Gain and phase margins based stability region analysis of time‐delayed shipboard microgrid with sea wave energy
2020; Institution of Engineering and Technology; Volume: 14; Issue: 8 Linguagem: Inglês
10.1049/iet-epa.2019.0762
ISSN1751-8679
Autores Tópico(s)Islanding Detection in Power Systems
ResumoIET Electric Power ApplicationsVolume 14, Issue 8 p. 1347-1359 Research ArticleFree Access Gain and phase margins based stability region analysis of time-delayed shipboard microgrid with sea wave energy Burak Yildirim, Corresponding Author Burak Yildirim byildirim@bingol.edu.tr orcid.org/0000-0002-2118-4297 Vocational and Tech. High School, Bingol University, Bingol, TurkeySearch for more papers by this author Burak Yildirim, Corresponding Author Burak Yildirim byildirim@bingol.edu.tr orcid.org/0000-0002-2118-4297 Vocational and Tech. High School, Bingol University, Bingol, TurkeySearch for more papers by this author First published: 29 June 2020 https://doi.org/10.1049/iet-epa.2019.0762Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract One of the used areas of wave energy conversion systems is shipboards. In shipboard microgrid with wave energy conversion systems, load frequency control is very important because of the over-variable nature of the energy generated by wave energy conversion system. In this study, new controller design for load frequency control of the shipboard microgrid with wave energy conversion systems is proposed. With the proposed controller design, the system is operated in suitable stable parameter space in order to guarantee the desired gain and phase margins and the system is more stable against the uncertainties of parameters that will occur. A virtual phase and gain margin tester is added to the shipboard microgrid with wave energy conversion systems in order to take into account the gain and phase margin in the stability region calculations. First, the characteristic equation of this system is obtained. After the appropriate conversion operations on the obtained characteristic equation, the stable parameter space of the controller is calculated by taking into account time delay, gain and phase margin. Eigenvalue and time-domain analysis studies are performed on the system to show the accuracy of the stability region results. 1 Introduction Electrical power systems have been used in marine vessels for more than a century, and many research works and studies have been carried out in the field of electrical power systems for marine vessels from the initial use in the 1880s to the current advanced electrical power systems. Nowadays, the electrical loads of the vessels are constantly increasing. The majority of auxiliary loads and propulsion systems are examples of electrical type loads [1, 2]. Global warming is an international problem that requires a reduction in fuel consumption and greenhouse gas emissions on all vessels, as is the case with all areas requiring the use of different types of energy. Due to the impact of vessels systems on global warming, International Convention for the Prevention of Pollution from Ships has imposed strict restrictions to reduce the impact of vessels on global warming [3, 4]. The system characteristics of the electrically integrated shipboard power system (SPS) and microgrid are very similar. A shore-anchored SPS can be connected to both the offshore power system and the grid. The switch from grid mode to islanded mode or from islanded mode to grid mode of SPS is not automatic and differs from microgrids with this feature. The SPS is similar to islanded microgrid although features such as automatic disconnection from the grid are not the same [5]. Many methods, equipment and components for SPSs and islanded microgrids are the same. In addition, many control strategies and design steps used for SPS are the same as those of microgrid. Power quality improvement strategies, voltage and frequency control schemes, energy management systems and power-sharing methods for multiple distributed generators are the examples of these [1, 6]. The wave energy represents a large renewable energy source (RES) [7]. Waves are generated by winds produced by solar energy. The power density of the solar energy is 0.1–0.3 kW/m2 and the power density of the wind energy is 0.5 kW/m2, while the power density of the wave energy is 2–3 kW/m2 and has a higher power density than the wind and the solar energy. The estimated global gross theoretical resource is ∼3.7 TW. In 2018, electricity consumption in the world is 22,694 TWh and 2.61 TW. Thus, as can be seen from the numbers, global electricity consumption can be met by the total wave energy source [8, 9]. SPSs are undoubtedly one of the areas with the greatest potential in the use of wave energy systems. There are new studies in the literature where wave energy is used as the main energy source in shipboard microgrids [10, 11]. Photovoltaic (PV), fuel cell (FC) and wind energy sources are other RES commonly considered on shipboards [3, 10]. The Shipboard microgrid has more uncertainty than terrestrial microgrid. One of the most important reasons for this is that it cannot be supported by the main power system as in terrestrial microgrids. Frequency control is very vital in these systems due to the variable structure of RESs in their structures, in both terrestrial microgrid and shipboard microgrid [12, 13]. The wave energy is highly variable and its structure affects the frequency and increases the risk of frequency instability. Therefore, shipboard microgrid load frequency control (LFC) is an important issue. In [10], a new method for LFC of the shipboard microgrid is proposed. In [14], Kalman filter is used for the shipboard microgrid LFC structure. In the LFC control of the microgrid, the control signals between the grid elements and the controller are transmitted via the communication network. There is a phenomenon of time delay inherent in communication networks and the time delays must be taken into account in the LFC of microgrid [15, 16]. In [17], time delays are taken into account when performing LFC control of microgrid. In [18, 19], the LFC of the shipboard microgrid system is performed taking into account the time delay. With the proposed controller design in this paper, ensuring that shipboard microgrid power systems remain in stable parameter space will provide effective shipboard microgrid LFC. Stable parameter space is determined by the stability boundary locus (SBL) method [20]. In addition, with the proposed controller design, the system will be more robust against the uncertainties that will occur in the system. In this study, gain margin (GM) and phase margin (PM) based stable region analysis of a time-delayed shipboard microgrid with sea wave energy (SWE) system is performed. In this study, SWE and PV are also considered as the main sources of a shipboard microgrid. Owing to the variable nature of these sources, a controller design has been proposed for shipboard microgrid LFC. By taking into account the time delays that may occur in the system for proposed controller design, it is ensured that the controller design guarantees the desired dynamic response. While there are studies that used SBL method for LFC in the literature [21, 22], there is no study in a shipboard microgrid system LFC. In [21], The SBL method is used for one area time delayed LFC. In [22], this method with GM and PM is also used for microgrid LFC. While the shipboard microgrid LFC requires a high on-line processing load in [10], the proposed controller design is set offline in this paper. Although the shipboard microgrid uses an offline method for the controller parameter in [23], the system does not have the structure to guarantee the desired dynamic response. The proposed controller design guarantees to meet the desired dynamic response with virtual gain and phase margin tester (GPMT) structure in this study. The main contributions of the framework presented in this study are given as follows: The gain values of the proposed controller are calculated offline using the SBL method and no online calculation is performed. Stable space of the system is determined and stable operation of the system under the desired conditions is guaranteed. The areas and the efficiency of the power system elements of vessels have been taken into consideration. In this context, the use of SWEs in the shipboard microgrid system is considered. At the design stage of the proposed controller, time delays that may occur in the system are considered. In this context, a controller design is provided which can compensate for time delays in the system. In controller designs made only with the stability of shipboard microgrid systems, the system can operate close to the stability limit and unwanted oscillations may occur in the system. GM and PM values are taken into consideration in the design of the proposed controller. In this context, a controller design in which the system can supply the desired dynamic response is provided. The other parts of this paper are presented as follows: the model of the shipboard microgrid with SWE is given in Section 2. The stability regions of time-delayed shipboard microgrid with SWE are obtained in Section 3. The shipboard microgrid performance and stability region results of the proposed controller design are presented in Section 4. Finally, the conclusion of this study is presented in Section 5. 2 Model of shipboard microgrid with SWE The system configuration of the shipboard microgrid used in this study is shown in Fig. 1. An islanded shipboard microgrid involves SWE, FC, PV, diesel generator (DG), as well as energy storage system (ESS) units such as battery ESS (BESS) and flywheel ESS (FESS). All shipboard microgrid units are controlled by the shipboard power management system (SPMS) through a bidirectional communication network. Fig. 1Open in figure viewerPowerPoint System configuration of shipboard microgrid When the load demand increases, BESS and/or FESS can give enough energy to the connected load in a short time. Due to the limits of rapid control of FC and DEG for significant changes in load, sufficient energy is provided by FESS and BESS in a short period of time [24]. FC and DG is a standby generator that can start automatically to power the system only when the total power generated by SWE and PV is insufficient [23, 25, 26]. The schematic representation of the proposed shipboard microgrid LFC is shown in Fig. 2. To control the frequency, the FC and DG are used as the manipulating units. Time delay in the bidirectional communication between the controller and the units is considered as a control input and time delay is indicated by in Fig. 2. In addition, the irradiation of PV, the wave of the SWE and the power of the load are determined as disturbance inputs, because they are assumed to be non-measurable and time-varying. The GPMT defined by the user to provide the desired GM and PM of the shipboard microgrid is shown with in Fig. 2 [23, 27]. Fig. 2Open in figure viewerPowerPoint Block diagram of shipboard microgrid system for LFC To fully simulate the dynamic behaviour of practical DG, FC, SWE, PV, BESS, FESS etc. should use non-linear high-order mathematical models. However, transfer functions or simplified models are usually used for large-scale power system simulations [26]. Therefore, the units that compose the shipboard microgrid in this paper are shown by transfer functions. The mathematical models of the units of the shipboard microgrid given in Fig. 2 are given below. 2.1 Diesel generator DGs have a fast response and low maintenance features. Due to these features, DGs are a suitable backup utility for shipboard microgrid systems. DGs adjust the fuel consumption to regulate the output power when they face with changes in the power of the RESs. The DG system is constructed by a first-order generator and a first-order governor. The power output of the DG unit may be linearised as (1) considering the actions of generator and governor [10] (1) 2.2 Fuel Cell FC systems convert chemical energy into electrical energy. In the shipboard microgrid, the FC unit contains three parts, FC, inverter and interconnection device. In order to achieve the transfer functions of FC portion, the power transient response characteristics of a FC are tested. With the measurement results, an approach curve of the dynamic features of proton-exchange membrane FC is created, and the first order transfer function is obtained [27]. The inverter and interconnection device are presented in the FC system with a first order lag transfer function that can achieve their transient responses. The power output of FC unit may be linearised as (2) considering the actions of FC, inverter and interconnection device [25] (2) 2.3 Energy storage system For the shipboard microgrid system, an ESS consisting of BESS and FESS is considered. It is believed that ESS units are capable of storing excess energy produced in this system and are able to respond to load increases in the system in a short time [24]. The BESS and FESS are realised with a first-order lag transfer function which can achieve their transient response characteristics in the shipboard microgrid system. Their linearised transfer function is shown in (3) [25]. (3) 2.4 Sea wave energy Wave energy converters (WECs) are devices that convert wave energy into electrical energy. There are two different anchoring methods for WEC devices, single-body case (moored system) and double-body case (drogue anchored system). A drogue (or sometimes also known as a sea anchor) will provide the required fixed anchor point to a WEC device, depending on wave motion. In this way, the wave energy can be transformed by a buoy that floats freely in very deep waters where mooring is not technically or economically feasible. With this anchoring method, WECs can be considered as a RES for shipboard microgrids [28]. A WEC consists of two main parts: floating buoy and linear generator. Sea waves captured by the buoy are converted into electrical energy by the linear generator. The linear generator connected to the Buoy with a rope plays the role of the power take-offs system. The electrical energy generated by the linear generator is carried to the shipboard microgrid via power cables and AC/DC/AC converter. Since the frequency and magnitude of the terminal voltage of the WEC vary, the AC/DC/AC converter is applied in the integration of WEC into the shipboard microgrid. An AC–DC converter block is integrated into the system to eliminate unnecessary oscillation from the output power, and also this DC output is converted to a frequency suitable for the shipboard microgrid system using an inverter block [9]. The power outputs of the WECs are affected by the marine condition, buoy characteristics and power take-offs damping. The first case depends on the place where the WEC is located, while the other two can be modified and optimised in the mechanical design and sizing of the buoy and the choice of load connection and generator type [29]. The power output of the SWE unit may be linearised as (4) considering the actions of buoy and generator [23]. (4) 2.5 Photovoltaic PV systems are devices that convert solar energy into electrical energy. In the shipboard microgrid, the PV unit contains two parts, PV and inverter. The PV portion is performed with a first-order degree transfer function with reference to a PV power plant operation as shown in [30]. The inverter is presented in the PV system with a first-order lag transfer function that can achieve its transient response characteristics. The power output of PV unit may be linearised as (5) considering the actions of PV and inverter [30]. (5) In order to understand the frequency response of a shipboard microgrid, a simplified frequency response model is given in Fig. 2 as a case study. The parameter values of this model are given in Table 1 [23, 25, 31]. The model is given in Fig. 2 is useful for illustrating and analysing the frequency behaviour of the shipboard microgrid system. The input and output signals of shipboard microgrid model are given in Table 2. Table 1. Shipboard microgrid system's parameters Symbol Nomenclatures Values R frequency droop factor of shipboard microgrid (Hz/p.u. MW) 3 , governor and generator time constants of DG unit 0.08, 0.4 s , , FC, inverter and interconnection device time constants of FC unit 0.26, 0.04, 0.004 s , bouy and linear generator time constants of SWE unit 4, 0.5 s , PV and inverter time constants of PV units 0.5, 4 s. , time constants of FESS and BESS units 0.1, 0.1 s M, D equivalent inertia and damping constants of shipboard microgrid 0.2, 0.012 Table 2. Shipboard microgrid system input and output signals Symbol Nomenclatures input manipulator of the DG input manipulator of the FC output power fluctuation of the DG output power fluctuation of the FC output power fluctuation of the SWE output power fluctuation of the PV output power fluctuation of the FESS output power fluctuation of the BESS power fluctuation of the load demand power fluctuation of the sea wave power fluctuation of the solar irradiation time delay frequency deviation of the shipboard microgrid 3 Computation stability region of time-delayed shipboard microgrid with SWE In this section, stable parameter spaces of the controller belonging to time-delayed shipboard microgrid are calculated. The GM- and PM-based PI controller structure for LFC of the shipboard microgrid is used. Time delays in the system are taken into account to ensure the reality of the proposed system. Virtual GPMT is added to the system to provide the controller with the desired dynamic response. The GM and PM values are taken into account when calculating the stable parameter spaces and the SBL method is used in these calculations. The characteristic equation is the denominator of the transfer function obtained from the shipboard microgrid model. The characteristic equation of shipboard microgrid without GPMT is given in (6). The characteristic equation is expressed by the P(s) and Q(s) polynomials in s with real coefficients in (6). Exponential expressions are inserted into the Q(s) polynomial. The remaining expressions are collected in the P(s) polynomial. (6) The polynomial coefficients in terms of the system parameters are given in Appendix. The characteristic polynomial with GPMT belonging to the shipboard microgrid is given in (7). (7) The characteristic equation of the shipboard microgrid model given in Fig. 2 is rearranged according to (8) for analysis and (9) is obtained. (8) (9) The expressions for the coefficients in (9) are given in (10) (10) The real and imaginary parts of the characteristic equation given in (9) are equalised to zero separately (11) The parameters and are calculated from (12) (12) The complex root boundary, real root boundary and infinite root boundary represent the existing boundaries according to the characteristic equation. These boundaries are defined for the system as follows: complex root boundary: , real root boundary: and infinite root boundary: . Based on these boundaries, the system stability analysis is as follows. The infinite root boundary is not in the plane because it is independent of the control parameters. is in the real root boundary and is in the complex root boundary. As a result, the boundary locus and the line split plane into stable and unstable regions [21, 22, 32, 33]. The steps of the proposed controller are shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Design stages of the controller 4 Numeric and simulation results The LFC model of the shipboard microgrid includes SWE, PV, FC, DG, FESS, BESS, controller and load and is shown in Fig. 2. The parameter values of the shipboard microgrid with SWE are given in Table 1 [23, 25, 31]. In this section, the SBL curves of the proposed controller design for the shipboard microgrid with SWE are shown and simulation studies are performed to prove the accuracy of these curves and to prove the success of the proposed controller design on the shipboard microgrid with SWE. First, the effects of GM and PM values on the stable parameter plane of the proposed controller design are shown. In addition, the effects of different time delay values on the stable parameter plane of the proposed controller design are also shown. Time domain and eigenvalue analysis studies are performed to show the accuracy of SBL curves obtained in the following section. Real-world sea wave and solar data are used in the shipboard microgrid with SWE to demonstrate the performance of the proposed controller design. The performance of the proposed controller design for different cases of the shipboard microgrid with SWE is also been compared with other controllers in the literature. 4.1 Stability regions results In this section, the SBL of the shipboard microgrid with SWE is shown. In addition, necessary analysis studies are presented to demonstrate the accuracy of the obtained SBL. The SBL of the shipboard microgrid with SWE is determination of the plane that will provide the characteristic equation given in (7) for the specific time delay value and the desired GM and PM values. The plane are determined using (12). The stable planes of according to τ's different values from 0.5 to 3 to 0.5 intervals for and are shown in Figs. 4 and 5. The system is stable for the point to be selected inside this curve. The system is critically stable for the point to be selected on this curve, while the system is unstable for the point to be selected outside this curve. As can be seen from the figures, the increase in the value of the time delay in the system causes a decrease in the parameter space where the system will operate stably. Furthermore, as can be seen from the figures, parameter plane is reduced according to the increase of desired GM values. Fig. 4Open in figure viewerPowerPoint Stable planes of according to τ's different values for Fig. 5Open in figure viewerPowerPoint Stable planes of according to τ's different values for Fig. 6 shows the frequency responses of three different values on curve , inside curve and outside curve for . Since the selected parameter values for are inside the stability region, it is appropriately stable in the time response of system for these parameters. Since the selected parameter values for are on the curve, it is critically stable in the time response of the system for these parameters. Finally, since the selected parameter values for are outside the stability region, the time response of the system for these parameters is suitably unstable. Fig. 6Open in figure viewerPowerPoint Frequency responses of three different values on curve , inside curve and outside curve for Fig. 7 shows the frequency responses of the time delay change belonging to selected on the SBL curve for . The system frequency response for becomes critically stable, while the system frequency response for (less than the margin time delay value) is stable and the system frequency response for (greater than the margin time delay value) is unstable. Fig. 7 clearly shows the accuracy of the theoretical margin time delay. Fig. 7Open in figure viewerPowerPoint Frequency responses of three different for and Fig. 8 shows the dominant roots belonging to the characteristic equation given in (7) of three different values on curve , inside curve and outside curve and . Since the selected parameter values for are inside the stability region, this system is appropriately stable. The eigenvalues of the system are and they are on the left half plane of the imaginary axis for these parameters. Since the selected parameter values for are on the curve, this system is critically stable. The eigenvalues of the system are and they are on the imaginary axis for these parameters. Finally, since the selected parameter values for are outside the stability region, this system is unstable. The eigenvalues of the system are and they are on the right half plane of the imaginary axis for these parameters. Fig. 8Open in figure viewerPowerPoint Dominant roots belonging to the characteristic equation given in (7) of three different values inside curve , on curve and outside curve and (a) Stable, (b) Critically stable, (c) Unstable Figs. 9 and 10 show the SBL curves of the shipboard microgrid system for , , , , and according to and . As can be seen from the figures, the size of curves belonging to the desired GM and PM values are smaller than the curve. Furthermore, as can be seen from the figure, parameter plane size is reduced according to the increase of desired GM and PM values. Fig. 9Open in figure viewerPowerPoint GM and PM based stability region for Fig. 10Open in figure viewerPowerPoint GM and PM based stability region for The system frequency responses of the values providing the GM and PM values shown in Fig. 9 are shown in Fig. 11. value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for and value inside SBL for are selected. As can be seen from the figure, all frequency responses are stable. The frequency response for the has unwanted oscillations and a slow time response. The results showed that the system frequency responses belonging to the values selected for the desired GM and PM values are faster and non-oscillatory. System frequency responses of the values providing the GM and PM values shown in Fig. 10 are shown in Fig. 12. As can be seen from the figure, all frequency responses are stable. The frequency response for the has unwanted oscillations and a slow time response. The results showed that the system response belonging to the values selected for the desired GM and PM values are faster and non-oscillatory. Fig. 11Open in figure viewerPowerPoint Frequency responses of value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for and value inside SBL for and Fig. 12Open in figure viewerPowerPoint Frequency responses of value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for , value inside SBL for and value inside SBL for and 4.2 Shipboard microgrid performance results of the proposed controller design To illustrate the validity of the proposed controller design in this section, a shipboard microgrid with SWE shown the LFC model with Fig. 2 is simulated. Simulations are performed in three different situations. 4.2.1 Case A In the first case, the shipboard microgrid is considered to have a constant load change. So only the power fluctuations of SWE and PV are changing in LFC system. The SWE fluctuation as per unit is presented in Fig. 13a and SWE data has been applied from the National Oceanographic Data Center [34]. PV data as is obtained from Keweenaw Research Center [35] and it is presented in Fig. 13b. The power ratios of the SWE and PV are considered in shipboard microgrid LFC as 0.95 and 0.45 p.u., respectively. Fig. 13Open in figure viewerPowerPoint Power fluctuations (a) SWE, (b) PV Fig. 14 shows the frequency responses of three different values on curve , inside curve and outside curve for and power fluctuation of SWE and PV. As can be seen from the figure, the system frequency response for selected values inside the stable region of the controller is stable, whereas system frequency response for values taken outside the stable region, the system is unstable. For the values taken on the SBL curve, the frequency response of the system shows oscillations against power changes. Fig. 14Open in figure viewerPowerPoint Frequency responses of three different values on curve , inside curve and outside curve for and power fluctuation of SWE and PV Fig. 15 shows the frequency response of the shipboard microgrid according to case A for different GM and PM values and . As can be seen in the figure, as a result of the increase in the desired GM and PM values, the frequency response of the system has less oscillation and responds faster to power changes. For shipboard microgrid with large variability systems such as wave energy, it is important to consider GM and PM values in the determination of controller parameters to ensure stable operation of these systems. Fig. 15Open in figure viewerPowerPoint Frequency responses of value inside SBL for , value inside SBL for , value inside SBL for and value inside SBL for , and power fluctuation of SWE and PV In Fig. 16, the performance comparison of the proposed controller design in this study with the controllers commonly used in LFC studies in the literature is performed. In Fig. 16, the frequency responses of the proposed controller design by the solid line, the [36] by the doted line, and the [37] by the dash line are presented. As can be seen from the figure, the proposed controller design in this paper has a lower frequency amplitude change than the proposed controllers in [36, 37] and the damping of unwanted oscillations is faster. Fig. 16Open in figure viewerPowerPoint Frequency response according to case A (the proposed controller design by the solid line, the [36] by the doted line, and the [37] by the dash line) 4.2.2 Case B In case of B, the multi-step load changes, as well as the SWE and PV power fluctuation, are applied to the shipboard microgrid LFC as a disturbance. Figs. 17 and 18 show the frequency response of the shipboard microgrid for case B. As can be seen from Fig. 17, the frequency responses of the selected values inside, on and outside the SBL are stable, critically stable and unstable, respectively. As can be seen in Fig. 18, increasing the GM and PM values improve the frequency response of the system. Fig. 17Open in figure viewerPowerPoint Frequency responses of three different values on curve , inside curve and outside curve for and power fluctuation of SWE, PV and load disturbances Fig. 18Open in figure viewerPowerPoint Frequency responses of value inside SBL for , value inside SBL for , value inside SBL for and value inside SBL for , and power fluctuation of SWE, PV and load disturbances The comparison of the proposed controller design with the other controllers available in the literature is performed for case B in Fig. 19. As can be seen from the figure, the dynamic behaviour of the proposed controller design is better than other methods. Fig. 19Open in figure viewerPowerPoint Frequency response according to case B (the proposed controller design by the solid line, the [36] by the doted line, and the [37] by the dash line) 4.2.3 Case C In order to demonstrate the robustness of the proposed controller design in this paper, some parameters of the shipboard microgrid have been modified. The values of the changes made in the parameters are given in Table 3. Table 3. Parameter change of shipboard microgrid system Parameter Variation range Parameter Variation range R +%33 +25% D −%10 −25% M −%25 +25% +%25 — — Fig. 20 shows the frequency response of the shipboard microgrid against parameter changes. As can be seen from the figure, in the case where GM and PM values are not controlled , the system is unstable in the parameter change that occurs in the system. However, in cases where the GM and PM values are controlled, the controller maintains system stability in the parameter change that occurs in the system. The control on the GM and PM when determining the controller parameters improves the robust stability of the shipboard microgrid. Fig. 20Open in figure viewerPowerPoint Frequency responses according to case C of the shipboard microgrid In Fig. 21, the performance of the proposed controller design is compared with other controllers available in the literature for case C. As can be seen from the figure, the proposed controller design is achieved a better dynamic response than the proposed controllers in [36, 37]. Furthermore, the system is unstable in case of parameter uncertainty in the shipboard microgrid for the proposed controllers in [36, 37]. Fig. 21Open in figure viewerPowerPoint Frequency response according to case C (the proposed controller design by the solid line, the [36] by the doted line, and the [37] by the dash line) 5 Conclusion In this paper, a GM and PM based controller design for LFC of shipboard microgrid with SWE is presented. Time delays in the system are taken into account to ensure the reality of the proposed system. A virtual GPMT is added to the system to provide the controller with the desired dynamic response. The stability region is shown in the parameter plane of controller for the desired GM and PM of the time delayed shipboard microgrid with SWE by the SBL method. The results have shown that undesired oscillations occur in the system only with the values obtained considering stability. With the addition of GMPT, the dynamic response of the system has improved and the time response of the system has been faster and non-oscillatory. The results also have shown that GM and PM values have significant effects on the stable region and cause a decrease in the dimensions of the stable region. The control on the GM and PM when determining the controller parameters improves the robust stability of the shipboard microgrid. 7 Appendix (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) 6 References 1Skjong, E., Volden, R., Rødskar, E., et al: 'Past, present, and future challenges of the marine vessel's electrical power system', IEEE Trans. Transp. Electr., 2016, 2, (4), pp. 522– 537 2Liu, Z., Wu, J., Hao, L.: 'Coordinated and fault-tolerant control of tandem 15-phase induction motors in ship propulsion system', IET Electr. Power App., 2017, 12, (1), pp. 91– 97 3Lan, H., Bai, Y., Wen, S., et al: 'Modeling and stability analysis of hybrid PV/diesel/ESS in ship power system', Inventions, 2016, 1, (5), pp. 1– 16 4Hou, J., Song, Z., Park, H., et al: 'Implementation and evaluation of real-time model predictive control for load fluctuations mitigation in all-electric ship propulsion systems', Appl. 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