^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Large hydropower project involves numerous stakeholders in the construction phase. The risk of delayed payment may escalate into an event of catastrophic economic loss through complex stakeholders’ relationships. This paper builds a complex stakeholder network by collecting all contract prices in large hydropower project construction (LHPC) and proposes a novel mathematical model based on Cellular Automata Susceptible-Infected-Susceptible (CA-SIS) framework to analyze the risk propagation of delayed payment (RPDP). In the model, to approximate actual loss diffusion, risk-resisting and risk-mitigating abilities of stakeholders are combined, and relevance between risk mitigation and time is taken into account. The rationality and feasibility of the novel model are illustrated through its application to an actual LHPC. The simulation results include the following: RPDP presents three phases: risk first slightly propagates, risk then exponentially breaks out, and at last propagation consequence will reach a stable state related to stakeholders’ risk-resisting ability closely; in respect of stakeholders’ risk-mitigating ability and property of the network, RPDP range decreases with the increase of number of partners and homogeneity of contract price distribution; in respect of stakeholders’ risk-resisting ability, RPDP range decreases with the decrease of number of partners. This research can serve as a powerful complement to analyze and control the delayed payment in LHPC.

Large hydropower project requiring substantial investment and tremendous public resources belongs to the critical infrastructure engineering and has more profound impact on local communities and even a nation than general infrastructure engineering [

Several previous works have studied the risk analysis and assessment in LHPC. Liu et al. [

Owing to the complexity of LHPC in structure, context, organization, and technology [

Although collaboration of interdependent stakeholders can enhance holistic and macroscopical operation of LHPC [

In the last decade, propagation of failure risk which is also called cascading failure is quite prevalent in physical networks, especially critical infrastructures, such as electric power networks [

Comparison between three epidemic models (SIR, SI, and SIS) from model description and propagation threshold.

Model name | Model description | Propagation threshold |
---|---|---|

SIR [ | The infected node is removed from the number of susceptible nodes, as recovery or failure completely. | There exists a threshold density of nodes. If the actual density surpasses this threshold value, one (or more) infected node will give rise to a spread. |

| ||

SI [ | The infected node will be in permanently infected state. | If susceptible probability is above the threshold, the failure becomes persistent. Below it, the failure exponentially dies out. |

| ||

SIS [ | The infected node has chance to vary from infected state to susceptible state. | When susceptible rate exceeds the threshold, the node will be infected, while there also exists the threshold of returning to susceptible state. |

The rest of the paper is organized as follows: Section

Large-scale hydropower projects are decomposed into a number of subprojects, requiring the involvement and participation of many interdependent stakeholders. The complex interrelationships among them are more appropriately viewed as a complex network. In the network building process, the first step is to identify the nodes representing the stakeholders in complex network. The stakeholders indicate any individuals and organizations who actively participate in the project or whose interests are positively or negatively affected by project execution, such as owners, designers, prime contractors, subcontractors, supervisors, and government departments. During the long period of construction, these project stakeholders are not constant, because with the advancement of construction schedule, the number of stakeholders shows a state of flux that increases at first and then decreases generally. It is noted that the set of project stakeholders is identified at a particular point in time. Yangfanggou hydropower station is the first large-scale hydropower engineering adopting the project management pattern of Design-Build (DB). The Prime Contractor was formed by Huadong Engineering Corporation Limited (designer) and Sinohydro Bureau 7 Co., Ltd (prime construction contractor). We collected all stakeholders of Yangfanggou hydropower station in October 2016 to identify the set of nodes of the stakeholder network.

Because of LHPC largely depending on the collaboration of multiple stakeholders throughout all project processes [

Contract prices between five stakeholders (Sup1, Sup8, CC20, CC22, and CC24).

Sup1 | Sup8 | CC20 | CC22 | CC24 | |
---|---|---|---|---|---|

Sup1 | 0.00 | 0.00 | 6.87 | 12.52 | 0.37 |

Sup8 | 0.00 | 0.00 | 218.85 | 398.98 | 11.81 |

CC20 | 6.87 | 218.85 | 0.00 | 0.00 | 0.00 |

CC22 | 12.52 | 398.98 | 0.00 | 0.00 | 0.00 |

CC24 | 0.37 | 11.81 | 0.00 | 0.00 | 0.00 |

Note. Unit of contract price is ten thousand RMB. This table is symmetric because of bidirectional property of contract, where 0.00 represents no contractual relationship.

Once the nodes (i.e., stakeholders) and edges (i.e., relationships) are identified, stakeholders of Yangfanggou hydropower station in October 2016 can be modeled as a complex stakeholder network. A bidirectional and weighted

A part of stakeholder network of construction project in Yangfanggou hydropower station (SNCPYHS). Sup1, Sup8, CC20, CC22, and CC24 represent the stakeholders. Edges represent contractual relationships and values on them are the contract prices in ten thousand RMB.

To complete the stakeholder network of Yangfanggou hydropower station, whole adjacent matrix in Excel spreadsheet, containing 64 stakeholders and contract prices among them, is exported to

Stakeholders, contractual relationships, and contract prices in Yangfanggou hydropower station are represented as nodes, edges, and weights of edges, respectively.

Nodes/Edges/Weights of edges | Stakeholders, contractual relationships, and contract prices |
---|---|

Owner | Hydropower Project Investor (Yalong River Hydropower Development Company, Ltd) |

PC | Prime Contractor formed by Designer and Prime Construction Contractor (Huadong Engineering Corporation Limited and Sinohydro Bureau 7 Co., Ltd) |

QI1 | Department 1 of Quality Inspection |

QI2 | Department 2 of Quality Inspection |

PS | Prime Supervisor |

S1-S6 | Sub-supervisor 1 ~ Sub-supervisor 6 |

PD | Prime Designer |

D1 ~ D5 | Designer 1 ~ Designer 5 |

PCC | Prime Construction Contractor |

CC1 ~ CC27 | Construction Contractor 1 ~ Construction Contractor 27 |

Sup1~Sup19 | Supplier 1 ~ Supplier 19 |

Edges | Contractual Relationships among Stakeholders |

Weights of edges | Contract Prices among Stakeholders (/ten thousand RMB) |

Complete stakeholder network of construction project in Yangfanggou hydropower station (SNCPYHS). Nodes (Owner, PC, QI1, etc.) represent all stakeholders. Edges represent contractual relationships between them.

We suppose that, in a central-position of SNCPYHS, the Owner makes a payment to the Prime Contractor. Thereafter, the Prime Contractor pays the amount to its subcontractors. If delayed payment happens, the risk will cause a cascading propagation. Upstream stakeholder transfers the irregular cash flow to downstream stakeholders, which makes them unable to maintain normal capital operation. The ‘downstream’ stakeholders in turn affect ‘upstream’ stakeholder through schedule delay, poor quality, and safety accidents. In the procedure of RPDP, the delayed payment risk that a stakeholder suffered accumulates with increase of quantity of loss-making stakeholders cooperating with it. When probability of risk occurrence exceeds the risk-resisting ability, the state of economic loss will appear. Moreover, each stakeholder has the ability to eliminate risk by capital pooling, management, and operation. The loss-making stakeholders can regain original state with a probability. Considering the stakeholder’s response to the delayed payment, the frame of Susceptible-Infected-Susceptible (SIS) is used for reference [

Variation process of stakeholder’s state and state variation process in SIS model. A stakeholder may be transferred from normal state at time_{d}. Normal state of the stakeholder is similar to susceptible state in SIS model. Risky state of the stakeholder resembles infected state in SIS model.

Process of risk propagation in conceptual stakeholder network. (a) Stakeholder 1, as risk propagation source, propagates the risk to its neighbor stakeholders 2, 3, and 5 with probability

Before we begin with the construction of the model, some assumptions must be made. The financial information of company is confidential to the public, which adds great burden of basic data collection. Additionally, numerous external and internal risk factors lead to computational complexity and inaccuracy in risk probability. All stakeholders with equal probability of risk mitigation are assumed at the beginning, and then let probability vary in a certain range to analyze multiple results. Assuming that all stakeholders have only two states, risky state regarded as economic loss and normal state, lessens the complicacy of the model construction. It must be noted that one stakeholder’s state is not solely restricted to binary. Both intermediate state and multistate can be also described according to risky extent [

Taking into account all of these possible concerns, we should make several assumptions as follows:

(1) All stakeholders have equal probability of risk mitigation.

(2) The states of all stakeholders have two aspects: the state of economic loss is 1 and the normal state is 0.

(3) The stakeholder is only affected by its direct partners.

(4) When economic loss situation occurs, they will not terminate the contractual relationship.

Cellular Automata (CA) are adopted to describe a complex discrete system consisting of a large number of connected cells and to simulate dynamical evolution of propagation process [

State transition function

(1) Each stakeholder has risk-resisting ability which is closely related to their own capability and partners’ impact. For calculating the risk-resisting threshold, Zhang et al. [

We use

(2) This rule represents a transformation of the risky state of stakeholders after a certain period of time. Each stakeholder has the ability to eliminate existing risk. Thus, except for the risk-resisting ability, risk-mitigating ability should be an indispensable factor. After the period of time

The threshold

At last, in order to measure the consequence caused by RPDP in the stakeholder network, the proportion of loss-making stakeholders at the end of time

It can dynamically express the evolutionary procedure of RPDP. When the proportion of loss-making stakeholders

In this section, SNCPYHS introduced in Section

Time interval, probability of risk occurrence, and probability without recovery are adjusted to prove the rationality of the improved model. The time interval (_{d}) related to extent of recovery is a critical factor in the model. It is varied from 1 to 7 for obtaining different results of RPDP.

Result in Figure _{d} from 1 to 7. We can observe that RPDP suddenly breaks out from time 2 to time 4 and it will finally maintain at a stable value_{d} =2, the value of_{d} grows. For instance, when_{d} = 1, 2, 3, 4, 5, 6, and 7, the values of_{d} = 1 means the RPDP without considering time factor. The stable values

Variation of loss-making stakeholders proportion (_{d}) from 1 to 7. One of curves reflects the variation of_{d} can be compared.

Figure _{d} =1, the two risk probabilities are increased at the growth rate 0.0020 to represent the variation of

Variation process of ultimate stable value (

Variation process of ultimate stable value (

In Figure

The above-mentioned analysis results are obtained under change of one probability. Figure

Variation process of ultimate stable value (

RPDP with and without risk mitigation are compared when_{d} =1 and shown in Figures

Variation of loss-making stakeholders proportion (

Variation process of number of loss-making stakeholders under situations with and without risk mitigation. Red nodes represent loss-making stakeholders and white nodes denote normal stakeholders. (a-1), (a-2), (a-3), and (a-4) show the diffusion process of delayed payment without risk mitigation from time 1 to 4. (b-1), (b-2), (b-3), and (b-4) present the diffusion process of delayed payment with risk mitigation from time 1 to 4.

Figure

The comparison in Figure

From the comparison between (a-4) and (b-4) in Figure

To explore the property of SNCPYHS under the delay payments, we assume a random stakeholder network which has same stakeholders, but contract price among stakeholders is completely random. In the random stakeholder network, the redistributed contract price can be set as _{d} =1.

Variation of ultimate stable values (

The result obtained from above figure shows that the whole curved surface of SNCPYHS is above the curved surface of the random stakeholder network. All

Probability distribution of nodes’ degrees of stakeholder network of construction project in Yangfanggou hydropower station (SNCPYHS). As a node’s degree is equal to the sum of contract prices owned by a stakeholder, probability distribution of nodes’ degrees of SNCPYHS represents the probability distribution of contract prices of SNCPYHS.

Probability distribution of nodes’ degrees of random stakeholder network. As a node’s degree is equal to the sum of contract prices owned by a stakeholder, probability distribution of nodes’ degrees of the random stakeholder network represents the probability distribution of contract prices of the random stakeholder network.

This paper presents a new perspective to analyze the risk of delayed payment in LHPC. By comparing the previous studies in risk analysis, the dynamic properties of propagation and diffusion from one stakeholder to another are fully taken into account in risk of delayed payment. The relationships among stakeholders are viewed as complex network to present the pattern of loss diffusion in actual project. We collected all contract prices to build the stakeholder network of LHPC and established a CA-SIS-based model considering relevance between risk mitigation and time to approximate RPDP. Because of close correlation between stakeholder’s status and importance in the whole project, a new threshold of risk mitigation was put forward to complete original propagation model. Subsequently, under delayed payment of the Owner in SNCPYHS, parameters of the model were varied to simulate the results of RPDP. The conclusions show the following: (1) decline of stable value_{d} and contrast between the model with and without risk mitigation prove the rationality and feasibility of the novel model; (2) regardless of change of parameters in the model, RPDP has three phases: (1) risk first slightly propagates, (2) risk then breaks out, and (3) in last phase value of

The conclusions above also provide theoretical basis for proactively controlling RPDP in the stakeholder network. For example, cooperating with several stakeholders could assist them to mitigate risk after suffering delayed payment. In addition, separating large contract price into numerous small contract prices can improve the homogeneity of contract price distribution. However, this suggestion increases the difficulty of risk resistance when the stakeholder’s partners are all in risky state. Therefore, in the future research, more reasonable allocation of contract price is very essential and meaningful to provide an approach to solve the problem.

The whole adjacent matrix data used to support the findings of this study are included within the Supplementary Materials.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to acknowledge the support of the National Natural Science Foundation of China (Grant no. 51779195).

MATLAB simulation is used in the paper. In order to verify the authenticity of the research and present the results easily for readers, the data of the study and codes of MATLAB simulation are provided. “matrix.xls” presents the whole adjacent matrix data of the article. “RPDP_with_mitigation” and “RPDP_without_mitigation” show the results of RPDP with and without risk mitigation. “Proba_analysis” includes the results of RPDP under change of two kinds of probabilities. “SNCPYHS_random_network” compares the propagation results in SNCPYHS and random stakeholder network, and “Distribution_analysis” analyzes the property of SNCPYHS. “Degree_Distribution” is a function calculating nodes’ degrees.