {\mathit{S}}_t,$$

(4)

$$abs\left({\mathit{A}}_l^T\mathit{P}\right)=abs\left(\Delta {\mathit{H}}_l\right)=D^2\left({\mathit{G}}_l\right)\cdot {\mathit{S}}_l,$$

(5)

where S_t is composed of tree PFPs, S_l is composed of cotree PFPs, satisfying S = [S_t, S_l]; ΔH_t is composed of head losses of tree pipes, ΔH_l is composed of head losses of cotree pipes, satisfying ΔH = [ΔH_t, ΔH_l]. Then, PFPs can be solved when head losses and pipe flows are available.

2.2. Expression of PFPs Estimation

A variable representing a relationship of head losses between different hydraulic conditions is introduced here. That is a critical process of solving PFPs calculation equations. Head losses relationship of cotree pipes between multiple hydraulic conditions can be written once the measured value of every node head is available. The first hydraulic condition satisfying the condition $\Delta h_l^{(1)}(i)\ne 0$ is chosen as a reference hydraulic condition. A reference hydraulic condition can be selected randomly and does not impact the estimation study. Then, a ratio of head losses in cotree pipe i between the kth and the first hydraulic conditions can be written as follows:

$$\Delta h_{lP}^{\left(k\right)}\left(i\right)=\Delta h_l^{\left(k\right)}\left(i\right)/\Delta h_l^{\left(1\right)}\left(i\right), (i=1, 2,\dots , b-n;k=1, 2,\dots , m)$$

(6)

where $\Delta h_l(i)$ is the head loss in cotree pipe i and $\Delta h_{lP}(i)$ is a ratio of head losses in cotree pipe i under the kth and the first hydraulic conditions. The superscript k is an order number of hydraulic conditions.

Variables in energy equations contain both node heads and pipe flows. To eliminate dependent variables in equations, a relationship between node heads and pipe flows under multiple hydraulic conditions needs to be introduced. Considering pipe flows are directional variables, a new vector variable $\Delta {\mathit{H}}_{lG}^{(k)}$ of dimension (b − n) × 1 is defined to express relationships of cotree pipe flows under different hydraulic conditions. A ratio of cotree pipe flow i between the kth and the first conditions (the ith element of $\Delta {\mathit{H}}_{lG}^{(k)}$) is:

$$\Delta h_{lG}^{(k)}(i)=\{\begin{array}{ll}abs{\left(\Delta h_{lP}^{(k)}(i)\right)}^{1/2},when \Delta h_l^{(k)}(i)\cdot \Delta h_l^{(1)}(i)\ge 0, \Delta h_l^{(1)}(i)\ne 0& (i=1, 2,\dots , b-n;\hfill \\ -abs{\left(\Delta h_{lP}^{(k)}(i)\right)}^{1/2},when \Delta h_l^{(k)}(i)\cdot \Delta h_l^{(1)}(i)<0& k=1, 2,\dots , m)\hfill \end{array}$$

(7)

By using Equation (7), a ratio of cotree pipe flow i is written by the ratio of head losses in the corresponding pipe between the kth and the first hydraulic conditions. Combining with cotree pipe flows under the first hydraulic condition, those under the kth hydraulic condition can be expressed as follows:

$$G_l^{\left(k\right)}=D\left(\Delta H_{lG}^{\left(k\right)}\right)\cdot G_l^{\left(1\right)}, (k=1, 2,\dots , m)$$

(8)

As dependent variables, pipe flows of a cotree do not appear except for the first hydraulic condition. Under the kth hydraulic condition, Equation (4) can be written, in view of Equation (8), as follows:

$$abs\left({\mathit{A}}_t^T{\mathit{P}}^{\left(k\right)}\right)=abs\left(\Delta {\mathit{H}}_t^{\left(k\right)}\right)=D^2\left(-{\mathit{A}}_t^{-1}{\mathit{A}}_{l}D\left(\Delta {\mathit{H}}_{lG}^{\left(k\right)}\right)\cdot {\mathit{G}}_l^{\left(1\right)}+{\mathit{A}}_t^{-1}{\mathit{Q}}^{\left(k\right)}\right)\cdot {\mathit{S}}_t (k=1, 2,\dots , m)$$

(9)

After eliminating cotree pipe flows under other hydraulic conditions except the first one, Equation (4) can be substituted by Equation (9). Considering Equation (9) expresses a non-linear equation, a new matrix variable is defined to transform Equation (9) into a linear equation. A vector $\Delta {\mathit{H}}_{tG}^{(k)}$ of dimension n × 1 is introduced here. An element of this vector is $\Delta h_{tG}^{(k)}$, whose absolute value equals head loss in a tree pipe to the half power under the kth hydraulic condition. Considering pipe flows are directional variables, the ith element of $\Delta {\mathit{H}}_{tG}^{(k)}$ is:

$$\Delta h_{tG}^{\left(k\right)}\left(i\right)=\{\begin{array}{l}{\left(\Delta h_t^{\left(k\right)}\left(i\right)\right)}^{1/2},when \Delta h_t^{\left(k\right)}\left(i\right)\ge 0\\ -{\left(-\Delta h_t^{\left(k\right)}\left(i\right)\right)}^{1/2},when \Delta h_t^{\left(k\right)}\left(i\right)<0\end{array}, (I=1, 2,\dots , n; k=1, 2,\dots , m)$$

(10)

where $\Delta h_t(i)$ is the head loss in tree pipe i. Equation (10) is preparation for transforming PFPs calculation equations into linear ones. A new vector S_tp is defined to express a vector related to a cotree PFPs. S_tp has the same dimension as the vector S_t. The relationship of elements can be written as $s_{tp}$ = $s_t^{-1/2}$, meaning $s_{tp}$ equals one divided by $s_t$ to the half power. The linear calculation matrix equation of PFPs is written as:

$$D\left(\Delta {\mathit{H}}_{tG}^{\left(k\right)}\right)\cdot {\mathit{S}}_{tp}+{\mathit{A}}_t^{-1}{\mathit{A}}_l\cdot D\left(\Delta {\mathit{H}}_{lG}^{\left(k\right)}\right)\cdot {\mathit{G}}_l^{\left(1\right)}={\mathit{A}}_t^{-1}{\mathit{Q}}^{\left(k\right)}, (k=1, 2,\dots , m)$$

(11)

In Equation (11), coefficient matrices can be represented by node heads and outflows. Unknown variables include cotree pipe flows and PFPs of a tree. The numbers of the two kinds of unknown variables are b − n and n, respectively. Equation (11) can substitute Equation (9) since it is a linear matrix equation and does not contain dependent unknowns.

Even though measured data of node heads and outflows under one hydraulic condition can be obtained, there are n equations and b independent unknown variables in Equation (11). As the number of pipes is greater than the number of nodes, Equation (11) expresses an equation with the number of unknown variables being more than that of equations. In this situation, accurate, unique solutions are unavailable. Thus, operational data of multiple hydraulic conditions are needed to get more accurate results for PFPs.

2.3. Linear Expression of PFPs

If operational data of m hydraulic conditions are available, an integrated matrix equation of PFPs calculation can be written as follows.

$$\left[\begin{array}{ll}\hfill D\left(\Delta {\mathit{H}}_{tG}^{\left(1\right)}\right)\hfill & \hfill {\mathit{A}}_t^{-1}{\mathit{A}}_l\cdot D\left(\Delta {\mathit{H}}_{lG}^{\left(1\right)}\right)\hfill \\ \hfill D\left(\Delta {\mathit{H}}_{tG}^{\left(2\right)}\right)\hfill & \hfill {\mathit{A}}_t^{-1}{\mathit{A}}_l\cdot D\left(\Delta {\mathit{H}}_{lG}^{\left(2\right)}\right)\hfill \\ \hfill \cdot \hfill & \hfill \cdot \hfill \\ \hfill \cdot \hfill & \hfill \cdot \hfill \\ \hfill D\left(\Delta {\mathit{H}}_{tG}^{\left(m\right)}\right)\hfill & \hfill {\mathit{A}}_t^{-1}{\mathit{A}}_l\cdot D\left(\Delta {\mathit{H}}_{lG}^{\left(m\right)}\right)\hfill \end{array}\right]\cdot \left[\begin{array}{l}{\mathit{S}}_{tp}\\ {\mathit{G}}_l^{\left(1\right)}\end{array}\right]=\left[\begin{array}{c}{\mathit{A}}_t^{-1}{\mathit{Q}}^{\left(1\right)}\\ {\mathit{A}}_t^{-1}{\mathit{Q}}^{\left(2\right)}\\ \cdot \\ \cdot \\ {\mathit{A}}_t^{-1}{\mathit{Q}}^{\left(m\right)}\end{array}\right]$$

(12)

There are m × n equations and b unknown variables. In Equation (12), if the number of independent equations can reach b, there must be unique solutions. If the number of independent equations is less than b, unique solutions cannot be reached. When the number of independent equations can reach b, unknown variables, including tree PFPs and cotree pipe flows under the first hydraulic condition, can be obtained.

Next, pursuing the values of PFPs can be transferred to solve Equation (12), which is a linear matrix equation. Depending upon the results, there will be unique estimation values of PFPs. To study Equation (12), the theory of matrix analysis is employed. There are two cases according to the number of independent equations in Equation (12). If the number of independent equations equals the number of unknown variables, a unique solution can be calculated directly; if not, a Moore–Penrose pseudo-inverse is introduced to express the solution of Equation (12) [13]. When the number of independent equations equals the number of unknown variables, the special Moore–Penrose pseudo-inverse solution can also express a unique solution of Equation (12) [22]. Estimation results of PFPs can be calculated through the unique solution.

Since Equation (12) is a linear matrix equation, whether there is a unique solution depends on the relationship between the numbers of algebraic equations and unknown variables. When there are m hydraulic conditions, Equation (12) contains m × n algebraic equations and b unknown variables. In those unknown variables, there are b − n cotree pipe flows and n tree PFPs. A necessary condition to obtain a unique solution is the number of algebraic equations greater than or equal to the number of unknown variables. However, it cannot be guaranteed that the number of independent equations can reach b. If hydraulic conditions of an HN are enough to express b independent equations, unique results of PFPs may be obtained; if not, accurate solutions cannot be obtained.

As mentioned above, if hydraulic conditions are sufficient to obtain a unique solution to Equation (12), the PFPs of a tree can be communicated directly. Cotree PFPs can be expressed by cotree pipe flows under the first hydraulic condition. Inversion processes of variables are written as follows:

$$s_t\left(i\right)={\left(s_{tp}\left(i\right)\right)}^{-2} (i=1, 2,\dots , n),$$

(13a)

$${\mathit{S}}_l=D^{-2}\left({\mathit{G}}_l^{\left(1\right)}\right)\cdot abs\left(\Delta {\mathit{H}}_l^{\left(1\right)}\right)=D^{-2}\left({\mathit{G}}_l^{\left(1\right)}\right)\cdot abs\left({\mathit{A}}_l^T{\mathit{P}}^{\left(1\right)}\right),$$

(13b)

According to the theory of linear equations, the number of independent equations being b is a sufficient and necessary condition of the existence of unique solutions. The number of independent equations in Equation (12) increases when adding hydraulic conditions, increasing the probability of adding the number of independent equations. If the number of hydraulic conditions can guarantee the number of independent equations can reach b, there is no need to keep on adding hydraulic conditions. At the beginning of PFPs estimation, the criteria mentioned here can be used to determine whether accurate, unique estimation values of PFPs can be obtained, bringing the advantage of not needing additional hydraulic conditions when there is no error with measured data.

With errors of measured data being considered, if measurement sensors are sufficient, more accurate estimation results of PFPs will be obtained by increasing hydraulic conditions. If not, increasing hydraulic conditions has no contribution to the accuracy of estimation results of PFPs. There is a possibility of a large difference between estimation results and real results of PFPs if measurement sensors are insufficient. The main processes of this method are shown as in Figure 1.

3. Case Study

To present application processes and demonstrate the effectiveness of the proposed method, two cases of HNs are introduced in this paper.

3.1. An Example HN

The estimation method is verified by an example HN. In the example, PFPs values can be calculated by utilizing measured data of node heads and outflows under two hydraulic conditions. The topology of the HN is shown in Figure 2. This example HN contains twelve pipes and nine nodes, wherein nine nodes represent three heat sources and six heat users. In this HN, setting values of PFPs in pipe 1, 3, 5, 8, 9 or 10 is 3.6 × 10⁻⁴ h²/m⁵, in pipe 2 or 4 is 6.0 × 10⁻⁴ h²/m⁵, and in pipe 6, 7, 11 or 12 is 7.2 × 10⁻³ h²/m⁵. By using the values of PFPs, node heads and outflows under two hydraulic conditions used in the next sections (estimation processes) are computed by hydraulic calculation equations. In this example, node heads and outflows of different hydraulic conditions are assumed as “known values”, and PFPs and pipe flows are considered as unknown variables. The known values under two hydraulic conditions are shown in Figure 2.

Pipes 6, 7, 11 and 12 in this HN are selected as cotree pipes. A linear equation is obtained by submitting “known values” in Equation (12). The example HN contains 12 pipes and nine nodes. According to Equation (12), when there are two hydraulic conditions available for PFPs estimation of the HN shown in Figure 2, there are 16 equations and 12 unknown variables, including eight variables related to PFPs of a tree and pipe flows of a cotree under the first hydraulic condition, illustrating a unique solution of Equation (12) exists. Further, in Equation (12), the number of independent equations can reach 12. According to the theory of linear equations, there are unique solutions to Equation (12). Those unknown variables can be solved by calculation tools included in Scilab 6.1.0. Then, the PFPs of the entire HN are calculated by Equation (13). The estimation results are presented in

Table 1. The estimation results of the example HN.

Pipe No.	Setting Values of PFPs (h2/m5) × 10−4	Estimation Values of PFPs (h2/m5) × 10−4	Estimation Deviation of PFPs (%)
1	3.6	3.58	0.56
2	6.0	6.0	0.00
3	3.6	3.63	0.83
4	6.0	6.04	0.67
5	3.6	3.60	0.00
6	72	68.4	5.00
7	72	73.4	1.94
8	3.6	3.63	0.83
9	3.6	3.55	1.39
10	3.6	3.57	0.83
11	72	73.6	2.22
12	72	70.0	2.78

.

By comparing with setting values of PFPs, the accuracy of PFPs estimation results can be evaluated. Estimation values of PFPs are found very close to setting values, except for the cotree pipes 6, 7, 11 and 12. The main reason can be concluded as follows. In the inversion process of variables, PFPs values of a tree and a cotree are calculated out by Equation (13). Since there is a one-to-one correspondence between PFPs values of a tree and variables representing the values in Equation (13a), other operational data will not be used in the inversion process. This process may give a small error. PFPs values of a cotree need to be calculated by Equation (13b), in which pipe flows and node heads need to be calculated through the inversion process. Not as direct as calculation of tree PFPs, this process brings about limited calculation errors, leading to calculation errors accumulating in PFPs values of a cotree. The maximum estimation deviation of every PFP is less than 5%, and the average estimation deviation is 1.42%, which is accurate enough for engineering.

3.2. A simple Actual HN

To further test the effectiveness of the proposed method, a simple actual HN, located in Harbin City, China, is employed here. This HN represents a primary network that connects heat sources and heat users (heating substations). The simple actual HN is shown in Figure 3.

As shown in Figure 3, this simple actual HN contains 17 pipes and 16 nodes (including a reference node, s1). In this HN, there are 14 nodes representing 2 heat sources and 12 heating substations, and pipe 10 and pipe 15 are selected as cotree pipes. All node elevations are the same at 12 m. In this heating system, there are two circulating pumps operating in parallel in both heat sources. Performance parameters of circulating pumps are shown in

Table 2. Performance parameters of circulating pumps.

Heat Source	Pump Series	Flow (m3/h)	Head (m)	Revolution (r/min)	Power (kW)
s1	KQSN/L350-M6	756	148	1480	710
s2	KQSN/L300-N4	563	96	1480	250

.

Setting values of all PFPs in the simple actual HN are shown in

Table 3. Details of pipes in the simple actual HN.

Pipe No.	From Node	To Node	Length (m)	Diameter (mm)	PFP (h2/m5) × 10−5
1	1	3	500	300	3.01
2	3	4	400	300	2.41
3	4	5	800	250	12.5
4	5	15	400	250	9.4
5	6	15	400	250	9.4
6	2	6	1000	300	4.81
7	2	7	400	350	1.07
8	7	8	300	300	1.8
9	8	9	300	250	4.7
10	10	9	500	200	25.3
11	16	10	500	250	7.83
12	11	16	500	200	25.3
13	12	11	700	250	11
14	1	12	700	300	4.21
15	13	16	700	200	35.4
16	14	13	400	250	6.27
17	15	14	400	300	2.41

.

In this paper, multiple hydraulic conditions can be achieved by adjusting the valve in every heating substation. However, the valves of pipes in both supply and return pipelines are not adjusted, which can keep PFPs unchanged under different hydraulic conditions. Based on previous study results, when there are measured data of two hydraulic conditions available for PFPs estimation of the simple actual HN shown in Figure 3, unique estimation results of PFPs could be obtained. In this estimation process, node heads and outflows under these two hydraulic conditions are considered “known values”, which are listed in

Table 4. “Known values” for the simple actual HN.

Node No.	Node Outflows (m3/h)		Node Heads (m)
	The First HC 1	The Second HC	The First HC	The Second HC
1	1055	700	200	180
2	900	552.36	173.10	170.09
3	112	56	193.01	177.36
4	46	32.2	189.71	175.97
5	68	61.2	176.54	170.54
6	211	84.4	170.42	168.96
7	256	163.84	168.38	168.39
8	167	56.78	165.38	167.39
9	172	92.88	162.65	165.89
10	321	250.38	161.44	164.04
11	79	26.86	177.23	170.44
12	287	246.82	186.18	173.14
13	100	79	167.14	166.59
14	136	102	168.46	167.40
15	0	0	170.37	168.52
16	0	0	166.42	166.16

¹ HC—hydraulic condition.

.

Details of the PFPs estimation processes are almost the same as those in Section 3.1. The estimation processes are not repeated here. The estimation results are presented in Figure 4.

As shown in Figure 4, estimation results of PFPs are close to those setting values shown in Table 3. As a cotree pipe, the relative estimation deviation of pipe 15 (grey mark in Figure 4) is 11.40% (the largest one). As the estimation result of pipe 15 cannot be reached by direct calculation, the relative estimation deviation of pipe 15 is larger than those of the rest of the pipes. This result seems not particularly ideal. If more accurate results are expected, this estimation method theoretically should employ operational data of more hydraulic conditions. In addition, pipe 10 (grey mark in Figure 4) is another cotree pipe with a relatively larger estimation deviation, whose relative estimation deviation is 3.05%. (Relative estimation deviations of tree pipes are smaller than those of cotree pipes.) The average estimation deviation of every PFP is 1.86%, which illustrates that the estimation results of PFPs are accurate enough for actual engineering. The overall accuracy of those estimation results of the simple actual HN is close to those of the example HN introduced in Section 3.1. (As a matter of fact, relative deviations of PFPs estimation results could be reduced further by adding operation data of more hydraulic conditions or choosing different tree and cotree pipes division plans.)

4. Discussion

The two HNs mentioned above show the unique values of PFPs cannot be obtained unless measured data of node heads and outflows under multiple hydraulic conditions are available. When hydraulic conditions are sufficient, the number of independent equations may reach the number of unknown variables. In the estimation processes, using the least number of hydraulic conditions to identify unique values of PFPs can relieve the difficulty of providing more hydraulic conditions. During the estimation processes, estimation results of tree PFPs calculated directly may be more accurate. Both estimation results of tree and cotree are accurate enough in engineering.

However, there are also two major limitations in this paper:

• In actual engineering, PFP estimation is inevitably affected by measurement errors. The estimation principle herein is similar to that of the publication [16], and the unique estimation results can be obtained theoretically. According to the research results of Liu et al. [16], the influence of measurement errors on this series of methods is also limited. Estimation results can satisfy the requirements of engineering applications. Thus, error analysis was not included here.
• In this paper, a large number of measurement sites are needed for obtaining operational data on HNs. Currently, only high monitoring levels or newly built heating systems could meet the above requirement. In order to achieve PFPs estimation of heating systems with lower monitoring levels or experiencing temporary failure in individual sensors, PFP estimation based on deep neural networks is worthy of being explored in the future.

5. Conclusions

In this paper, PFPs, representing the resistance of an HN, are introduced. Then, a framework is proposed to identify PFPs values by measured data of nodes heads, but not pipe flows. The estimation processes can be achieved by solving matrix equation, which makes a theoretical basis for obtaining accurate resistance models for actual HNs.

In the estimation processes, mass and energy conservation equations were divided by a tree and a cotree. By eliminating dependent unknown variables and expressing relationships between pipe flows and head losses under different hydraulic conditions, a non-linear PFPs calculation equation was developed. This equation is then transformed into linear algebraic equations, providing the probability of further analysis. From studying these linear algebraic equations under multiple hydraulic conditions, the numbers of independent equations and unknown variables are determined, which can be used as a tool to judge a PFPs estimation problem. Estimation values of PFPs can be accomplished by rapidly solving the matrix equations. Utilizing operational data under multiple hydraulic conditions is highly advantageous to PFPs estimation because the number of independent algebraic equations in the PFPs calculation matrix equation increases with the addition of hydraulic conditions. If hydraulic conditions are sufficient, unique estimation results of PFPs can be achieved, and no more hydraulic conditions need to be provided.

This method can find unique results of PFPs directly by solving matrix equations under the least number of hydraulic conditions (usually from two to three hydraulic conditions), being essentially different from previous estimation or calibration methods. The PFPs estimation method mentioned in this paper improves the overall regulation and control performance of heating systems and energy efficiency in buildings.