Optimal operation for marine hybrid power system: Novel optimization algorithm and comprehensive analysis for penalty functions

Abstract

In order to optimally control the marine hybrid power system (HPS) under increasingly complex regulation constraints or hardware constraints, an efficient power-flow scheduling model and optimization algorithm are of great importance. This work focuses on the optimal power-flow scheduling of marine HPS, especially on the efficiency improvement of the penalty functions for satisfying complex constraints. To be specific, an optimal operation model of marine HPS is discussed, and the complex model constraints are described as various penalty functions. Secondly, a novel optimization algorithm, namely adaptive multi-context cooperatively coevolving differential evolution algorithm with random topology and mutated context vector (AMCCDE_{- rt - mcv}) is developed to optimize the aforementioned model. In order to ensure the satisfaction of the complex model constraints, the detailed forms for penalty functions are researched and the optimal parameters for penalty functions are comprehensively compared, analysed and tested by a set of numerical experiments. Finally, the developed methodologies are tested by simulation experiments. Experimental results show that the damping factor, exponent parameter and punish strength constant effect the efficiency of penalty functions a lot, and the developed penalty functions can effectively satisfy all the model constraints with fast response speed. With the integration of penalty functions, the developed methodology can obtain promising performance on the optimal scheduling of the evaluated marine HPS.

Keywords

Hybrid power system optimal energy management penalty function optimization algorithm differential evolution

Nomenclature

η_c, η_d

Coefficients of charging and discharging efficiency

ρ (t)

Price of shore power at the tth time-unit

P _r

Rated power of onboard diesel generator

n_a, n_b

Fuel intercept and the fuel slop coefficients

F _d

Hourly fuel consumption of onboard diesel

P _f

Price of fuel oil

ρ _p

Price of shore power for peak period

ρ _s

Price of shore power for standard period

ρ _o

Price of shore power for off-peak period

SOC(t)

SOC of battery bank at the tth time-unit

SOC _h

Upper bound of SOC

SOC _l

Lower bound of SOC

P_d (t)

Diesel output at the tth time-unit

SOC(0)

The initial SOC of battery bank

SOC(N)

The final SOC of battery bank

Proportion determined by the port–s regulation

P _base

Historical baseline of ship hoteling load

P_mppt (t)

Maximum PV output at the tth time-unit

P _i-h

Upper bound of the power flowP_i

L(t)

The electric load demand in the tth time-unit

P _bd

Maximum discharging power of battery bank

C _s

Cost paid for shore power service

C _f

Cost of fuel oil consumption for running diesel generator

C _b

Battery degradation cost

C _p

Entire penalty item

N _P

Population size of AMCCDE_{- rt - mcv}

N _SP

Number of sub-populations in AMCCDE_{- rt - mcv}

P _j

The jth subcomponent in AMCCDE_{- rt - mcv}

C ₀

Positive constant to control the punish strength

ζ_Pj (i)

Active index of the power flows P_j (i)

The damping factor

The exponent parameter

S _pv

Sizing of PV system

S _bb

Sizing of battery banks

M ₀

Total investment cost of battery bank

P _br

Rated charging power of battery

Max_FES

Maximum number of fitness evaluations

Dynamic group size for random grouping

T _mut

Iteration number to mutate context vectors

1 Introduction

In recent years, the regulations related to the energy consumption and pollution emissions in marine transport become more and more stringent. For example, the International Maritime Organization (IMO) adopted a particularly important strategy to reduce the annual greenhouse gas (GHG) emissions from ships by at least 50% by 2050 compared with 2008 [1]. In addition, the California Air Resources Board (CARB) has required that at least 50% of a fleet’s visits to California ports should use the shore power to feed the hoteling load demand, or at least reduce their use of onboard diesel generator by 50% compared to a historical baseline. In 2017 and 2020, this requirement increased to 70% and 80% respectively [2]. Driven by those regulations, a lot of green energy resources are installed on large ships to obtain energy saving and emission reduction [3, 4]. In the marine hybrid power system (HPS), the photovoltaic (PV) system, battery banks and other kinds of power resources are employed as supplement of onboard diesel to supply power for the ship [5 –7]. In addition, the shore-power can be also connected to charge the battery or feed the electric load when the ship berths in port [8, 9]. However, when these various power resources are installed on ships and work as power supplier, some new issues are needed to be solved.

1.1 Literature review

The application of renewable energy system (RES) on large ships attracts attentions of worldwide scholars. The hot topic in this field includes the integration of RES on marine electric power system, the environmental effect on power generation of onboard RES, the optimal operation control of marine HPS and so on. For example, the moving and rocking of ship will influence the output of onboard PV system, which is focused and studied in detail by Liu et al. [10]. They established a mathematical model to describe the fluctuation of ship PV output, introduced the super-capacitor and developed a binary energy storage scheme to both stabilize the PV power fluctuations and slow the aging of actual battery caused by rocking. According to their study, the application of super-capacitor and energy storage scheme could reduce the battery replacement rate for about 24.8% to 35.0%. In addition, Lee et al. developed a hybrid PV/diesel generator ship, which consisted of a PV generation system, diesel generator, batteries, stand-alone/grid-connected inverter and a control system [11]. In terms of the installation tilt-angle of marine PV panels, Lan et al. studied the optimal PV panel tilt-angle by evaluating a real PV system on a large oil tanker ship. In addition, they also developed a methodology to optimize the capacity of onboard energy storage system [12]. Karatug et al. designed a PV system for a ro-ro ship, and its energy saving was also estimated by case studies [13]. Fang et al. focused on the operation of PV integrated all-electric ship microgrids, and a data-driven robust coordination of generation and demand-side is developed to address the uncertainty of marine PV system, and also reduce the fuel cost of all-electric ships [14]. Yu et al. developed a power management strategy for marine micro-grid system using fuzzy control method. Their evaluated system contained the marine PV array, onboard diesel generator and lithium battery banks. According to the provided experimental results, their developed approach greatly improved the operational reliability and stability of the evaluated ship [15]. Balsamo et al. developed an optimal design and management strategies for the hybrid energy storage system in marine applications, in order to improve the battery pack durability and ensure a smooth profile of the required current [16]. Han et al. concerned the fluctuation control of marine hybrid energy storage system. In their study, an energy management strategy based on a two-stage fuzzy logical controller was developed, in order to redistribute the marine current power and abate marine current power fluctuation [17].

To protect the environment around port, shore power systems are widely equipped on both the ship-side and the port-side in recent years. Shore power supplies power to the ship so its onboard diesel generators can be turned off. As a result, the shore power is also called “cold-ironing” [18 –20]. Lu et al. developed a multi-period dual-objective shore power deployment optimization model, in which the operational costs and CO₂ emissions were minimized. In addition, the impact of government subsidies on shore power construction and ship shore power use was analysed in their study in detail [21]. Peng et al. developed a cooperative optimization method to optimize the berth and shore power allocation problem for the stochastic arriving ship. In their developed optimization model, the total costs of installing and using shore power system were minimized, and the environmental benefit for reducing air pollution and GHG emissions were maximized at the same time. In addition, the multi-objective particle swarm optimization (PSO) algorithm was adopted to optimize their developed model [22]. Abu Bakar et al. developed a hybrid system for seaport microgrid, in which the PV system, wind turbines, energy storage system and cold-ironing facilities were included. In addition, the best configuration of their developed HPS was analysed by considering the economic feasibility, energy reliability and environmental impacts [23].

1.2 Research gap and motivation

As discussed in the literature review part, the existing studies in marine HPS mainly concerns the integration of distributed RES, the environmental effect on power generation of RES, the optimal sizing and operation control of marine HPS. However, the cooperation of the integrated RES, the shore power system and the traditional marine electric power system is rarely addressed. In fact, the power flow dispatching problem for marine HPS (consists of the onboard RES, onboard energy storage system, traditional onboard diesel generator, onboard electric load and the shore power system) is of great importance for the operational safety and efficiency of the entire marine HPS.

In addition, for the power flow dispatching problem of marine HPS, the complexity significantly increases with the model dimensionality. In our previous work, the evolutionary algorithms (EA) are employed to solve the aforementioned dispatching problem to obtain promising optimization performance [24, 25]. However, a lot of constraints should be considered in the marine HPS dispatching problem, e.g., the power balance constraint, SOC boundary constraint, diesel output constraint, PV output constraint, ports’ regulation constraint, battery charging/dis-charging constraint and so on. Obviously, how to introduce these constraints into the optimization model significantly affects the final performance of EA. However, the existing studies rarely concerns the effect of these constraints and the detailed definitions of the related penalty functions. As a result, this study developed a novel EA-based algorithm to overcome the high-dimensional characteristic of the power flow dispatching problem. In addition, a set of penalty functions for describing the various constraints are developed, and the robustness of the penalty functions are analysed in detail.

1.3 Contribution and paper organization

This study mainly focuses on the optimal power-flow dispatching of marine HPS, and the detailed effect of penalty functions on dispatching performance is also concerned. The contribution of this study can be summarized as the following bullets:

Optimal operation of marine HPS is modelled as a large-scale global optimization problem.

A novel evolutionary algorithm namely AMCCDE_{- rt - mcv} is developed to effectively optimize the high-dimensional marine HPS operation model.

The various constraints for marine HPS are modelled as a set of efficient penalty functions, and the efficiency and robustness for the penalty functions are comprehensively analyzed.

The remainder of this work is organized as follows. In Section 2, the structure and optimal operation model of the evaluated marine HPS are discussed. Section 3 develops a novel optimization algorithm namely AMCCDE_{- rt - mcv} to optimize the aforementioned model. In Section 4, various penalty functions for ensuring the satisfaction of constraints are developed and analysed in detail, and the effect of different penalty functions and their parameters are comprehensively tested by numerical experiments. Then, the case studies and analysis are conducted in Section 5. Finally, this work is concluded in Section 6.

2 Marine hybrid power system and its optimal operation model

2.1 Structure of marine hybrid power system

For an HPS integrated green ship, the ship power can be supplied by the onboard PV system, diesel generator, battery bank and the connected shore power. In addition, the onboard battery bank can be charged by PV, diesel and shore power. Structure of the marine HPS with shore power connection is shown in Fig. 1, in which the power flows P₂, P₃, P₅, and P₇ denote the output power from PV system, battery bank, diesel generator and shore power to feed the ship load, respectively. The power flows P₁, P₄, and P₆ denote the power flows to charge the battery from PV system, diesel generator and shore power, respectively.

Fig. 1

Structure of marine HPS with shore power connection.

2.2 Optimal operation model of marine HPS

In order to optimally schedule the power flows of HPS (i.e., P₁ to P₇ in Fig. 1), the mathematical model of each distributed power resource and the optimal operation model of the entire HPS are required. In this study, the employed mathematical model of each power resource is listed in Table 1.

Table 1
Mathematical model of each distributed power resource

Power resource Mathematical model Parameter Parameter meaning

Battery bank $\begin{matrix} SOC (t) = SOC (0) + η_{c} \cdot \sum_{i = 1}^{t} [P_{1} (i) + P_{4} (i) + P_{6} (i)] \end{matrix}$ η_c η_d Coefficients of charging and discharging efficiency

$\begin{matrix} - \frac{1}{η_{d}} \cdot \sum_{i = 1}^{t} P_{3} (i) \end{matrix}$

Onboard diesel ${\begin{matrix} C_{f} (t) = \frac{P_{f} \cdot F_{d} (t)}{P_{d} (t)} = \frac{P_{f} \cdot [n_{a} \cdot P_{r} + n_{b} \cdot P_{d} (t)]}{P_{d} (t)} \\ P_{d} (t) = P_{4} (t) + P_{5} (t) \end{matrix}$ P _r Rated power of onboard diesel generator

n_a, n_b Fuel intercept and the fuel slop coefficients

F _d Hourly fuel consumption of onboard diesel

P _f Price of fuel oil

Shore power $ρ (t) = {\begin{matrix} ρ_{p}, t \in [7 : 00, 10 : 00) \cup [18 : 00, 20 : 00) \\ \begin{matrix} ρ_{s}, t \in [6 : 00, 7 : 00) \cup [10 : 00, 18 : 00) \\ \cup [20 : 00, 22 : 00) \end{matrix} \\ ρ_{o}, t \in [0 : 00, 6 : 00) \cup [22 : 00, 24 : 00) \end{matrix}$ ρ (t) Price of shore power at the tth time-unit

ρ _p Price of shore power for peak period

ρ _s Price of shore power for standard period

ρ _o Price of shore power for off-peak period

Power resource	Mathematical model	Parameter	Parameter meaning
Battery bank	$\begin{matrix} SOC (t) = SOC (0) + η_{c} \cdot \sum_{i = 1}^{t} [P_{1} (i) + P_{4} (i) + P_{6} (i)] \end{matrix}$	η_c η_d	Coefficients of charging and discharging efficiency
	$\begin{matrix} - \frac{1}{η_{d}} \cdot \sum_{i = 1}^{t} P_{3} (i) \end{matrix}$
Onboard diesel	${\begin{matrix} C_{f} (t) = \frac{P_{f} \cdot F_{d} (t)}{P_{d} (t)} = \frac{P_{f} \cdot [n_{a} \cdot P_{r} + n_{b} \cdot P_{d} (t)]}{P_{d} (t)} \\ P_{d} (t) = P_{4} (t) + P_{5} (t) \end{matrix}$	P _r	Rated power of onboard diesel generator
		n_a, n_b	Fuel intercept and the fuel slop coefficients
		F _d	Hourly fuel consumption of onboard diesel
		P _f	Price of fuel oil
Shore power	$ρ (t) = {\begin{matrix} ρ_{p}, t \in [7 : 00, 10 : 00) \cup [18 : 00, 20 : 00) \\ \begin{matrix} ρ_{s}, t \in [6 : 00, 7 : 00) \cup [10 : 00, 18 : 00) \\ \cup [20 : 00, 22 : 00) \end{matrix} \\ ρ_{o}, t \in [0 : 00, 6 : 00) \cup [22 : 00, 24 : 00) \end{matrix}$	ρ (t)	Price of shore power at the tth time-unit
		ρ _p	Price of shore power for peak period
		ρ _s	Price of shore power for standard period
		ρ _o	Price of shore power for off-peak period

As discussed in Section 1, operation of the marine HPS should satisfy a lot of complex constraints, e.g., the power balance, the state-of-charge (SOC) of batteries, the PV output constraints, the diesel output constraints, the battery charging/dis-charging constraint and so on. In addition, for protecting the surrounding environment, most of the ports have issued the regulations to restrict the use of onboard diesel generator during the berthing period. For example, the port regulation always requires that the ships should reduce the use of onboard diesel by a certain proportion compared to a historical baseline [2]. In this study, the constraints employed in the evaluated marine HPS are listed in Table 2.

Table 2

Complex constraints and their parameters of the evaluated marine HPS

Constraints	Equations	Parameters	Parameter meanings
SOC boundary	SOC_l ⩽ SOC (t) ⩽ SOC_h	SOC_h, SOC_l	Upper and lower bounds of SOC
		SOC(t)	SOC of battery bank at the tth time-unit
Diesel output	P_d (t) = P₄ (t) + P₅ (t) ⩽ P_r	P_d (t)	Diesel output at the tth time-unit
		P _r	The rated output power of diesel
SOC terminate state	SOC (N) ⩾ SOC (0)	SOC(0)	The initial SOC of battery bank
		SOC(N)	The final SOC of battery bank
Port’s regulation	$\sum_{t = 1}^{N} [P_{4} (t) + P_{5} (t)] ⩽ η \cdot P_{base}$	η	A proportion determined by the port’s regulation
		P _base	Historical baseline of ship hoteling load
PV output	P₁ (t) + P₂ (t) ⩽ P_mppt (t)	P_mppt (t)	Maximum output power of PV array at the tth time-unit
Battery charging/dis-charging	$\begin{matrix} P_{1} (t) \cdot P_{3} (t) + P_{4} (t) \cdot P_{3} (t) + \\ P_{6} (t) \cdot P_{3} (t) + P_{1} (t) \cdot P_{4} (t) + \\ P_{1} (t) \cdot P_{6} (t) + P_{4} (t) \cdot P_{6} (t) = 0 \end{matrix}$	/	/
Power boundary	0 ⩽ P_i (t) ⩽ P_i-h (i = 1, 2, . . . , 7)	P _i-h	Upper bound of the power flow P_i

In this study, the sequences of all the power flows (i.e., P₁ to P₇) during the entire scheduling period (i.e., t = 1, 2, ... , N, where N denotes the length of the scheduling period) are directly employed as the optimization variables. To be specific, the power flows P₁(t) to P₆(t) (t = 1, 2, ... , N) are employed to compose the optimization vector, and the rest power flow P₇(t) is calculated using Equation (1) to ensure the power balance. $P_{7} (t) = L (t) - P_{2} (t) - P_{3} (t) - P_{5} (t)$ (1) where L(t) denotes the electric load demand in the tth time-unit. As discussed above, the optimization vector P is defined as $\begin{matrix} P = [P_{1} (1), . . ., P_{1} (N), P_{2} (1), . . ., P_{2} (N), \\ P_{3} (1), . . ., P_{3} (N), P_{4} (1), . . ., P_{4} (N), \\ P_{5} (1), . . ., P_{5} (N), P_{6} (1), . . ., P_{6} (N)] \end{matrix}$ (2)

In this study, the evolutionary algorithms are employed to optimize the variables listed in Equation (2). As a result, an appropriate coding scheme is required to connect the real power flows in Equation (2) and the variables to be evolved in the evolutionary algorithm. The coding scheme is defined as follows:

Define the evolving variables $x_{i}^{j}$ (i = 1, 2, .... , 6; j = 1, 2, ... , N) to represent the corresponding power flows in Equation (2), i.e., P_i (j), (i = 1, 2, ... , 6; j = 1, 2, ... , N);

All the evolving variables are bounded within the interval [0, 100];

The coding relationships between the optimization variable P_i (j) and the corresponding evolving variable $x_{i}^{j}$ are listed in Table 3.

Table 3

Coding relationships between the optimization variables and evolving variables

Optimization variable	Coding relationship	Parameter	Parameter meaning
P₁(t)	$P_{1} (t) = {\begin{matrix} 0, if x_{1}^{t} \in [0, 50) \\ P_{mppt} (t), if x_{1}^{t} \in [50, 100] and P_{mppt} (t) < P_{br} \\ P_{br}, if x_{1}^{t} \in [50, 100] and P_{mppt} (t) ⩾ P_{br} \end{matrix}$	P_mppt (t)	Maximum output power of PV array at the tth time-unit
		P _br	The rated charging power of battery bank
P₂ (t)	$P_{2} (t) = {\begin{matrix} 0, if x_{2}^{t} \in [0, 50) \\ P_{mppt} (t), if x_{2}^{t} \in [50, 100] and P_{mppt} (t) < L (t) \\ L (t), if x_{2}^{t} \in [50, 100] and P_{mppt} (t) ⩾ L (t) \end{matrix}$	P_mppt(t)	Maximum output power of PV array at the tth time-unit
		L(t)	The electric load demand in the tth time-unit
P₃(t)	$P_{3} (t) = {\begin{matrix} 0, if x_{3}^{t} \in [0, 50) \\ P_{bd}, if x_{3}^{t} \in [50, 100] and P_{bd} < L (t) \\ L (t), if x_{3}^{t} \in [50, 100] and P_{bd} ⩾ L (t) \end{matrix}$	P _bd	The maximum discharging power of battery bank
		L(t)	The electric load demand in the tth time-unit
P₄(t)	$P_{4} (t) = {\begin{matrix} 0, if x_{4}^{t} \in [0, 50) \\ P_{br}, if x_{4}^{t} \in [50, 100] \end{matrix}$	P _br	The rated charging power of battery bank
P₅ (t)	$P_{5} (t) = x_{5}^{t} and x_{5}^{t} \in [0, P_{r}]$	P _r	The rated output power of diesel
P₆(t)	$P_{6} (t) = {\begin{matrix} 0, if x_{6}^{t} \in [0, 50) \\ P_{br}, if x_{6}^{t} \in [50, 100] \end{matrix}$	P _br	The rated charging power of battery bank

In this study, the total cost of marine HPS is employed as the minimization goal of the model. The total cost of marine HPS includes the following three parts: 1) cost of fuel oil consumption for running diesel generator (denote as C_f); 2) the cost paid for shore power service (denote as C_s); 3) the battery degradation cost (denote as C_b). In addition, in order to ensure the satisfaction of all the constraints listed in Table 2, a penalty item (denote as C_p) is added to the total cost. That is to say, the penalty item C_p is equal to 0 only when the solution satisfies all the constraints. Otherwise, the penalty item C_p will be equal to a large value. The penalty item is discussed in detail in Section 4.

As discussed above, the overall fitness function employed in this study is defined as Equation (3-4). $min C_{t} (\vec{P}) = C_{f} + C_{s} + C_{b} + C_{p}$ (3) ${\begin{matrix} C_{f} = \sum_{t = 1}^{N} \frac{P_{f} \cdot {n_{a} \cdot P_{r} + n_{b} \cdot [P_{4} (t) + P_{5} (t)]}}{P_{4} (t) + P_{5} (t)} \\ C_{s} = \sum_{t = 1}^{N} ρ (t) \cdot [P_{6} (t) + P_{7} (t)] \\ C_{b} = \sum_{i = 1}^{K} N_{eq - i} \cdot \frac{M_{0}}{N (D_{eq - i} = 1)} \\ C_{p} = \sum_{t = 1}^{N} [P_{SOC - l} (t) + P_{SOC - h} (t) + P_{diesel} (t) + P_{battery} (t) + P_{boundary} (t) + P_{pv} (t)] + P_{reg} + P_{SOC - t} \end{matrix}$ (4) where K denotes the number of charging and discharging cycles of battery bank counted by the rainflow method [26-28].

3 A novel optimization algorithm

As discussed in Section 2, optimal operation of the evaluated marine HPS is modelled as a constrained optimization problem. As shown in Equation (2), dimensionality of the aforementioned optimization problem is equal to 6 · N, where N denotes the length of scheduling period. For example, for a berthing plan with 50 hours and the time-unit is 15 minutes, the model dimensionality is 6 × 50 × 4 = 1200. Obviously, this ultra-high dimensional problem is too complex to be optimized by using traditional optimization algorithms. In order to overcome this problem, a novel evolutionary algorithm called the adaptive multi-context cooperatively coevolving differential evolution algorithm with random topology and mutated context vector (AMCCDE_{- rt - mcv}) is developed.

3.1 Cooperatively coevolving

With the development of artificial intelligence techniques, some advanced intelligent algorithms are employed to solve the real-world engineering problems [29, 30]. The cooperatively coevolving (CC) inspired by the “divide and conquer” philosophy, is developed for optimizing the high-dimensional problems [31, 32]. To be specific, CC decomposes the D-dimensional problem into several sub-problems and each with a certain number of optimization variables. Then, all the sub-problems are evolved in turn in each evolutionary cycle. As the individual within each sub-problem only represents several dimensionalities of the original problem, it is impossible to compute its fitness function value without a context. For this purpose, a D-dimensional context vector (CV) is defined to provide references of other dimensionalities.

Principle of the basic CC framework can be illustrated as the following steps:

For the original D-dimensional problem Problem₀, initialize the D-dimensional population with N_P individuals. Each individual represents an optimization vector listed in Equation (2). Then, decompose the original problem Problem₀ into K sub-problems Sub - Problem_i (i = 1, 2, ... , K), i.e., Problem₀= [Sub - Problem₁, Sub - Problem₂, ... , Sub - Problem_K]. Note that the dimensionality of each sub-problem is equal to D/K. For a D-dimensional individual x , x = ( x ¹, x ², ... , x ^K), where x ⁱ represents the corresponding optimization variables (i.e., dimensionalities) belongs to the ith sub-problem Sub - Problem_i.

Define the global best individual y as the context vector. Then, the ith sub-problem (i.e., Sub - Problem_i) in CC is defined as $min f^{i} (x, y), x \in R^{S}$ (5) where fⁱ (x, y) = f (y¹, … y^i-1, x, yⁱ⁺¹, …, y^k), f represents the fitness function; R^S represents the solution space.

Start an evolutionary circle. All the sub-problems are optimized using an evolutionary algorithm in turn. Note that the context vector is updated using the global best individual in every iteration.

Proceed another cycle if the stopping criteria are not satisfied; Otherwise, stop the cooperative co-evolution process and output the current global best individual.

3.2 AMCCDE_{- rt - mcv} algorithm

In our developed AMCCDE_{- rt - mcv} algorithm, the CC mechanism discussed in Section 3.1 is integrated with the classic differential evolution algorithm (DE). In addition, the adaptive multi-context mechanism developed in our previous work is also introduced [33]. In order to further improve the performance of AMCCDE_{- rt - mcv} on optimizing the ultra-high dimensional operation model as established in Section 2, the random topology and mutated context vector mechanisms are proposed and integrated with DE.

Random topology of sub-populations

In AMCCDE_{- rt - mcv}, all the N_P individuals within the original population Population₀ are divided into N_sp sub-populations (i.e., Sub - population₁ to Sub - population_Nsp). Obviously, each sub-population has N_p/N_sp individuals. For example, all the 50 (i.e., N_P= 50) individuals within the original Population₀ are numbered Individ₁ to Individ₅₀. Then, the sub-populations Sub - population₁ to Sub - population₅ (i.e., N_SP= 5) is established as Equation (6). ${\begin{matrix} Sub - {population}_{1} = [{Individ}_{1}, {Individ}_{2}, . . ., {Individ}_{10}] \\ Sub - {population}_{2} = [{Individ}_{11}, {Individ}_{12}, . . ., {Individ}_{20}] \\ Sub - {population}_{3} = [{Individ}_{21}, {Individ}_{22}, . . ., {Individ}_{30}] \\ Sub - {population}_{4} = [{Individ}_{31}, {Individ}_{32}, . . ., {Individ}_{40}] \\ Sub - {population}_{5} = [{Individ}_{41}, {Individ}_{42}, . . ., {Individ}_{50}] \end{matrix}$ (6)

Denote the best individual within the jth sub-population Sub - population_j as Sub - best_j (j = 1, 2, ... , N_sp). Then, when evolve each individual within Sub - population_j, the context vector is randomly selected from Sub - best_j and all the $Sub - {best}_{j}^{topology}$ , where $Sub - {best}_{j}^{topology}$ denotes the neighbour sub-populations of Sub - population_j under a certain topology. In random topology of AMCCDE_{- rt - mcv}, all the sub-populations are randomly connected to form the unidirectional neighbour-relationships. To be specific, for the jth Sub - population_j, randomly generate a variable Nei_j ∈ [1, N_sp - 1]. Then, randomly select Nei_j sub-populations from all the sub-populations except Sub - population_j itself, and employ the selected Nei_j sub-populations as the neighbour sub-populations of Sub - population_j. Note that the neighbour-relationships are randomly re-generated at the beginning of each iteration.

In addition, in order to keep the population diversity, the unidirectional topology strategy is employed. That is to say, in the ith iteration, if the kth Sub - population_k is selected by the jth Sub - population_j, check whether the jth Sub - population_j is selected by the kth Sub - population_k in this iteration. If so, eliminate the kth Sub - population_k from the Nei_j sub-populations selected by Sub - population_j in the ith iteration. Then, randomly generate the context vector form the rest selected sub-populations of Sub - population_j. If the selected sub-populations of Sub - population_j is empty, then randomly select an individual from the entire population as the context vector for Sub - population_j in the ith iteration.

The random topology in AMCCDE_{- rt - mcv} is illustrated as Fig. 2. Note that in random topology, each sub-population has at least one and at most N_sp - 1 neighbour sub-populations. For example, Sub - population₃ in Fig.2 has 2 sub-populations (i.e., Sub - population₂ and Sub - population₄). As a result, for each individual in Sub - population₃, the employed context vector is randomly selected from Sub - best₂, Sub - best₄, and Sub - best₃. For the Sub - population₅ in Fig.2, it has only one sub-population (i.e., Sub - population_Nsp). So the context vector for Sub - population₅ will be randomly selected from Sub - best_Nsp and Sub - best₅.

Fig. 2

Schematic of random topology in AMCCDE_{- rt - mcv}.

Mutated context vector mechanism

As the energy storage unit, the battery bank plays an important role in marine HPS. In order to protect the batteries, the battery bank is prohibited to be charged and discharged at the same time. In addition, at most one power resource is allowed to charge the battery at each time. In another word, at most one variable in P₁ (t), P₃ (t), P₄ (t) and P₆ (t) is allowed to be greater than 0 at each time. In the mutated context vector mechanism, the best individual within each sub-population (i.e., Sub - best_j, j = 1, 2, ... , N_sp) is mutated to Sub - best - mut_j, j = 1, 2, ... , N_sp, in which the component [P₁ (t) , P₃ (t) , P₄ (t) , P₆ (t)] (t = 1, 2, ... , N) is randomly mutated to a “zero-vector” or “one-hot” vector. The developed mutated context vector mechanism is illustrated as the following steps:

Step 1: In every T_mut iterations, execute the mutated context vector mechanism for one time. In the mutated context vector mechanism, for each Sub - best_j (j = 1, 2, ... , N_sp), randomly mutate the component [P₁ (t) , P₃ (t) , P₄ (t) , P₆ (t)] (t = 1, 2, ... , N) within Sub - best_j to [r_0-p1, r_0-p3, r_0-p4, r_0-p6], [r_0-p1, r_0-p3, r_0-p4, r_1-p6], [r_0-p1, r_0-p3, r_1-p4, r_0-p6], [r_0-p1, r_1-p3, r_0-p4, r_0-p6] and [r_1-p1, r_0-p3, r_0-p4, r_0-p6], in which r_0-p1, r_0-p3, r_0-p4, r_0-p6 are randomly generated within [0, 50), and r_1-p1, r_1-p3, r_1-p4, r_1-p6 are randomly generated within 50, 100]. Denote the mutated vector for Sub - best_j (j = 1, 2, ... , N_sp) as Sub - best - mut_j (j = 1, 2, ... , N_sp), respectively.

Step 2: If the fitness function value of Sub - best - mut_j is better than that of Sub - best_j, then replace Sub - best_j by Sub - best - mut_j.

Step 3: For each Sub - best_j (j = 1, 2, ... , N_sp), randomly select another Sub - best_k (k = 1, 2, j-1, j + 1 ... , N_sp). Then, randomly exchange the component related to a certain power flow, i.e., [P₁ (t)] , [P₂ (t)] , [P₃ (t)] , [P₄ (t)] , [P₅ (t)] or [P₆ (t)] (t = 1, 2, ... , N) of Sub - best_j and Sub - best_k. Denote the mutated vector for Sub - best_j after exchanging as Sub - best - exc_j.

Step 4: If the fitness function value of Sub - best - exc_j is better than that of Sub - best_j, then replace Sub - best_j by Sub - best - exc_j.

Flow of AMCCDE_{- rt - mcv}

By introducing the aforementioned CC mechanism, adaptive multi-context mechanism, random topology and mutated context vector mechanisms into the classic differential evolution algorithm, the developed AMCCDE_{- rt - mcv} algorithm can be illustrated as Table 4. Note that in the real engineering application, the developed model and optimization algorithm work as the strategy layer, and the output of AMCCDE_{- rt - mcv} transmits to each power control device to obtain the power flow control. Schematic of the AMCCDE_{- rt - mcv} based power flow optimal control in real application is illustrated as Fig. 3.

Table 4

Pseudo code of AMCCDE_{- rt - mcv}

Algorithm: AMCCDE_{- rt - mcv}
Initialize the D-dimensional Popolation₀ with N_P individuals, then divide Popolation₀ into N_sp sub-populations.
repeat
Generate the neighbour-relationships using the random
and unidirectional topology strategy.
Compute and update the best individual for each
sub-population (Sub - best_j).
Decompose the original D-dimensional optimization
vector into several subcomponents using random-grouping
method [33]. Denote the jth subcomponent as P_j.
foreach subcomponentP_jdo
Coevolve the corresponding dimensions using
AMCCDE [32].
end
ifreach integral multiple ofT_mutiterationsthen
foreachSub - best_jdo
Mutate each Sub - best_j to Sub - best - mut_j.
ifC_t (Sub - best - mut_j)< C_t (Sub - best_j) then
Update Sub - best_j using Sub - best - mut_j.
Mutate each Sub - best_j to Sub - best - exc_j.
ifC_t (Sub - best - exc_j)< C_t (Sub - best_j) then
Update Sub - best_j using Sub - best - exc_j.
end
until the stopping criteria are satisfied.

Fig. 3

Schematic of AMCCDE_{- rt - mcv} based power flow optimal control.

4 Detailed analysis for penalty functions

As discussed in Section 2, operation of the marine HPS should satisfy a lot of complex constraints, for example, the power balance constraint, SOC constraint of battery bank, PV and diesel output constraints, battery charging/discharging constraint, power boundary constraint and some other constraints from the port’s regulations. Influenced by these constraints and together with the ultra-high model dimensionality, to effectively optimize the marine HPS operation model becomes extremely difficult. As discussed in Section 1, the evolutionary algorithms have been employed in the power-flow scheduling of marine HPS. According to the results of existing studies, the definition of penalty functions significantly affects the final optimization performance of evolutionary algorithms. However, the efficient definition of penalty functions for satisfying the increasingly complex constraints is rarely addressed.

4.1 Penalty function: satisfaction for complex constraints

In this study, the penalty functions are developed and analysed in detailed to effectively integrate all the constraints to the operation model. The penalty functions discussed in this study includes the penalty function for SOC boundary constraint, penalty function for diesel output constraint, penalty function for port’s regulation constraint, penalty function for PV output constraint and the penalty function for battery charging/dis-charging constraint.

Penalty function for SOC boundary constraint

As discussed in Table 2, SOC of the battery bank should be within a certain interval [SOC_l, SOC_h], in which SOC_l and SOC_h represent the lower and upper bounds of battery SOC, respectively. The constraint of SOC boundary can be formulized as $SOC (t) ⩾ {SOC}_{l}$ (7) $SOC (t) ⩽ {SOC}_{h}$ (8) where SOC(t) denotes the battery SOC in the tth time-unit (t = 1, 2, ... , N).

In order to ensure the satisfaction of Equation (7), the power scheduling solutions which breaks Equation (7) should be punished and the solutions should be guided to increase its charging power flows and decrease the discharging power flows, in order to improve the SOC to satisfy Equation (7). For example, the SOC(t) (t = t₀) in a solution P is lower than SOC_l, i.e., SOC (t₀)< SOC_l, then the fitness function value of solution P , i.e., C_t ( P ) = C_f ( P ) + C_s ( P ) + C_b ( P ) + C_p ( P ) will have a large value because of the punish item C_p ( P ) to obtain the punishment. Then, the solution P will be guided to decrease the SOC at t = t₀ to satisfy Equation (7). As SOC (t₀) is directly related to the charging and discharging power flows before t = t₀, the charging power flows P₁(t < t₀), P₄(t < t₀) and P₆(t < t₀) should be increased to improve SOC(t = t₀). In addition, the discharging power flows P₃(t < t₀) should be decreased to improve SOC(t = t₀). As a result, the penalty function related to Equation (7) is developed as ${Penalty}_{SOC - l} (t) = {\begin{matrix} \frac{[\sum_{i = 1}^{t} ζ_{P 3} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{3} (i)]^{λ}}{[\sum_{i = 1}^{t} ζ_{P 1} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{1} (i)]^{λ} + [\sum_{i = 1}^{t} ζ_{P 4} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{4} (i)]^{λ} + [\sum_{i = 1}^{t} ζ_{P 6} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{6} (i)]^{λ} + 1} \cdot C_{0} + C_{0} if SOC (t) < {SOC}_{l} \\ 0 else \end{matrix}$ (9) where λ is the introduced exponent parameter; α is the damping factor to decide the active depth of power flows to influence the SOC; C₀ is an positive constant to control the punish strength; ζ_P1 (i), ζ_P3 (i), ζ_P4 (i) and, ζ_P6 (i) represents the active index of the power flows P₁(i), P₃(i), P₄(i) and P₆(i), respectively. Equations of ζ_P1 (i), ζ_P3 (i), ζ_P4 (i) and, ζ_P6 (i) are defined as follows $ζ_{Pj} (i) = \frac{P_{j} (i)}{\sum_{p = 1}^{6} P_{p} (i) + 1} (j = 1, 3, 4, 6)$ (10)

Similarly, when the SOC(t) (t = t₀) in a solution P is higher than SOC_h, i.e., SOC (t₀)> SOC_h, the charging power flows P₁(t < t₀), P₄(t < t₀) and P₆(t < t₀) should be decreased and the discharging power flows P₃(t < t₀) should be increased to reduce SOC(t = t₀) and satisfy Equation (8). As a result, the punish function for Equation (8) is developed as ${Penalty}_{SOC - l} (t) = {\begin{matrix} \frac{[\sum_{i = 1}^{t} ζ_{P 1} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{1} (i)]^{λ} + [\sum_{i = 1}^{t} ζ_{P 4} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{4} (i)]^{λ} + [\sum_{i = 1}^{t} ζ_{P 6} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{6} (i)]^{λ}}{[\sum_{i = 1}^{t} ζ_{P 3} (i) \cdot (\frac{i^{α}}{\sum_{j = 1}^{t} j^{α}}) \cdot P_{3} (i)]^{λ} + 1} \cdot C_{0} + C_{0} if SOC (t) > {SOC}_{h} \\ 0 else \end{matrix}$ (11)

Penalty function for diesel output constraint

The diesel output constraint describes that the total power flows output by diesel at each time (i.e., P₄(t) and P₅(t), t = 1, 2, ... , N) should be lower than the diesel’s rated power. If the diesel output constraint at t = t₀ is broken, i.e., P₄(t₀)+P₅(t₀)> P_r, the diesel output power flows at t = t₀, i.e., P₄(t₀) and P₅(t₀), should be decreased to satisfy this constraint. As a result, the penalty function related to the diesel output constraint is defined as

${Penalty}_{diesel} (t) = {\begin{matrix} [ζ_{P 4} (t) \cdot P_{4} (t)^{λ} + ζ_{P 5} (t) \cdot P_{5} (t)^{λ}] \cdot C_{0} + C_{0} if P_{4} (t) + P_{5} (t) > P_{r} \\ 0 else \end{matrix}$ (12) where the active indices of P₄(t) and P₅(t) are defined as $ζ_{Pj} (t) = \frac{P_{j} (t)}{\sum_{p = 1}^{6} P_{p} (t) + 1} (j = 4, 5)$ (13)

Penalty function for port’s regulation constraint

The port’s regulation always restricts the use of onboard diesel and encourages the ships to use shore-power to protect the air around port. According to the port’s regulation, the power generated by onboard diesel during the berthing period should be less than a certain proportion of the historical baseline [2]. The port’s regulation constraint can be formulized as $\sum_{t = 1}^{N} [P_{4} (t) + P_{5} (t)] ⩽ η \cdot P_{base}$ (14) where η denotes the proportion decided by the port’s regulation; P_base denotes the baseline of the ship hoteling load demand. Obviously, when the port’s regulation constraint is broken, the diesel output power flows (i.e., P₄ and P₅) during the entire scheduling period should be deceased to satisfy Equation (14). As a result, the penalty function related to the port’s regulation constraint is formulized as ${Penalty}_{port} = {\begin{matrix} {[\sum_{t = 1}^{N} ζ_{P 4} (t) \cdot P_{4} (t)]^{λ} + [\sum_{t = 1}^{N} ζ_{P 5} (t) \cdot P_{5} (t)]^{λ}} \cdot C_{0} + C_{0} if \sum_{t = 1}^{N} [P_{4} (t) + P_{5} (t)] > η \cdot P_{base} \\ 0 esle \end{matrix}$ (15) where N denotes the entire time-unit in the scheduling period; the definitions of active indices of P₄(t) and P₅(t) are the same with Equation (13).

Penalty function for PV output constraint

The PV output constraint restricts that the total power flows output by the PV system at each time (i.e., P₁(t) and P₂(t), t = 1, 2, ... , N) should be lower than the PV’s maximum output power. If the PV output constraint is broken at t = t₀, the power flows P₁(t₀) and P₂(t₀) should be decreased to satisfy this constraint. As a result, the penalty function related to the PV output constraint is defined as

${Penalty}_{PV} (t) = {\begin{matrix} [ζ_{P 1} (t) \cdot P_{1} (t)^{λ} + ζ_{P 2} (t) \cdot P_{2} (t)^{λ}] \cdot C_{0} + C_{0} if P_{1} (t) + P_{2} (t) > P_{mppt} (t) \\ 0 else \end{matrix}$ (16) where P_mppt(t) denotes the real-time maximum output power of onboard PV system; the active indices of P₁(t) and P₂(t) are defined as $ζ_{Pj} (t) = \frac{P_{j} (t)}{\sum_{p = 1}^{6} P_{p} (t) + 1} (j = 1, 2)$ (17)

Penalty function for battery charging/dis-charging constraint

As discussed in Section 3.2, the battery bank is prohibited to be charged and dis-charged at the same time. In addition, at most one power resource is allowed to charge the battery at each time. In another word, at most one power flow in P₁(t), P₃(t), P₄(t) and P₆(t) is allowed to be greater than 0 at each time. The battery charging/dis-charging constraint is formulized as

$P_{1} (t) \cdot P_{3} (t) + P_{4} (t) \cdot P_{3} (t) + P_{6} (t) \cdot P_{3} (t) + P_{1} (t) \cdot P_{4} (t) + P_{1} (t) \cdot P_{6} (t) + P_{4} (t) \cdot P_{6} (t) = 0$ (18)

If the battery charging/dis-charging constraint is broken, the decreasing of all the battery-related power flows is good for the satisfaction of this constraint. As a result, the penalty function for battery charging/dis-charging constraint is developed as $P_{battery} (t) = {\begin{matrix} [ζ_{P 1} (t) \cdot P_{1} (t)^{λ} + ζ_{P 3} (t) \cdot P_{3} (t)^{λ} + ζ_{P 4} (t) \cdot P_{4} (t)^{λ} + ζ_{P 6} (t) \cdot P_{6} (t)^{λ}] \cdot C_{0} + C_{0} if Eq . (18) \neq 0 \\ 0 else \end{matrix}$ (19)

4.2 Numerical analysis of penalty functions

In order to verify the effectiveness of the developed penalty functions, a comprehensive set of numerical experiments are conducted in this section. The detailed parameters settings of the evaluated HPS are listed in Table 5. Profiles of the maximum PV output (P_mppt) and the ship load (L) for 4 days are plotted in Fig. 4. The time-unit is set to 15 minutes. Note that the system parameters as listed in Table 5 directly depend on the detailed specification of the evaluated HPS. However, the model parameters (i.e., the damping factor α, exponent parameter λ and punish strength constant C₀) should be set to appropriate values to ensure good optimization performance. In the following experiments, the optimal values for each introduced model parameter are comprehensively tested.

Table 5
Parameter settings of the evaluated HPS

Parameter Meaning Value Parameter Meaning Value

S_pv Sizing of PV array 210 kW P_r Rated power of on-board diesel 350 kW

S_bb Sizing of battery bank 288 kWh M ₀ Total investment cost of battery bank 6548 $

η _c Battery charging efficiency 85% P_br Rated charging power of battery 60 kW

η _d Battery discharging efficiency 100% ρ _p Shore power price for peak period 0.31538 $/kWh

SOC _l Lower bound of SOC 144 kWh ρ _s Shore power price for standard period 0.15948 $/kWh

SOC _h Upper bound of SOC 288 kWh ρ _o Shore power price for off-peak period 0.06558 $/kWh

η On-board diesel restriction 40% P_f Fuel price 0.78571 $/kWh

SOC(0) Initial SOC 200 kWh P_bd Maximum discharging power of battery 100 kW

Parameter	Meaning	Value	Parameter	Meaning	Value
S_pv	Sizing of PV array	210 kW	P_r	Rated power of on-board diesel	350 kW
S_bb	Sizing of battery bank	288 kWh	M ₀	Total investment cost of battery bank	6548 $
η _c	Battery charging efficiency	85%	P_br	Rated charging power of battery	60 kW
η _d	Battery discharging efficiency	100%	ρ _p	Shore power price for peak period	0.31538 $/kWh
SOC _l	Lower bound of SOC	144 kWh	ρ _s	Shore power price for standard period	0.15948 $/kWh
SOC _h	Upper bound of SOC	288 kWh	ρ _o	Shore power price for off-peak period	0.06558 $/kWh
η	On-board diesel restriction	40%	P_f	Fuel price	0.78571 $/kWh
SOC(0)	Initial SOC	200 kWh	P_bd	Maximum discharging power of battery	100 kW

Fig. 4

Profiles of the maximum PV output and ship load.

(1) Analysis of α: the damping factor in penalty function

As the SOC boundary constraint is closely related to two-thirds of the optimization variables, the definition of penalty function related to this constraint significantly influences the final optimization performance. In the penalty functions listed in Equations (11), the damping factor α decides the active depth of power flows to influence the SOC. The following numerical experiments are conducted to verify the optimal value of damping factor α.

In the following experiments, parameters settings of the evaluated marine HPS are listed in Table 5. Assume the ship arrival time is 11 : 00, July 2, and its departure time is 13 : 00, July 4. Obviously, the entire scheduling period is 50 hours, that is to say, the total number of time-units within the scheduling period (N) is equal to 50 hours/15 minutes = 200.

The AMCCDE_{- rt - mcv} developed in Section 3 is employed as the optimization algorithm. Parameters setting of AMCCDE_{- rt - mcv} is as follows: The population size N_p is set to 50; The maximum number of fitness evaluations is set to Max _ FES = 5 ×10⁶; The dynamic group size for random grouping is set to S = {5, 10, 20, 30, 50}; The number of sub-populations is set to N_SP= 5; The iteration number to mutate context vectors is set to T_mut= 10; The punish strength constant C₀ is set to 1000; The exponent parameter λ is set to 2.

For each α value, the damping curves for the charging/dis-charging power flows to influence the battery SOC are plotted in Fig. 5 (t = 50% N=100). The detailed effectiveness factors for different α values are compared in Table 6 (t = 50% N=100). The SOC variations for each α value are plotted in Fig. 6. The final results (i.e., C_t in Equation (3)) for different α values are compared in Table 7.

Fig. 5

Damping curves for each α value (t = 50% N=100).

Fig. 6

SOC variations for each α value.

Table 6

Detailed effectiveness factors for different α values (t = 50%, N = 100)

α	Effectiveness factors for different depth (time-unit)
	100th	99th	98th	97th	96th	95th	94th	93th	80th	50th	10th
0.1	0.0110	0.0110	0.0109	0.0109	0.0109	0.0109	0.0109	0.0109	0.0107	0.0102	0.0087
0.5	0.0149	0.0148	0.0147	0.0147	0.0146	0.0145	0.0144	0.0144	0.0133	0.0105	0.0047
1.0	0.0198	0.0196	0.0194	0.0192	0.0190	0.0188	0.0186	0.0184	0.0158	0.0099	0.0020
5.0	0.0582	0.0554	0.0526	0.0500	0.0475	0.0451	0.0427	0.0405	0.0191	0.0018	0.0000
10.0	0.1042	0.0942	0.0851	0.0768	0.0693	0.0624	0.0561	0.0504	0.0112	0.0001	0.0000
30.0	0.2666	0.1972	0.1454	0.1069	0.0783	0.0572	0.0417	0.0302	0.0003	0.0000	0.0000
100.0	0.6361	0.2328	0.0844	0.0302	0.0107	0.0038	0.0013	0.0004	0.0000	0.0000	0.0000
150.0	0.7795	0.1726	0.0376	0.0081	0.0017	0.0004	0.0001	0.0000	0.0000	0.0000	0.0000

Table 7

Comparisons for different values of damping factor α

α	Final optimization result ($)	α	Final optimization result ($)
0.1	29069.1132	10.0	574.1174
0.5	10613.9803	30.0	598.4840
1.0	7345.2198	100.0	32012.5638
5.0	6004.8092	150.0	40942.9601

As shown in Fig. 5 and Table 6, when the penalty functions are given a small damping factor value (e.g., α = 0.1 or α = 0.5), the power flows from the 1st time-unit to the current time-unit (i.e., t = 100th time-unit) are considered to influence the current SOC(t) to the similar degrees. As a result, when the SOC boundary constraint at t = 100th time-unit is broken, the charging and dis-charging power flows P₁(t), P₃(t), P₄(t) and P₆(t) (t = 1, 2, ... , 100) are all adjusted to ensure the satisfaction of the constraint at t = 100 time-unit. However, the effectiveness factors in the long-ago time-units decrease significantly with the increasing of α. That is to say, when the penalty functions are given a large damping factor value, the charging and dis-charging power flows in the long-ago time-units are almost ignored. Therefore, the satisfaction of SOC boundary constraint at t = 100th time-unit is mainly obtained by adjusting the power flows in recent time-unit, for example, P₁(t), P₃(t), P₄(t) and P₆(t) (t = 90, 91, ... , 100).

As shown in Table 7, for the small damping factor values (e.g., α = 0.1 or α = 0.5), the obtained total costs are very large (29069.1132 $ for α = 0.1 and 10613.9803 for α = 0.5). That is because some constraints in the final scheduling solution are not satisfied. The SOC variations for each α value are plotted in Fig. 6. It can be seen from Fig. 6(a) and (b) that the SOC boundary constraints at t = 19th, 80th and 93th time-units (for α = 0.1) and at t = 19th and 131th (for α = 0.5) time-units are broken. The reason behind this phenomenon is that when a small α value is employed, the charging and dis-charging power flows in the long-ago time-units (e.g., 80 time-units ago) are employed to adjust the current SOC. Thus, effect of the power flows in recent time-units (e.g., 1 or 2 time-units ago) is significantly reduced.

Similarly, as shown in Table 7, the large damping factor values (e.g., α = 100 or α = 150) also obtain poor performance (32012.5638 $ for α = 100 and 40942.9601 for α = 150). As shown in Fig. 6 (g) and (h), the SOC boundary constraints at t = 23th, 77th and 101th time-units (for α = 100) and at t = 17th, 131th and 133th (for α = 150) time-units are broken. The reason is that some useful charging and dis-charging power flows (e.g., 3 time-units ago) are ignored and only the power flows in 1 or 2 time-units ago are employed to adjust the current SOC.

As shown in Table 7, the appropriate values of α are good for the satisfaction of SOC constraints. For example, the total costs for α = 1 . 0 and α = 5 . 0 are much better than that of α = 0 . 1. That is because the number of broken constraints is decreased in the final solution. In addition, the settings of α = 10 and α = 30 obtain the best performance (574.1174 $ for α = 10 and 598.4840 for α = 30). As shown in Fig. 6(e) and (f), all the SOC boundary constraints for α = 1 . 0 and α = 5 . 0 are strictly satisfied.

The convergence graphs for each α value are compared in Fig. 7. As shown in Fig. 7, the convergence speeds for α = 0 . 1, 0 . 5, 1 . 0 and5 . 0 are almost the same. But the final performance for α = 1 . 0 and α = 5 . 0 is much better than that of α = 0 . 1 and α = 0 . 5. For the large α values, the convergence speeds for α = 100 and α = 150 are slightly faster than that of α = 10 and α = 30 at the beginning of evolution. However, the continuous evolution ability in the middle and late evolution stages for α = 10 and α = 30 are better, and the final performance for α = 10 and α = 30 is much better than that of α = 100 and α = 150.

Fig. 7

Convergence graphs for different values of damping factor α.

(2) Analysis of λ: the exponent parameter in penalty function

The exponent parameter λ exists in the penalty functions for SOC boundary constraint (Equation (9) and Equation (11)), diesel output constraint (Equation (12)), PV output constraint (Equation (16)) and the battery charging/dis-charging constraint (Equation (19)). In order to obtain the most efficient penalty function form, the exponent parameter λ is tested in detail by numerical experiments. In the following experiments, assume the ship arrival time is 05 : 00, July 2, and its departure time is 15 : 30, July 3. Obviously, the entire scheduling period is 36 hours, that is to say, the total number of time-units within scheduling period (N) is equal to 28 hours/15 minutes = 144. The damping factor α is set to 10. The other parameter settings are the same with the above experiments. The final results for different settings of λ are compared in Table 8. The convergence graphs are plotted in Fig. 8.

Table 8

Comparisons for different values of the exponent parameter λ

λ	Final optimization result ($)	λ	Final optimization result ($)	λ	Final optimization result ($)
0.1	27459.8961	1	552.2450	5	229591.9092
0.5	6309.5902	2	431.5624	10	13125325.9021

Fig. 8

Convergence graphs for different values of the exponent parameter λ.

As shown in Table 8, the performance on final optimization result is poor when λ < 1. For example, the final optimization results obtained by λ = 0.1 and λ = 0.5 are 27459.8961 and 6309.5902 respectively, which are significantly greater than the normal values. Obviously, some of the constraints are broken when λ is set to 0.1 and 0.5. However, when the values of λ are set too large (e.g., λ = 5 and λ = 10), the obtained results are also unacceptable because of the broken constraints. That is because the penalty of constraints is overly amplified or narrowed by these improper λ values.

As shown in the second column of Table 8, the total costs obtained by λ = 1 and λ = 2 are 552.2450 $ and 431.5624 $, respectively. Obviously, all the constraints are strictly satisfied in these scheduling solutions.

(3) Analysis of C₀: the punish strength constant

In all the developed penalty functions, a positive constant C₀ is introduced to control the punish strength of each constraint. The value of C₀ also directly affects the final optimization performance. As a result, different values of C₀ are compared by numerical experiments. In the following experiments, assume the ship arrival time is 03 : 30, July 2, and its departure time is 17 : 00, July 3. Obviously, the entire scheduling period is 37.5 hours, that is to say, the total number of time-units within the scheduling period (N) is equal to 37.5 hours/15 minutes = 150. The damping factor α is set to 10. The exponent parameter λ is set to 2. The other parameter settings are the same with the above experiments. The final optimization results for different values of C₀ are compared in Table 9, in which the “broken” and “satisfied” in the brackets indicate some of the constraints are broken and all the constraints are satisfied, respectively.

Table 9

Comparisons for different values of the punish strength constant C₀

C ₀	Final optimization result ($)	C ₀	Final optimization result ($)	C ₀	Final optimization result ($)
10^-2	283.9716 (broken)	10³	408.7048 (satisfied)	10⁵	13755890.7821 (broken)
10^-1	415.1827 (broken)	10⁴	438.0349 (satisfied)	10⁷	169199037.8637 (broken)

It can be seen from Table 9 that the appropriate value of punish strength constant C₀ is between 10³ to 10⁴. When the value of C₀ is set too small (e.g., C₀ = 10^-1 or 10^-2), the penalty imposed on the model is not strength enough to satisfy the broken constraints. As shown in Table 9, when C₀ is set to 0.1 and 0.01, the algorithm loses its efficacy and some of the constraints are broken. Note that because the employed C₀ _< 1, the penalty item is narrowed by multiplying C₀. As a result, the final optimization results for C₀ = 10^-1 and 10^-2 in Table 9 seem very small. The variations of SOC for C₀ = 10^-1 and 10^-2 are plotted in Fig. 9. As shown in Fig. 9, for C₀ = 10^-1, the SOC boundary constraint in at least 5 time-units are broken. For C₀ = 10^-2, the performance is even worse: the SOC boundary constraint in the 46th time-unit is broken, and the battery is thoroughly not used after this time-unit.

Fig. 9

SOC variations for C₀ = 10^-1 and C₀ = 10^-2.

In addition, when the value of C₀ is set too large (e.g., C₀ = 10⁵, 10⁸ or 10¹⁰), the optimization performance is unacceptable because the magnitude of the imposed penalty is much greater than the fitness function itself. As a result, it is too difficult for the individuals to avoid all the penalties and satisfy all the constraints. The convergence graphs for different C₀ values are plotted in Fig. 10.

Fig. 10

Convergence graphs for different values of the punish strength constant C₀.

5 Case studies

In order to verify the overall performance of the developed methodology, the aforementioned AMCCDE_{- rt - mcv} algorithm and the optimal operation model are tested by a comprehensive set of case studies.

5.1 Parameter settings

In the following experiments, parameter settings of the evaluated marine HPS are the same with Section 4.2. According to the experimental result provided in Section 4, the optimal values of penalty function parameters (i.e., the damping factor α, exponent parameter λ and punish strength constant C₀) are set to α = 10, λ = 2 and C₀ = 1000, respectively. Profiles of the maximum PV output (P_mppt) and the ship load (L) for 5 days are plotted in Fig. 11. The time-unit is set to 30 minutes.

Fig. 11

Profiles of P_mppt and L employed in the following case studies.

The ship berthing plans employed in the following experiments are listed in Table 10. Note that the last column of Table 10 denotes the dimensionality of each problem. Obviously, the complexity constantly increases from case 1 to case 4 due to the growing dimensionality.

Table 10

Berthing plans employed for the case studies

Case number	Arrival time	Departure time	Length of berthing time	Problem dimensionality
Case 1	01 : 30, April 3	23 : 30, April 3	22 hours	264
Case 2	17 : 30, May 15	12 : 00, May 17	42.5 hours	510
Case 3	12 : 00, September 1	13 : 00, September 4	73 hours	876
Case 4	15 : 30, October 18	23 : 00, October 22	103.5 hours	1242

5.2 Case study for AMCCDE_{- rt - mcv}

In this section, the developed AMCCDE_{- rt - mcv} is employed to solve the optimal operation problems for the cases listed in Table 10. The final optimization results obtained by AMCCDE_{- rt - mcv} are listed in Table 11.

Table 11
Final optimization result obtained by AMCCDE_{- rt - mcv} for each case

Case number Final optimization result ($) Case number Final optimization result ($)

Case 1 131.6025 Case 3 486.0509

Case 2 234.1878 Case 4 601.6413

Case number	Final optimization result ($)	Case number	Final optimization result ($)
Case 1	131.6025	Case 3	486.0509
Case 2	234.1878	Case 4	601.6413

As shown in Table 11, the developed AMCCDE_{- rt - mcv} algorithm can effectively optimize the scheduling problem of the evaluated marine HPS, and all the constraints are strictly satisfied. For example, in Case 1, the dimensionality of the model is 264, the obtained final result is 131.6025 $. In Cases 2 and 3, the model dimensionalities increase to 510 and 876, the final optimization results obtained by AMCCDE_{- rt - mcv} are 234.1878 $ and 486.0509 $, respectively. In Case 4, the model dimensionality is even larger than 1200, the AMCCDE_{- rt - mcv} still performs well and obtains the final result of 601.6413 $.

The detailed optimal power flows scheduled by AMCCDE_{- rt - mcv} are plotted in Fig. 12. As shown in the figures, the input and output power flows of the PV system, battery bank, onboard diesel, shore power system and the ship load demand are adequately scheduled during the entire scheduling period. It can be also seen from the figures that in the daytime, most of the PV output is used for feeding the ship load (i.e., the power flow P₂), and the surplus PV power is used to charge the batteries (P₁). However, when the PV output is deficient for feeding the ship load demand, the onboard diesel (P₅), the shore power system (P₇) and even the battery (P₃) are activated to supply power for the ship load.

Fig. 12

Optimal power-flows obtained by AMCCDE_{- rt - mcv} for each case.

In addition, as shown in Fig. 12, the battery bank is mostly charged by PV power (P₁) and shore power (P₆), but rarely charged by onborad diesel (P₄). This implies that to charge the battery by running onboard diesel is not efficient, so the AMCCDE_{- rt - mcv} never use P₄ to charge the battery bank.

The detailed variations of SOC for each case are compared in Fig. 13. Obviously, the battery banks are effectively utilized during the scheduling process and all the SOC constraints are strictly satisfied for Case 1 to 4.

Fig. 13

SOC variations of battery bank for each case.

5.3 Comparisons and analysis

In order to further verify the effectiveness of the developed methodology, the AMCCDE_{- rt - mcv} is further compared with some state-of-the-art algorithms, including: CPSO-SK_- rg - aw [34], CCPSO2 [31], and AMCCPSO[33]. The population sizes of all of the compared algorithms are set to 50; The maximum number of fitness evaluations is set to Max_FES = 5×10⁶; The dynamic group size for random grouping is set to S = {5, 10, 20, 30, 50}; The number of sub-populations is set to N_SP = 5; The iteration number to mutate context vectors is set to T_mut = 10; The values of penalty function parameters are set to α = 10, λ = 2 and C₀ = 1000, respectively. The parameters setting for the compared algorithms follows their original studies. In the comparison experiments, 10 independent runs for each algorithm are conducted, and the average optimization results of the algorithms for Case 1 to 4 are compared in Table 12. The average success rate, the best and worst performance of each algorithm for the 10 independent runs are listed in Table 13 in detail.

Table 12
Average optimization results of the compared algorithms for Case 1 to 4

Case number CPSO-SK_- rg - aw CCPSO2 AMCCPSO AMCCDE_{- rt - mcv}

Case 1 3131.4048 152.5741 138.4482 127.2264

Case 2 35250.8978 277.7162 269.8537 239.2550

Case 3 155161.3872 3871.6093 498.0860 501.3845

Case 4 184980.0000 25697.8000 7695.4306 600.4064

Case number	CPSO-SK_- rg - aw	CCPSO2	AMCCPSO	AMCCDE_{- rt - mcv}
Case 1	3131.4048	152.5741	138.4482	127.2264
Case 2	35250.8978	277.7162	269.8537	239.2550
Case 3	155161.3872	3871.6093	498.0860	501.3845
Case 4	184980.0000	25697.8000	7695.4306	600.4064

Table 13

Detailed analysis of compared algorithms for 10 independent runs

Case number	Indicator	CPSO-SK_- rg - aw	CCPSO2	AMCCPSO	AMCCDE_{- rt - mcv}
Case 1	Success rate	90%	100%	100%	100%
	Best	138.6447	146.7832	130.0411	123.3959
	Worst	30009.1283	163.6469	147.5643	131.7639
Case 2	Success rate	10%	100%	100%	100%
	Best	289.9782	268.9852	265.6152	232.2503
	Worst	49875.6092	289.9830	277.9871	243.9561
Case 3	Success rate	0%	80%	100%	100%
	Best	117411.9731	502.2902	490.9839	497.0927
	Worst	201942.9830	17359.7391	505.7832	509.0912
Case 4	Success rate	0%	0%	30%	100%
	Best	167260.2716	20937.1387	689.0976	593.8367
	Worst	209380.9036	31408.8932	14808.0000	606.0382

Note that the solutions which can escape from all the penalty functions are set in bold.

As shown in Tables 12 13, the developed AMCCDE_{- rt - mcv} outperforms its competitors for all the cases. To be specific, for the 264-dimensional model in Case 1, AMCCDE_{- rt - mcv}, AMCCPSO and CCPSO2 can successfully escape punishment from all the penalty functions and obtain the 100% success rate. However, the success rate of CPSO-SK_- rg - aw for Case 1 is just 90%. In addition, the average optimization result obtained by AMCCDE_{- rt - mcv} for Case 1 is 127.2264 $, better than 138.4482 $ obtained by AMCCPSO and 152.5741 $ obtained by CCPSO2 respectively. For the 510-dimensional model in Case 2, the success rate of CPSO-SK_- rg - aw furtherly decreases to 10%, but AMCCDE_{- rt - mcv}, AMCCPSO and CCPSO2 can still keep 100% success rate. Specifically, the average optimization result obtained by AMCCDE_{- rt - mcv} is 239.2550 $, which outperforms AMCCPSO (269.8537 $) and CCPSO2 (277.7162) significantly. In addition, the best and worst performance obtained by AMCCDE_{- rt - mcv} also outperforms its competitors to different degrees.

For Case 3 and Case 4, the model dimensionality furtherly increases to 876 and 1242. The success rates of CPSO-SK_- rg - aw and CCPSO2 for Case 3-4 decrease to 0%, and the success rates of AMCCPSO also decreases to 30%. However, with the integration of random topology and mutated context vector mechanisms, AMCCDE_{- rt - mcv} can keep 100% success rate, and the average optimization results for Case 3-4 are 501.3845 $ and 600.4064 $ respectively, which outperform its competitors significantly.

As discussed above, the advantages of the developed methodology can be summarized as: On one hand, it can effectively solve high-dimensional scheduling problem with more than 1000 dimensionalities. The final optimization results significantly outperform the compared state-of-the-art algorithms, and the robustness on success rate is well. On the other hand, by defining the effective penalty functions, the methodology can incorporate many complex constraints, which is very suitable for the marine HPS application. However, there also exists some drawbacks need to be further improved. For example, the penalty functions are sensitive to the parameters like damping factor, exponent parameter and punish strength constant. As a result, performance of the developed methodology heavily relies on the settings of these parameters.

6 Conclusion

In this study, the optimal power-flow scheduling model and optimization algorithm for marine hybrid power system (HPS) are discussed. In the existing studies, the optimal power-flow scheduling problem is always solved using the traditional linear or nonlinear programming methods, but these methods have high requirements for the detailed form of the scheduling model. In addition, some existing studies use the advanced intelligent methods like particle swarm optimization and its variants to solve this problem, but these methods always lose efficacies for the high-dimensional problem due to the exponentially increasing complexity and complex constraints. In this study, the aforementioned issue is modelled as a high-dimensional optimization problem, and the various constraints are described as a set of effective penalty functions. In addition, a novel optimization algorithm namely AMCCDE_{- rt - mcv} is developed to optimize the developed model. Moreover, the detailed form for each penalty function is analysed and comprehensively verified by a set of numerical simulations. Result of case study shows that the damping factor, exponent parameter and punish strength constant effect the efficiency of penalty functions a lot. The improper settings of these parameters will significantly deteriorate the algorithm performance. The optimal values of these parameters are suggested to be set as: 10 to 30 for the damping factor, 1 to 2 for the exponent parameter and 10³ to 10⁴ for the punish strength constant.

In the real engineering application, the developed methodology can provide as the upper level control strategy for the marine HPS. Output of the methodology can be employed as the control target for the power control equipment between each distributed power source. In the future, more efforts will be made to further improve the efficiency of the operation model and optimization algorithm, especially for the more efficient and robust forms of penalty functions.

7 Conflict of interest

The authors declare that they have no conflict of interests regarding the publication of this paper.

8 Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Footnotes

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (WUT: 2018IVB003).

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Optimal operation for marine hybrid power system: Novel optimization algorithm and comprehensive analysis for penalty functions

Abstract

Keywords

Nomenclature

1 Introduction

1.1 Literature review

1.2 Research gap and motivation

1.3 Contribution and paper organization

2 Marine hybrid power system and its optimal operation model

2.1 Structure of marine hybrid power system

3.1 Cooperatively coevolving

4.1 Penalty function: satisfaction for complex constraints

5.1 Parameter settings

Table 11 Final optimization result obtained by AMCCDE - rt - mcv for each case Case number Final optimization result ($) Case number Final optimization result ($) Case 1 131.6025 Case 3 486.0509 Case 2 234.1878 Case 4 601.6413

7 Conflict of interest

8 Data availability statement

Footnotes

Acknowledgments

References

Table 11
Final optimization result obtained by AMCCDE_{- rt - mcv} for each case

Case number Final optimization result ($) Case number Final optimization result ($)

Case 1 131.6025 Case 3 486.0509

Case 2 234.1878 Case 4 601.6413