The medium-voltage direct-current (MVDC) ship power system has been extensively investigated and discussed in recent years. This paper presents an alternating current/direct-current (AC/DC) power flow algorithm based on complex affine arithmetic for the zonal MVDC shipboard power system in the presence of power variation. The power converter effect is considered, and an affine power converter model is proposed in the power flow model. An affine-analysis-based sequential method is adopted to

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... e AC/DC power flows. With the proposed algorithm, the bounds of bus voltages can be effectively obtained. A relative influence factor is defined to quantitatively assess the effect of power variation on voltage profile. Four cases are studied in detail to demonstrate the applicability of the proposed algorithm. The proposed algorithm is useful in network architecture design, planning, and online operations of MVDC power systems when such decisions are subject to power variation. Energies 2018, 11, 1697 2 of 16 requested for this system. It is important to maintain both the voltage and the line power flow within limits to keep steady state security under all kinds of operating conditions [7], especially for navy ships. Power flow analysis is the most fundamentally used tool to assess the static security of a power system for specified input values, referred to here as a deterministic power flow which has been widely applied in early non-electric propulsion ship systems. Deterministic power flow algorithm is studied for MVDC shipboard distribution systems in [8, 9] . However, it is remarkable that load powers are changeable in the different operational conditions for MVDC shipboard power systems. For instance, propulsion loads, which make up nearly two-thirds of the whole system load, vary with large range under full speed, cruise, and docking mode. Service loads, cable lifter, and other loads also have power variation according to the different operating mode. Furthermore, renewable energies have been recommended for use in large ship systems to save energy and reduce emissions; fluctuating power is characteristic of renewable energy generation. Thus, it can be seen that the power fluctuation is more complicated and changeable for the MVDC shipboard power system compared with the traditional ship power system. In order to fully understand and master the impact of power fluctuation to the system, numerous power flow scenarios need to be analyzed that require a large investment of time and computational resources. In addition, the future warship is envisioned to have an approximate 70% (or greater) decrease in the numbers of crew and a subsequent increase in automation. Thus, it is crucial that the ship's crews are provided with quick decision support, and action could be taken more quickly and effectively in some circumstances, such as network reconfiguration due to battle damage or equipment failure. A rapid and effective method is therefore needed to analyze power flow under power uncertainty, which is convenient and useful for shipboard power network topology design, reconfiguration, network optimization, voltage control, and others [10, 11] . There are three major methods to model uncertainty power flow: the probabilistic method [12], the fuzzy method [13] , and the interval method. The first two methods depend on the statistical dependence or probability distributions of input data. However, it is hard to obtain statistical load change data due to complex operational conditions or lack of prior experience for ship zonal MVDC power systems. The interval algorithm (IA) is able to solve the power flow when the upper and lower bounds of uncertain input data are known. The application of IA to power flow analysis has been investigated by various authors [14] [15] [16] . However, its conservativeness results in impractical bounds, especially in complicated expressions or long iterative computations. To overcome the IA limitations, an affine algorithm (AA) is proposed. This algorithm uses the affine form instead of the interval form to describe the uncertainties of power injections and has been researched widely in recent years [17] [18] [19] [20] [21] [22] [23] [24] [25] . References [18, 21] demonstrate that the AA-based power flow method returns tighter bounds on power flow results than those obtained via IA and has a better computational performance. In addition, AA can trace the impacts of individual input uncertainties on power flow solutions; it is helpful for shipboard zonal-distribution topology-structure design and network reconfiguration. Affine arithmetic was first applied to transmission system power flow according to ref. [18]; it was shown that affine arithmetic better handles uncertainty compared to the traditional and widely used IA approaches. As well, balanced and unbalanced three-phase radial distribution system power flow based on affine arithmetic have been presented [19, 20] ; an index of relative influence is proposed in ref. [20] to study the impacts of uncertainties on power flows. AA is applied to an optimization-based power flow model for computing reliable enclosures of uncertain power flow in ref. [21, 22] . Evidence theory and AA are combined to analyze uncertain power flow, which can deal with probabilistic, possibilistic, and interval inputs [23] . AA is further used to explore the application for optimal power flow, regional control of unscheduled power fluctuation, and stochastic information management [22, 24, 25] . At present, these papers mainly focus on uncertain power flow of AC transmission and distribution systems. To the best of the authors' knowledge, AA has not been presented for an AC/DC power system with converters, and in particular, for an MVDC shipboard power system, in which the widespread use of converters complicates the network structure [26] . The MVDC shipboard power system is an AC/DC hybrid system, where generators transmit power