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The purpose of this paper is to propose a new simplified numerical model, based on a very compact semi-empirical formulation, able to simulate the fluid dynamics behaviors of an electrohydraulic servovalve taking into account several effects due to valve geometry (e.g. flow leakage between spool and sleeve) and operating conditions (e.g. variable supply pressure or water hammer).

The proposed model simulates the valve performance through a simplified representation, deriving from the linearized approach based on pressure and flow gains, but able to evaluate the mutual interaction between boundary conditions, pressure saturation and leak assessment. Its performance was evaluated comparing with other fluid dynamics numerical models (a detailed physics-based high-fidelity one and other simplified models available in the literature).

Although still showing some limitations attributable to its simplified formulation, the proposed model overcomes several deficiencies typical of the most common fluid dynamic models available in the literature, describing the water hammer and the nonlinear dependence of the delivery differential pressure with the spool displacement.

Although still based on a simplified formulation with reduced computational costs, the proposed model introduces a new nonlinear approach that, approximating with suitable precision the pressure-flow fluid dynamic characteristic of a servovalve, overcomes the shortcomings typical of such models.

The paper aims to illuminate the platform created by a cluster organization to facilitate its internationalization and thereby enhance its regional innovation system partners' competitiveness by providing access to global value chains and boosting innovativeness.

The study draws upon the interaction approach, focusing on the interaction process, interaction partners, relationship atmosphere, and relationship environment. A qualitative study was conducted at Future Position X, a Swedish cluster organization. A total of 58 interviews were conducted, including 48 face-to-face in-depth interviews between 2017 and 2019 with six key informants at FPX, representatives from 28 SMEs, ten members of regional innovation systems to which FPX belongs, and four process leaders of regional and local networks, in addition to online interviews with ten members of the regional innovation systems conducted via Microsoft Teams in March 2021. The time span of the study provides a longitudinal perspective.

The FPX cluster collaborates with actors in the quadruple helix, maintaining a mindset that has led to a number of new partner agreements in the global arena to secure the resources and expertise necessary for cluster activities, and thereby ensuring firms in FPX networks access to platforms for international expansion. Internationalization thus expands the cluster's knowledge base beyond the traditional environment of its member firms.

Very few innovations arise from the isolated work of a lone genius. Instead, most innovation is achieved through complex, interactive, iterative and cumulative learning processes in which a variety of actors are involved. The FPX cluster organization's internationalization platform is therefore vital to the internationalization of its partners since cluster actors lack the time, resources, knowledge, experience, and networks required to break into international markets singlehandedly.

This study suggests that, for practitioners and researchers alike, the growing importance and relevance of the regional innovation system cannot be overemphasized. It also holds policy and societal implications in that FPX's global network helps regional SMEs to internationalize, in addition to inspiring international firms to establish operations in the Gävleborg region, thereby helping to strengthen the overall GIS environment. Internationalization also expands the FPX cluster's knowledge base beyond the traditional environment of its firms, an example of this being the construction start of a Microsoft data centre in the region in 2020.

FPX is financed through taxation and grant funding. By initiating projects, creating relationships and building collaborations, FPX thus contributes to collaboration between business, academia and the public sector. FPX also contributes to knowledge development of new technology by creating meeting places and networks around digital issues, such as GIS, AI, the IoT and blockchain technology.

While earlier research has concentrated on endogenous gaps critical to cluster dynamics, comparatively little attention has been paid to exogenous gaps, i.e. linkages between regional clusters and innovation partners elsewhere in the world. This study showcases the richness of interactions in the cluster against the background of wider, global innovation interactions. Future research should examine other vital questions that remain unanswered, e.g. by measuring and exploring the extent to which regional innovation systems can contribute to long-term economic growth for society.