Optimal Power Management for Failure Mode of AC/DC Microgrids in AllElectric Ships
Abstract
Optimal power management of shipboard power system for failure mode (OPMSF) is a significant and challenging problem considering the safety of system and person. Many existing works focused on the transienttime recovery without consideration of the operating cost and the voyage plan. In this paper, we formulate the OPMSF problem considering the longtime scheduling and the faults at bus and generator. For reducing the fault effects, we adopt twoside adjustment methods including the load shedding and the reconfiguration. To address the formulated nonconvex problem, we transform the travel equality constraint into an inequality constraint and obtain a convex problem. Then, considering the infeasibility scenario affected by faults, a further relaxation is adopted to obtain a new problem with feasibility guaranteed. Furthermore, we derive a sufficient condition to guarantee that the new problem has the same optimal solution as the original one. Because of the mixedinteger nonlinear feature, we develop an optimal algorithm based on Benders decomposition (BD) to solve the new one. Due to the slow convergence caused by the timecoupled constraints and the tailingoff effect, we also propose a lowcomplexity nearoptimal algorithm based on BD (LNBD). The results verify the effectivity of the proposed methods and algorithms.
I Introduction
Ia Motivation
Shipboard power system (SPS) is selfpowered by distributed electrical power generators operating collectively, which can be considered as an isolated microgrid. From the perspective of electrical design of allelectric ship (AES), there are three architectures of SPS to date, i.e., medium voltage DC (MVDC), medium voltage AC (MVAC), and higher frequency AC (HFAC). As the everincreasing DCbased loads, it is likely that AES will feature a medium voltage primary distribution system in the future [hebner2015technical]. Due to the intensive coupling and finite inertia feature, the consequences of a minor fault in a system component can be catastrophic. Since the AES mostly targets at military applications, it is highly susceptible to be damaged. Distinguished from terrestrial systems, the system failure of SPS is more disastrous due to the personal safety on the shipboard. Thus, optimal power management of SPS for failure mode (OPMSF) is essential to guarantee the system safety, while meeting the load demand.
The time scales of the OPMSF problem includes transienttime, shorttime, and longtime. Most of existing works about OPMSF focused on the recovery at a transienttime scale [Bose2012Analysis, Srivastava2007Probability, das2013dynamic, jiang2012novel, Seenumani2012Real, Feng2015Multi] or a shorttime scale [seenumani2011reference]. Their objective is to improve the restored power of loads and guarantee the power balance. However, due to the damaged system structure by faults, the power supplydemand relationship is changed. Consequently, the original operating schedule of generators is not suitable for the remaining voyage owing to the increase of operating cost and the risk of system safety. Hence, the longtime scheduling OPMSF problem is essential and meaningful.
The adjustment methods for longtime scheduling can be classified into two categories, i.e., the supply side and the demand side. On the supply side, the generators that are the primary generation equipment cannot operate at original optimal state caused by faults. Additionally, energy storage module compensation (ESMC) is a potential solution for improving the energy efficiency of SPS [inventions2017Effect]. Hence, the generation scheduling including ESMC in the remaining voyage has to be reorganized according to faults. On the demand side, the load adjustment also plays a key role in optimal power management. The load of SPS includes the propulsion modules and service loads. The power of propulsion modules constitutes a large proportion of SPS, which dynamically changes with operation mode switching. The propulsion power adjustment (PPA) can achieve significant energy efficiency improvement [kanellos2014optimal]. However, once the capacity of generators is not enough to cover the load demand in the corresponding zones after faults happening, load shedding of service loads and reconfiguration of power network are required for guaranteeing the system safety and completing the voyage. Hence, it is necessary to consider load shedding and reconfiguration besides PPA on the demand side. To sum up, twoside adjustment methods including load shedding and reconfiguration are meaningful and have significant effects on the OPMSF problem for guaranteeing the system safety, completing the voyage, and reducing the operating cost.
IB Literature Review
Many efforts have been devoted to study the OPMSF problem [Bose2012Analysis, das2013dynamic, amba2009genetic, Nelson2015Automatic, Mitra2011Implementation, jiang2012novel, mashayekh2015integrated, Srivastava2007Probability, Seenumani2012Real, Feng2015Multi, kanellos2014optimal, seenumani2011reference, Kanellos2016Smart, Kanellos2017cost, Kanellos2014Optimaldemand, shang2016economic]. In the transienttime scale, their main objective is to maximize the weighted sum of restored loads [Bose2012Analysis, das2013dynamic, jiang2012novel, Srivastava2007Probability, Seenumani2012Real, Feng2015Multi, amba2009genetic, Nelson2015Automatic, Mitra2011Implementation]. Additionally, there are other considerations including obtaining the correct order of switching[das2013dynamic], probabilitybased prediction of fault effects[Srivastava2007Probability], minimizing the number or cost of switching actions[Bose2012Analysis, jiang2012novel], realtime management[Seenumani2012Real, Feng2015Multi], etc. In the shorttime scale (warmup time of the backup power sources), the authors in [seenumani2011reference] developed a reference governorbased control approach to support the noncritical loads as much as possible while maximizing the battery usage. However, they focused on the recovery of power supply for loads in a transienttime or shorttime scale, without consideration of the voyage plan and operating cost after faults in a longtime scale.
In the longtime scale, the algorithms based on the particle swarm optimization (PSO) method are developed to solve the optimal power management (OPMS) problem for normal mode in [kanellos2014optimal, Kanellos2016Smart, Kanellos2017cost]. In the three works, PPA and ESMC are considered for improving energy efficiency. Additionally, energy efficiency operation indicator (EEOI) is taken into account for the limitation of greenhouse gas (GHG) emission. In [Kanellos2014Optimaldemand], dynamic programming (DP) algorithm is adopted to solve the same problem. In [shang2016economic], the authors formulated a multiobjective problem that considers the reduction of fuel consumption and GHG mitigation together. In the above works, they do not consider the failure mode for improving the system safety and reliability. To the best of our knowledge, there is no work focused on the OPMSF problem in a longtime scale. Meanwhile, due to the damaged power network, and the usage of reconfiguration and load shedding, the algorithms in above works cannot be directly adopted to solve the OPMSF problem.
IC Challenges
The main target of this work is to solve the OPMSF problem in a longtime scale. There are three main challenges to solve this problem. Firstly, since reconfiguration and load shedding are adopted, how to coordinate them with the other adjustment methods for meeting the load demand in the first place. Secondly, considering the nonconvex feature of the proposed problem and the infeasibility scenario affected by faults, appropriate relaxation methods have to be developed to make the problem tractable. Thirdly, there are more variables in the twoside adjustment methods for failure mode than that for normal mode. Furthermore, the variables in the travel and ESM constraints of this mixedinteger nonlinear programming (MINLP) problem are coupled in time. Hence reducing the complexity needs to be considered in the algorithm design.
ID Contributions
In this paper, we deal with OPMSF problem in MVDC SPS to guarantee the system safety, complete the voyage and reduce the operating cost. Twoside adjustment methods including load shedding of service loads and reconfiguration of power network are all considered in this work. To make this problem more tractable, we develop twostep relaxations and obtain a new convex problem. Then, we design an optimal and a nearoptimal low complexity algorithms to solve the new problem. The contributions of this paper are summarized below.

With the objective of minimizing the total operating cost that contains the cost of generation, energy storage, and load shedding, we formulate the OPMSF problem based on the analysis of faults. Different from existing works, we add load shedding and reconfiguration into the adjustment methods considering the fault effects. A sufficient condition is established to guarantee that load shedding is adopted only when generator scheduling (GS) and ESMC cannot solve the OPMSF problem.

We firstly relax the nonconvex travel constraint in the original problem to obtain a convex problem. Then, considering the nonfeasibility scenarios caused by faults, we formulate a new MINLP problem with feasibility guarantee by further relaxing the travel constraint and adding a penalty term in the objective. Lastly, a sufficient condition is derived to guarantee that if the original problem is feasible, the new one has the same optimal solution; if not, the maximum allowed distance can be further obtained to assist rescue mission.

To address the new problem, we design an optimal algorithm based on Benders decomposition (BD) that splits it into two more tractable problems (subproblem and master problem). Due to the slow convergence caused by the timecoupled constraints and the tailingoff effect, we propose a lowcomplexity nearoptimal algorithm based on BD (LNBD) by decomposing the timecoupled constraints in the subproblem and adding the accelerating constraints in the master problem.
The paper is organized as follows: in Section II, MVDC SPS and the main modules are introduced, and the OPMS problem is formulated; Section LABEL:sec:problem_transformation reformulates the OPMSF problem according to different faults; Section LABEL:sec:algorithm_design details the proposed offline and online algorithms; the performance of the proposed method is evaluated in Section LABEL:sec:simulation. Finally, the conclusion is drawn in Section LABEL:sec:conclusion.
Ii System Models and Problem Formulation
SPS is an integrated power system, which consists of power generators, energy storage modules (ESMs), loads, converters, and transmission power network. In this section, these main models are introduced in detail. Then, the OPMS problem for normal mode is formulated based on these models. Particularly, the main notations used in this work are summarized in Table IIA.
Iia System Structure Overview
The classic architecture of MVDC SPS is shown in Fig.1. This architecture adopts a zonal approach with a starboard bus (SB) and a port bus (PB), and the SPS is parted into electric zones. The generators are classified into two types: main turbine generator (MTG) and auxiliary turbine generator (ATG). The distributed layout of generators contributes to enhancing the overall service capability. In each zone, the service loads are managed by the power control module (PCM) and power distribution module (PDM). The structure is shown in Fig. LABEL:fig:ESM. The DC zones are powered by a set of generators and converters denoted by . The loads are powered by a set of buses which run longitudinally along the PB and SB. Here propulsion modules are considered separately from service loads because of large energy consumption. In this work, we assume that the SPS operates in discrete time with and time interval .
Notation  Physical interpretation 

, ,  Set, number and index of generators and converters 
, ,  Set, number and index of DC zones 
, ,  Set, number and indexes of propulsion modules 
, ,  Set, number and index of ESMs 
, ,  Set, number and index of island parts 
, ,  Set, number and index of time 
,  Superscript denoting minimum and maximum 
Output power of generator at time  
Status of generator , (1/0 = online/offline)  
Startup status of generator , (1/0 = startup/shutdown)  
Output power of the ESM in zone at time  
Capacity of the ESM in zone at time  
Propulsion power of the th propulsion module  
Total output power of generators in island part at time  
Total output power of the ESMs in island part at time  
Total output power of the propulsion modules in island part at time  
Total Load demand in island part at time  
Power demand of vital and semivital loads at time  
Power demand of nonvital loads at time  
Lower bound of the loadshedding amount of  
Ship speed at time  
Total operating cost at time  
Fuel cost of generator at time  
Startup/shutdown cost of generator at time  
Operating cost of the ESM at time  
Load shedding cost at time  
Input power of converter at time  
Output power of converter at time  
Redundancy switches of PB in zone at time  
Redundancy switches of SB in zone at time  
Travel distance, reduced travel distance  