Identification Method for Resistance Coefficients in Heating Networks Based on an Improved Differential Evolution Algorithm

Abstract

The intelligent upgrade of heating systems faces the challenge of accurately identifying high-dimensional pipe-network resistance coefficients; difficulties in accomplishing this can lead to hydraulic imbalance and redundant energy consumption. To address the limitations of traditional Differential Evolution (DE) algorithms under high-dimensional operating conditions, this paper proposes an Improved Differential Evolution Algorithm (SDEIA) incorporating chaotic mapping, adaptive mutation and crossover strategies, and an immune mechanism. Furthermore, a multi-constrained identification model is constructed based on Kirchhoff’s laws. Validation with actual engineering data demonstrates that the proposed method achieves a lower average relative error in resistance coefficients and exhibits a more concentrated error distribution. SDEIA provides a high-precision tool for multi-heat-source networking and dynamic regulation in heating systems, facilitating low-carbon and intelligent upgrades.

1. Introduction

Accelerating urbanization in northern China has led to significant increases in heating loads and carbon emissions, demanding higher efficiency in heating systems. Traditional decentralized, small-scale heat sources, characterized by low efficiency and severe pollution, have become inadequate to meet growing demand and environmental requirements. Consequently, the development of efficient, low-carbon, large-scale, centralized heating systems has become imperative. To enhance system reliability and coverage capacity, multi-source heating configurations are increasingly being adopted. Within such systems, dynamic variations in regional thermal loads compel heat source switching—requiring flexible adjustment of each source’s supply coverage. This results in shifts in their respective pipe-network service areas, which passively disrupts the established hydraulic balance. This disruption readily induces hydraulic maldistribution, causing energy waste, uneven heating distribution, and potential safety hazards. Furthermore, active regulation by end-users exacerbates hydraulic fluctuations within the network. Consequently, accurate analysis and prediction of hydraulic conditions has become critically important. Therefore, high-precision pipe-network resistance identification technology is essential for ensuring the stable operation and optimized control of heating systems.

The problem of identifying resistance coefficients in heating systems is a system parameter identification problem. Given a known mathematical model, model parameters are determined by adjusting the parameters themselves to minimize the error between the model output and actual measured data. Nash et al. [1] proposed a method for solving water supply network models based on the least-squares principle, aiming to address model calibration. Datta et al. [2] utilized weighted least squares to solve the pipe-network parameter identification problem for cases in which flow and pressure measurement data are partially available. Reddy [3] presented a weighted least-squares method based on the Gauss–Newton algorithm, aiming to minimize deviations between observed and calculated values. Zhang [4], Kang [5], and Gao [6] improved the calculation and optimization of resistance coefficients, using methods such as Jacobian matrices with grouping, least squares with gradient iteration, and linearization of non-linear problems, respectively. Although these methods remain applicable to resistance coefficient identification, they typically require high-quality initial estimates and often fail to converge to the global optimum or tend to become trapped in local optima when applied to large-scale networks [7]. Li [8], Bahlawan [9], and Bekibayev [10], respectively, employed multi-condition inversion methods based on graph theory and least squares, health index optimization, and data-driven iterative approaches to accurately reflect the actual resistance characteristics of pipeline networks. However, these methods heavily rely on dense sensor networks and costly multi-condition tests, while also suffering from computational complexity and poor real-time performance. Luo et al. [11] integrated matrix equation theory to determine decision conditions and solution methods in order to obtain unique estimates of pipe-section resistance coefficients. Case studies have demonstrated that estimates satisfying practical engineering requirements could be obtained using measured data under various hydraulic conditions. However, this resistance identification method, relying on dissimilar hydraulic conditions, requires that both nodal pressures and flows be observable, which is often not feasible in heating networks, due to a lack of sufficient pressure measurement points, thus imposing significant limitations.

With the rise of intelligent optimization algorithms, Genetic Algorithms (GAs) have been widely applied to resistance coefficient identification in pipe-network systems due to their global search capability. Savic [12] and Lingireddy [13], among others, employed GAs for pipe-network resistance identification, verifying the effectiveness of the algorithms in various typical scenarios. However, traditional GAs exhibit limitations in identification accuracy and convergence efficiency when confronted with topologically complex aging networks. Consequently, Dini [14], Zheng [15], and others adopted improvements using Ant Colony Optimization and Particle Swarm Optimization algorithms, but these methods suffered from error fluctuations in high-dimensional non-linear scenarios. Fan [16] and Zhao [17] et al. employed heuristic three-parent Genetic Algorithms and Differential Evolution algorithms, yet these methods still exhibited error fluctuations as the pipe-network scale increased. Greco [18], Zhou [19], and their colleagues proposed multi-condition iterative identification methods to reduce relative error through constraints from multiple sets of hydraulic data. Wang et al. [20] further optimized identification results based on multi-condition collaborative control. Nevertheless, these methods depend on large-scale condition calculations, resulting in low computational efficiency and difficulty in meeting the real-time control demands of high-dimensional networks. To address dynamically changing resistance coefficients, scholars have attempted to enhance identification efficiency and accuracy by combining different algorithms. For instance, Kapelan et al. [21] combined the Levenberg–Marquardt algorithm with GAs, Song et al. [22] used a Simulated Annealing–GA hybrid, and Wang et al. [23] employed a Genetic-Effective Set method, all achieving some success. However, challenges remain under high-dimensional complex conditions, creating an urgent need for novel collaborative optimization frameworks.

To overcome the limitations mentioned above, this study focuses on the Differential Evolution (DE) algorithm [24]. DE is a highly efficient global optimization algorithm, and research has proven its superior global search capability and optimization accuracy compared to GAs under high-dimensional conditions [17]. However, DE still faces bottlenecks such as low accuracy, strong parameter sensitivity, and a tendency to converge to local optima in high-dimensional complex scenarios. Therefore, this research concentrates on deep improvements to the DE framework. Existing studies have revealed several key directions for improvement: Chaotic initialization strategies can overcome the limitations of traditional random initialization, enhancing population diversity and avoiding local search traps in the early stages [25,26,27,28]. Adaptive parameter adjustment mechanisms can dynamically optimize the mutation factor and crossover probability, significantly improving algorithm convergence efficiency [29,30,31,32]. However, the existing enhanced DE variants may still exhibit insufficient convergence precision in high-dimensional complex problems, sensitivity to parameter settings, and a propensity to converge to local optima. Studies have demonstrated that functions based on immune mechanisms—such as the establishment of vaccine libraries, immune memory, and immunization—can, when hybridized with other algorithms, effectively enhance global search capabilities, maintain population diversity, and improve robustness [33]. Consequently, this study introduces core immune mechanisms into the DE framework to synergistically address its local optima entrapment and robustness issues, thereby enhancing overall algorithmic performance and adaptability.

This paper aims to address the inherent defects of the DE algorithm itself in high-dimensional network identification. An Improved Differential Evolution Algorithm (SDEIA), integrating chaotic mapping, adaptive mutation/crossover strategies, and an immune mechanism, is proposed for pipe-network resistance coefficient identification. To validate the algorithm’s effectiveness, resistance coefficient identification methods based on the standard Differential Evolution (DE), the improved SDEIA, standard Particle Swarm Optimization (PSO), and Genetic Algorithm Simulated Annealing (GASA) are applied to an actual heating network in Luoyang City. Comparative experiments are conducted in a multi-condition test environment. The results demonstrate that SDEIA out-performs DE, PSO, and GASA in both identification accuracy and solution stability within this real-world heating network application.

Based on Kirchhoff’s laws, a hydraulic calculation model is established as the theoretical foundation. By integrating characteristic parameters of the pipe-network system, a resistance coefficient identification model is formulated. After discussion of the limitations of traditional DE in high-dimensional parameter identification, its optimization mechanism is explored; this leads to the development of SDEIA. The paper is organized as follows: Section 2 focuses on modeling the drag coefficient identification problem, aiming to establish a bridge between practical problems and algorithms. Section 3 details the SDEIA algorithm design and the resistance identification process, introducing chaotic sequences to enhance population diversity, designing adaptive mutation/crossover operators to improve search efficiency, and incorporating an immune optimization strategy to strengthen global optimization capability. Section 4 presents the engineering validation, identifying four operating conditions using 31 measurement points in the actual network. A comprehensive performance comparison is conducted between the proposed SDEIA method and benchmark algorithms including Differential Evolution (DE), Particle Swarm Optimization (PSO), and Genetic Algorithm Simulated Annealing (GASA). Section 5 discusses the ways in which SDEIA’s high-dimensional superiority arises from the organic integration of three innovative improvements, along with the algorithm’s broader implications. Section 6 concludes the paper.

2. Modelling of the Resistance Coefficient Identification Problem

Modeling of the drag coefficient identification problem aims to establish a bridge between practical applications and algorithmic solutions, enabling the algorithm to comprehend and adapt to the characteristics of the actual engineering problem in order to achieve improved resistance coefficient identification. The mathematical model for this problem consists of an objective function and constraints. Due to the scarcity of pressure measurement points in actual heating networks and the difficulty in obtaining pressure data, the current objective function is based solely on flow error.

2.1. Objective Function for Resistance Coefficient Identification

The objective function is established by using the least-squares method [1] to compute the sum of squared errors between the system output values and the actual values. In minimizing this objective function, the resistance coefficient values that yield the smallest objective function are sought, enabling accurate identification of the heating network resistance. The expression for the objective function is given in Equation (1):

$$minf\left(S\right)=\sum _{n=1}^{NG}(G{M}_n-G{C}_n{)}^2$$

(1)

where F(S) is the objective function, representing the sum of squared errors between the calculated and observed flow rates at NG flow measurement points. S is the vector of pipe-section resistance coefficients generated by the optimization algorithm. The dimension of the vector equals the number of pipe sections, with each element representing the resistance coefficient value of a pipe section (Pa/(t/h)²). GMn is the calculated flow rate at measurement point n(t/h) (generated through iterative algorithm calculations). GCn is the observed flow rate at measurement point n(t/h) (measured in the actual operating network at a specific time). NG is the number of flow measurement points.

2.2. Constraints for Resistance Coefficient Identification

The mathematical model for resistance coefficient identification encompasses implicit and explicit constraints. Firstly, there are implicit constraints that arise from the principles of mass conservation and energy conservation and these must be satisfied by the heat carrier flowing within the pipes. These implicit constraints are automatically satisfied when the fundamental loop method is applied to solve the hydraulic calculation model. Secondly, the explicit constraints specify that the resistance coefficients to be identified must lie within a certain range.

For outdoor heating network pipe sections, this range is typically centered around the design resistance coefficient of the section. The specific deviation depends on the actual network conditions and is set based on engineering experience. The calculation is as follows:

$${\phi }_1{S}_n\le S_n\le {\phi }_2{S}_n$$

wherea Sn is the design resistance coefficient value for pipe section n (Pa/(t/h)²). Each element in the resistance coefficient vector s generated by the optimization algorithm is constrained by this condition. ${\phi }_1,{\phi }_2$ is the permissible deviation range for the pipe-network resistance coefficient in practical engineering; this range is determined based on actual conditions (commonly 0.8 to 1.2). The range ${\phi }_1$ = 0.8, ${\phi }_2$ = 1.2 is derived from empirical observations of aging networks in district heating systems, in which resistance deviations typically remain within ±20% of design values under conditions of short-term operational stability [7].

For resistance coefficients within heat exchange stations and downstream of unit heating inlets, the upper limit Smax is determined based on the pump’s total head and the minimum flow rate at the measurement point, while the lower limit is set to 0. From a technical perspective, the equivalent resistance of each dynamically adjusted user branch (with unique and unmeasurable internal topology) varies continuously due to real-time end-user valve control, making precise lower-bound estimation impossible. From a physical standpoint, zero represents the theoretical, frictionless minimum that covers all possible states.

Assuming the heating system consists of n + 1 nodes and b branches, its hydraulic condition calculation model can be established as follows.

The matrix form of Kirchhoff’s current law (KCL) [34] is given by Equation (2):

$$A·G=Q$$

(2)

where A is the fundamental correlation matrix (n × b), $A=(a_{ij}{)}_{n\times b}$, and $a_{ij}$ is the flow direction sign function. G is the b × 1 column vector of pipe-section flow rates. Q is the n × 1 column vector of nodal flow rates.

The matrix form of Kirchhoff’s voltage law (KVL) [34] is given by Equation (3):

$$B·H=0$$

(3)

where B is the fundamental loop matrix, $B=(b_{ij}{)}_{(b-n)\times b}$, and bij is the branch flow direction sign. H is the b × 1 column vector of pipe-section pressure drops.

Combining the above elements yields the mathematical model for the resistance coefficient identification problem used in this study.

3. SDEIA Algorithm Design and Resistance Identification Process

3.1. Population Initialization Optimization

Traditional initialization methods, due to their randomness, lead to unstable convergence speed, variable solution quality, insufficient diversity with respect to global search, and increased tuning difficulty due to parameter sensitivity. The Chebyshev chaotic map, employed in initialization, enhances population diversity, strengthens exploration capability, avoids local optima, and improves convergence and global search ability, making it suitable for complex, non-linear, and multi-modal optimization problems [35]. The main steps for population initialization using the Chebyshev chaotic map are as follows:

(1)

Initialize Population: Create a matrix $X_{chebyshev}$ of size NP × Dim, where NP is the population size and Dim is the dimensionality of each individual. For each individual I, generate a random value within the interval [−1, 1] to initialize each dimension of the population, as shown in Equation (4):

$$X_{chebyshev}\left(i,:\right)=2 · rand\left(1,Dim\right)-1$$

(4)

Here, $rand\left(1,Dim\right)$ generates random numbers in [0, 1], transformed to [−1, 1].

(2)

Apply Chebyshev Chaotic Map: For each dimension $X_{chebyshev}\left(i,j\right)$ of each individual, apply the Chebyshev map for non-linear transformation, as shown in Equation (5):

$$X_{chebyshev}\left(i,j\right)=cos\left(2\pi ·X_{chebyshev}\left(i,j\right)\right)$$

(5)

  

The cosine function maps each dimension’s value within [−1, 1], introducing non-linear variation to diversify the population.

  

Map to Search Space: Linearly map the values obtained from the Chebyshev map to the actual search space [$X_{min}$, $X_{max}$], as shown in Equation (6):

$$X=X_{min}+\left(X_{maX}-X_{min}\right)·{\displaystyle \frac{{ 1+X}_{chebyshev}}{2}}$$

(6)

  

Here, values are transformed from [−1, 1] to [0, 1], which are then scaled to [$X_{min}$, $X_{max}$].

The mapped results are shown in Figure 1 (Point Distribution Plot). The transformed population of individuals is uniformly distributed across the entire feasible search space. This distribution characteristic, achieved through the non-linear transformation of the Chebyshev map (Step 2), exhibits a bimodal pattern with higher density at the ends (Figure 2 Histogram), ensuring both diversity and ordered rule-based global search capability.

3.2. Adaptive Mutation and Crossover Mechanism

In the traditional Differential Evolution algorithm, the mutation scale F and crossover probability CR are typically fixed values. However, fixed F and CR limit search capability, prevent full utilization of search characteristics, reduce adaptability to different problems, and can cause premature convergence or search divergence. Furthermore, they restrict information exchange, affecting the generation of new high-quality individuals and the retention of good individual structures. To overcome this limitation, this study proposes a dynamic adjustment mechanism based on population diversity.

Diversity is quantified by calculating the Euclidean distance between individuals within the population [36]. In DE algorithms, population diversity reflects whether premature convergence is occurring. Therefore, a diversity function for the current population is introduced as the basis for adaptively adjusting the mutation factor F and crossover probability CR.

Diversity Measure: The diversity of the population is quantified using the Euclidean distance between individuals, as shown in Equation (7):

$$d={\displaystyle \frac{1}{N\left(N-1\right)}}\sum _{i=1}^N\sum _{j=i+1}^N\left|\left|X\left(i,:\right)-X\left(j,:\right)\right|\right|$$

(7)

The Euclidean distance is given by Equation (8):

$$\left|\left|X\left(i,:\right)-X\left(j,:\right)\right|\right|=\sqrt{\sum _{k=1}^D(X\left(i,k\right)-X\left(j,k\right){)}^2}$$

(8)

where d is the diversity measure, $X\left(i,:\right)$ and $X\left(j,:\right)$ are the solution vectors of the i-th and j-th individuals, N is the population size, and D is the dimension of the solution vector.

Normalization: Perform normalization processing on the diversity measure to constrain subsequent adaptive operators within a specified range, using the maximum possible Euclidean distance in the solution space, as shown in Equation (9):

$$dmax=\sqrt{\sum _{k=1}^D(X\left(max,K\right)-X\left(min,K\right){)}^2}$$

(9)

where $X\left(max,K\right)$ and $X\left(min,K\right)$ are the upper and lower bounds of the search space for dimension k.

Parameter Adaptive Adjustment: F and CR are adjusted based on the normalized diversity value, as shown in Equations (10) and (11):

$$F=F_{min}+\left(F_{max}-F_{min}\right)\times {\displaystyle \frac{d}{d_{max}}}$$

(10)

$$CR={CR}_{min}+\left({CR}_{max}-{CR}_{min}\right)\times {\displaystyle \frac{d}{d_{max}}}$$

(11)

Here, when diversity d is large, F and CR increase; conversely, they decrease when the diversity variable decreases.

This mechanism dynamically adjusts the mutation factor (F) and crossover probability (CR) based on diversity. When population diversity is high, F and CR are increased to promote exploration and avoid local optima. As the population converges, these parameters are appropriately decreased to intensify local search capability. Experimental results indicate that this dynamic adjustment mechanism effectively enhances the algorithm’s global search ability, overcoming the limitations of fixed parameter selection, and demonstrates superior convergence and solution quality, particularly in complex high-dimensional optimization problems.

3.3. Integration of Immune Vaccination Strategy

The Immune Algorithm (IA) is a heuristic optimization algorithm inspired by the self-learning and adaptive capabilities of the immune system [37]. The specific integration of IA with DE is performed as follows: after the initial population’s fitness is computed by the DE algorithm, the population is sorted, and individuals with higher fitness are selected to form a vaccine bank. Following the mutation operation, the fitness of the mutated individuals is calculated, and the individuals are sorted. Individuals in the mutated population with lower fitness are replaced with individuals from the vaccine bank. Subsequently, the standard DE operations of crossover and selection are performed. If the convergence criterion is not met, the vaccine bank is updated based on the population selected to participate in the next vaccination cycle.

(1)

Sort Individuals by Fitness: Solutions to the optimization problem are represented as “antibodies” (individuals in the population). Fitness values indicate the quality of the current solution. Sorting is performed to obtain the global best individual. Assuming a population of N individuals, the fitness of the i-th individual is f(xi), where xi is its solution vector. The mathematical process is as follows:

1.

Calculate fitness f(xi) for each individual i (i = 1, 2, …, N).

2.

Sort fitness values in descending order, yielding a sequence i1, i2, …, iN such that f(xi) ≥ f(xi2) ≥ … ≥ f(xiN).

   

High-fitness individuals (quality antibodies) are prioritized for subsequent operations, accelerating convergence toward the optimal solution.

(2)

Establish a Vaccine Bank: Select the most promising individuals from the initial population to form the vaccine bank. Specifically, individuals ranked in the top NP/6 by fitness are selected, where NP denotes the population size. This preserves high-quality solutions for guiding the evolutionary process.

(3)

Mutation Operation: Mutate individuals in the current population using the method described in Section 3.2, resulting in an intermediate population of new resistance coefficient vectors. Calculate the fitness of this intermediate population.

(4)

Immune Vaccination: Replace individuals in the mutated intermediate population that have low fitness with individuals from the vaccine bank.

This hybrid framework leverages DE’s global stochastic search capability to ensure broad exploration of the parameter space while utilizing the IA’s vaccination mechanism to achieve two key optimizations: (1) targeted preservation of high-fitness individuals, aiming to maintain population diversity; and (2) guidance of the search direction through immune memory in order to avoid blind search.

3.4. Resistance Coefficients Identification Process Based on the Improved Differential Evolution Algorithm

The SDEIA algorithm first initializes parameters, variables, and the population, using the Chebyshev chaotic map to generate initial solutions; this is followed by fitness evaluation. Next, promising individuals are selected through sorting to form a candidate set and the vaccine bank. If an individual in the initial population meets the precision requirement, it is output directly as the optimal solution. Otherwise, the mutation is performed to generate an intermediate population, which is then subjected to immune vaccination using individuals from the vaccine bank. Subsequently, a new population is generated through crossover and boundary condition handling according to DE principles, and fitness is recalculated. A new generation population is formed based on the selection criterion. If the termination condition is not satisfied, the vaccine bank is updated, and the iteration continues. Finally, SDEIA outputs the best individual and its fitness value. The resistance coefficient identification process based on SDEIA is illustrated in Figure 3.

4. Engineering Validation

4.1. Application Case Description

To validate the effectiveness of the proposed improvements, operational data from an actual district heating network in Luoyang City were employed, establishing a comprehensive benchmarking framework that incorporated three optimization algorithms—the classical Differential Evolution (DE) as the baseline reference; Particle Swarm Optimization (PSO), representing swarm intelligence methodologies; and Genetic Simulated Annealing (GASA), exemplifying advanced hybrid metaheuristics—while rigorously maintaining identical experimental conditions, including population sizes and termination criteria, to objectively assess the engineering robustness of the SDEIA-based resistance coefficient identification in practical applications.

The heating system operates with proximal constant pressure, with a set pressure of 100,000 Pa. The identification targets were 61 outdoor pipe sections and 31 variable-resistance terminal sections. Figure 4 shows the topological structure of the case network, where solid lines represent supply pipes, dashed lines represent return pipes, dash–dot lines represent heat sources, and dash–double-dot lines represent heat users. Heat meters installed at user points upload flow information to a data management platform. To simplify the identification of the objects, supply and return pipes were assumed to be mirror-symmetric, with equal resistance coefficient search ranges for symmetric pairs. The terrain where the outdoor pipes are located is relatively flat, so all node elevations were considered to be identical.

In the actual heating network, variable local resistance components, such as electric control valves and balancing valves, are installed within the user units. Consequently, the resistance coefficient values within the units change in real time due to user self-regulation; these sections are termed variable-resistance sections. Outdoor pipe sections, lacking such variable components, experience resistance changes primarily due to alterations in network structure or internal corrosion over time, remaining relatively stable in the short term; these are termed stable sections. Therefore, the network was divided into stable sections and variable-resistance sections (considering the section downstream of the unit heating inlet as a single pipe) for identification. For stable sections [7], the upper and lower bounds of the resistance coefficient were determined based on pipe information, using Equation (12):

$$S=6.88\times {10}^{-9}{\displaystyle \frac{K^{0.25}}{d^{5.25}}}\left(l+l_d\right)\rho$$

(12)

where S is the pipe-section resistance coefficient (Pa/(t/h)²). d is the internal pipe diameter (m). l is the pipe length (m). $l_d$ is the equivalent length for local resistances of valves, bends, etc. (m). ρ is the fluid density (kg/m³). K is the absolute pipe roughness (m), typically K = 0.0005 m for hot water heating systems.

Specific pipe-section information, including basic data and resistance coefficient search bounds, is provided in

Table 1. The search range for the resistance coefficients in the stabilized pipe sections.

ID	Pipe Length/m	Piping Specifications	Lower Limit of Resistance Factor Search/Pa/(t/h)2	Drag Coefficient Search Limit/Pa/(t/h)2	ID	Pipe Length/m	Piping Specifications	Lower Limit of Resistance Factor Search/Pa/(t/h)2	Drag Coefficient Search Limit/Pa/(t/h)2
E1	17.5	DN250	0.026	0.040	E32	7.9	DN150	0.170	0.268
E2	29.3	DN250	0.043	0.068	E33	27.3	DN50	186.114	303.297
E3	40.8	DN200	0.194	0.306	E34	6.3	DN150	0.136	0.214
E4	67.1	DN100	12.021	19.293	E35	26.3	DN50	179.297	292.187
E5	7.1	DN80	4.104	6.638	E36	40.5	DN150	0.874	1.375
E6	11.4	DN80	6.590	10.658	E37	22.2	DN50	151.345	246.637
E7	30.3	DN150	0.654	1.029	E38	11.5	DN150	0.248	0.390
E8	14.9	DN100	2.669	4.284	E39	27.7	DN50	188.841	307.741
E9	3.0	DN80	1.734	2.805	E40	7.3	DN150	0.158	0.248
E10	5.5	DN80	3.179	5.142	E41	27.3	DN50	186.114	303.297
E11	64.7	DN125	3.636	5.765	E42	3.7	DN150	0.080	0.126
E12	5.4	DN50	36.814	59.993	E43	22.4	DN50	152.709	248.859
E13	6.9	DN125	0.388	0.615	E44	32.9	DN150	0.710	1.117
E14	5.4	DN50	36.814	59.993	E45	22.3	DN50	152.027	247.748
E15	35.4	DN125	1.990	3.154	E46	11.5	DN125	0.646	1.025
E16	5.8	DN50	39.541	64.437	E47	27.7	DN50	188.841	307.741
E17	6.9	DN125	0.388	0.615	E48	7.2	DN125	0.405	0.642
E18	5.8	DN50	39.541	64.437	E49	27.4	DN50	186.796	304.408
E19	49.5	DN125	2.782	4.411	E50	34.8	DN100	6.234	10.006
E20	5.7	DN50	38.859	63.326	E51	15.6	DN50	106.351	173.313
E21	6.7	DN125	0.377	0.597	E52	54.5	DN100	9.763	15.670
E22	5.7	DN50	38.859	63.326	E53	3.5	DN80	2.023	3.272
E23	32.5	DN100	5.822	9.344	E54	17.4	DN80	10.059	16.268
E24	11.8	DN50	80.445	131.095	E55	39.6	DN200	0.189	0.297
E25	55.7	DN100	9.978	16.015	E56	3.0	DN100	0.539	0.864
E26	15.0	DN80	8.671	14.024	E57	21.6	DN150	0.468	0.736
E27	21.2	DN80	12.255	19.820	E58	3.0	DN100	0.540	0.866
E28	20.5	DN150	0.442	0.696	E59	51.0	DN150	1.106	1.740
E29	19.8	DN50	134.984	219.974	E60	3.0	DN100	0.541	0.868
E30	11.2	DN150	0.242	0.380	E61	29.0	DN100	5.232	8.397
E31	27.8	DN50	189.523	308.852

.

The variable-resistance pipe segment represents the equivalent of the resistance in the user branches. Its physical dimensions dynamically change due to valve adjustments; thus, only the resistance range is provided. When determining the resistance search range for such segments, the upper limit of the resistance coefficient Smax is defined based on the total pump head and the minimum flow rate at the measurement points [7]. This upper resistance coefficient limit is given by Equation (13):

$$∆P=S{Q}^2$$

(13)

where $∆P$ represents the total differential pressure at the pump, Pa; S is the resistance coefficient, Pa/(t/h)²); and Q denotes the flow rate, t/h.

The lower limit of the resistance coefficient, Smin, for the variable-resistance pipe section is set to 0. Specific information is provided in

Table 2. The search range for the resistance coefficients in the variable resistance pipe sections.

ID	Lower Limit of Resistance Factor Search/Pa/(t/h)2	Drag Coefficient Search Limit/Pa/(t/h)2	ID	Lower Limit of Resistance Factor Search/Pa/(t/h)2	Drag Coefficient Search Limit/Pa/(t/h)2
u1	0	730.51	u17	0	17,175.716
u2	0	1165.441	u18	0	16,199.882
u3	0	1294.249	u19	0	6847.353
u4	0	965.964	u20	0	13,288.923
u5	0	56,604.868	u21	0	6428.135
u6	0	13,677.563	u22	0	5046.598
u7	0	10,840.486	u23	0	46,927.242
u8	0	14,140.822	u24	0	26,444.614
u9	0	24,608.463	u25	0	27,770.921
u10	0	13,774.723	u26	0	833.987
u11	0	12,568.107	u27	0	698.007
u12	0	1641.990	u28	0	849.909
u13	0	2678.603	u29	0	932.900
u14	0	27,809.931	u30	0	3784.566
u15	0	16,567.559	u31	0	2028.323
u16	0	9786.581

.

4.2. Algorithm Parameter Setting and Experimental Design

To achieve optimal performance of the four optimization algorithms (SDEIA, DE, PSO, and GASA) in the resistance coefficient identification, we conducted multiple experimental iterations to determine the topology-specific parameter configurations that maximize algorithmic efficacy for this network. The systematically calibrated parameters for each algorithm are formally documented in

Table 3. SDEIA algorithm: Parameter setting table.

Parameter Type	Setpoint
Mutation Operator	$\mathrm{Adaptive} \mathrm{operator} \mathrm{F}\left(\mathrm{t}\right)$
Crossover Operator	$\mathrm{Adaptive} \mathrm{operator} \mathrm{C}\mathrm{R}\left(\mathrm{t}\right)$
Population Size	6 × Dim
Crossover Strategy	Randomized selection of individual difference variant strategies
Select Operation	Greed criterion
Termination Conditions	Adaptation value less than 0.1 or 300 iterations
Number of Vaccines	NP/6

,

Table 4. DE algorithm: Parameter setting table.

Parameter Type	Setpoint
Mutation Operator	0.4
Crossover Operator	0.3
Population Size	6 × Dim
Crossover Strategy	Randomized selection of individual difference variant strategies
Select Operation	Greed criterion
Termination Conditions	Adaptation value less than 0.1 or 300 iterations

,

Table 5. PSO algorithm: Parameter setting table.

Parameter Type	Setpoint
Inertia Weight (w)	0.6
Cognitive Factor (c1)	1.5
Social Factor (c2)	1.5
Population Size	max (10, 6 × Dim)
Select Operation	Greedy update (for personal and global best)
Termination Conditions	Adaptation value less than 0.1 or 300 iterations

and

Table 6. GASA algorithm: Parameter setting table.

Parameter Type	Setpoint
Mutation Operator	Temperature-dependent adaptive mutation
Crossover Operator	0.6
Population Size	6 × Dim
Crossover Strategy	Single-point crossover
Select Operation	Roulette wheel selection
Initial Temperature (T0)	1000
Final Temperature (T_end)	1 × 10−5
Cooling Rate (alpha)	0.95
Termination conditions	Adaptation value less than 0.1 or 300 iterations or Temperature < T_end

.

4.3. Analysis of Identification Results

To validate the algorithm’s effectiveness, four operating conditions, corresponding to 2:00, 8:00, 14:00, and 20:00 on a day in mid-December 2024, were identified for the case network. As flow measurement points were only installed at unit heating inlets in this case, the actual resistance values could not be calculated for direct comparison with the identified values to assess algorithm performance.

Therefore, the identified resistance coefficient values were substituted into the hydraulic calculation model to obtain calculated flow rates at the measurement points. The accuracy of the algorithm’s identification results was judged by comparing the relative error between these calculated flow rates and the measured values. The stability of the results was assessed by examining the distribution of these errors. As shown in Figure 5, Figure 6, Figure 7 and Figure 8, the average relative errors of the flow rates at all measurement points for the four different operating conditions identified by the SDEIA algorithm are all less than 5%, whereas the other three algorithms exhibit average relative errors greater than 5% at some measurement points. Furthermore, out of the 31 measurement points, the SDEIA algorithm has smaller errors at 21 points compared to the other three algorithms. This indicates that the SDEIA algorithm achieves higher accuracy under complex operating conditions, compared to the DE, PSO, and GASA algorithms.

By comparing the distribution characteristics of the identification result errors, we can evaluate the stability of the algorithm’s identification results. The distribution of flow rate relative errors at various measurement points under the four operating conditions can intuitively reflect the stability of the algorithm in complex operating conditions. This paper uses box plots to display the distribution of flow rate relative errors identified by the SDEIA algorithm, DE algorithm, PSO algorithm, and GASA algorithm at each measurement point. In the box plot, the central line of the box represents the median of the data. The dispersion of the data is judged by the length of the box and the whiskers; typically, a longer box and longer whiskers indicate greater data dispersion. The distributions of flow rate relative errors at each measurement point under the four operating conditions are shown in Figure 9, Figure 10, Figure 11 and Figure 12. The figures show that the flow rate relative error distribution obtained by the SDEIA algorithm is more concentrated, demonstrating that the SDEIA algorithm achieves greater stability in resistance coefficient identification under complex operating conditions.

It can be concluded that the SDEIA-based resistance coefficient identification method demonstrates superior accuracy and stability in practical engineering applications, com-pared to alternative optimization approaches such as DE, PSO, and GASA. This conclusion highlights its enhanced performance in handling complex engineering problems. Therefore, the resistance coefficient distribution identified by SDEIA can be reliably used for analyzing the hydraulic conditions of heating networks. This provides a basis for hydraulic analysis and regulation, leading to the more effective and accurate assessment of network hydraulic states, ensuring safe system operation and adequate heat supply to end-users.

5. Discussion

5.1. Interpretation of Results in Context

The validation results in Section 4.3 demonstrate SDEIA’s superiority in high-dimensional resistance identification. This performance stems from three synergistic innovations:

• Chaotic Initialization (Equations (4)–(6)) seeds the population with structured diversity (bimodal distribution in Figure 1 and Figure 2), ensuring comprehensive coverage of the 154-dimensional search space. This prevents early convergence to suboptimal regions—providing the raw material for adaptive evolution.
• Diversity-Driven Adaptation leverages Euclidean distance metrics (Equations (7)–(9)) to dynamically tune F and CR (Equations (10) and (11)). Crucially, it responds to population states generated by chaotic initialization: When population diversity is high, F and CR are increased to promote exploration and avoid local optima. As the population converges, these parameters are appropriately decreased to intensify local search capability.
• Immune–Vaccine Mechanism then preserves and propagates high-fitness patterns: Top NP/6 elites from the adaptively evolved population are stored in the vaccine bank. These vaccines inject prior knowledge into subsequent generations by replacing low-fitness mutants—effectively curbing diversity loss while accelerating convergence.

This organic combination resolves high-dimensional conflicts: chaotic maps enable exploration of extreme ranges, adaptive parameters translate exploration into targeted exploitation, and immune mechanisms stabilize exploitation outcomes through experiential learning.

5.2. Broader Implications

Digital Twin Reliability: Identified coefficients can enable physics-informed models utilized for predictive control, to reduce the energy consumption of the water pump.

Anomaly Detection Framework: Resistance deviations that vary by 20% from the design values signal pipe corrosion or valve failure; this determination can trigger proactive maintenance.

Multi-Source Integration: The stable error distribution (Figure 9, Figure 10, Figure 11 and Figure 12) ensures hydraulic balance during heat source switching, addressing the concern about “topology-induced maldistribution”.

Carbon-energy synergy: By integrating resistance data with thermodynamic models, pump scheduling can be optimized, potentially reducing CO₂ emissions in regional systems.

6. Conclusions

This paper addressed the challenge of accurately identifying resistance coefficients in heating networks under high-dimensional complex conditions by proposing an Improved Differential Evolution Algorithm (SDEIA). The algorithm integrates the Chebyshev chaotic map, a dynamic adaptive mechanism, and an immune regulation strategy: the chaotic map enhances the global distribution of the initial population; the diversity-based adaptive parameter tuning strategy balances global exploration and local exploitation capabilities; and the immune mechanism effectively suppresses premature convergence. Based on Kirchhoff’s laws, a multi-constrained identification model was established and applied to an actual heating network system, with validation performed using four typical operating conditions. Analysis of the identification results yielded the following:

• Superior Accuracy: SDEIA achieves better precision in resistance coefficient identification. In complex networks, precise identification is fundamental for achieving accurate hydraulic balance and optimization, and aids in reducing energy losses and improving overall system efficiency.
• Enhanced Stability: SDEIA exhibits a more uniform and concentrated error distribution, indicating its superior ability to adapt to and handle the variable conditions encountered in complex networks. This is crucial for ensuring the stability and reliability of hydraulic regulation in pipe networks.

In summary, the SDEIA algorithm demonstrates excellent performance in heating network resistance identification and is suitable for application in practical engineering. While validated on a Luoyang case study, SDEIA’s algorithm design is independent of climatic or geographic factors. Its applicability extends to regions with comparable heating network complexity, provided hydraulic data from standard sensors are available.

Future Outlook

• Ultra-large-scale network validation: The current case network size (154 pipe sections) validates the high-dimensional characteristics of the problem. However, due to practical constraints, the algorithm’s performance on ultra-large-scale networks (>300 pipe sections) was not explored. Future work will test SDEIA’s performance on networks exceeding 300 pipe sections, leveraging parallel computing and hierarchical decomposition techniques to enhance computational efficiency.
• The current objective function is based solely on flow error, as pressure measurement points are scarce in actual heating networks (the case in Figure 4 only includes flow meters), making pressure data difficult to obtain. Future work will explore an optimization model that integrates pressure error.
• Real-time identification framework: A digital twin-based online resistance identification system could be developed to dynamically adapt to user-regulated hydraulic fluctuations and enable proactive network control.
• Multi-objective extension: Energy consumption and carbon emission goals could be integrated into the resistance identification model to establish a collaborative optimization framework for low-carbon regulation.