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The coming interaction between a growing electrified vehicle fleet and the desired growth in renewable energy provides new insights into the economic dispatch (ED) problem. This paper presents an economic dispatch model that considers electric vehicle charging, battery exchange stations, and wind farms. This ED model is a high-dimensional, non-linear, and stochastic problem and its solution requires powerful methods. A new finite action-set learning automata (FALA)-based approach that has the ability to adapt to a stochastic environment is proposed. The feasibility of the proposed approach is demonstrated in a modified IEEE 30 bus system. It is compared with continuous action-set learning automata and particle swarm optimization-based approaches in terms of convergence characteristics, computational efficiency, and solution quality. Simulation results show that the proposed FALA-based approach was indeed capable of more efficiently obtaining the approximately optimal solution. In addition, by using an optimal dispatch schedule for the interaction between electric vehicle stations and power systems, it is possible to reduce the gap between demand and power generation at different times of the day.

Economic dispatch (ED) is defined as the allocation of generation levels to different electrical generation units, so that the system load may be supplied entirely and most economically. To solve the ED problem, one seeks to find the optimal allocation of the electrical power output from various available generators. Electric vehicle (EV) fleets and renewable energy sources (such as wind power) have brought two new dimensions to this problem, along with the challenges introduced by their uncertainty.

EVs have the potential to be a revolutionary invention for both transportation and the electricity industry [

The integration of EVs and wind turbines into the grid creates new challenges for power system operators. A large number of EVs will introduce significant uncertainty into power systems because of the random nature of their charging behaviors. If the EVs are disorderly when charging, an extra burden will be placed on the power system, which will expose vulnerabilities. However, a fair charging schedule can significantly reduce those negative effects [

From a mathematical point of view, ED is an optimization problem. Various algorithms have been applied to solve classic ED problems. They include mixed integer programming [

In this paper, we present an ED model that includes wind power and a high penetration of EVs, which we solve using a finite action-set learning automata (FALA) method. The organization of this paper is as follows. We describe the modeling of EVs, wind power and load in

We consider the demand/supply of EVs, the output of wind farms, and the regular load as stochastic variables. This is because it is difficult to forecast these variables and they are related to uncertainties such as driver behavior. In this section, we analyze and model all of these stochastic factors.

There are three mainstream interactive modes of EVs: normal interaction (NI), fast charging (FC), and battery exchange (BE). FC mode is detrimental to the battery, and introduces harmonic pollution to power systems. Therefore, this mode is not likely to be widely applied in the future, and our ED model only considers NI mode and BE mode.

The behavior characteristics of EV owners are determined by different uncertain factors (such as travel habit, vehicle type, and interactive mode), so their charging demand tends to be uncertain and difficult to be estimated with precision. Therefore, simplification is essential. In this paper, we use the following conditions and assumptions.

The analyses around EVs are derived from a prototype of the Nissan Leaf [

The energy an EV has consumed is in direct proportion to the distance it has traveled.

The probability distributions of an EV’s arrival time and the distance the EV has traveled are derived from driving pattern data collected in the National Household Transportation Survey (NHTS) [

The peaks of battery exchange demand are during the morning rush hour, lunch time, and afternoon rush hour.

We will now discuss the interactive modes of the EVs separately.

NI EVs are widely distributed in the daytime, making it difficult for them to follow scheduling instructions, so this model only considers night-time dispatch. Suppose some EV owners (proportion τ) sign the user agreement so that their car is connected to the power grid as soon as they finish the last trip in a day. In addition, suppose that they obey the dispatch plan so that the battery will be fully charged at 6:00 a.m. The NI EVs dispatch is a bi-level model: the upper dispatcher gives the plan to the NI stations, and the stations control the charging (or discharging), of the EVs. The stations communicate the condition of the EV to the upper dispatcher, such as unsatisfied demand of battery charging. The dispatch time is between 20:00 and 6:00 the next day, indicated as “NI schedulable period”.

To simplify our model, we assume that each EV charges/discharges in one specified NI station. For the EVs that sign the agreement with the NI station connected with bus k (indicated as “NI station _{k}_{,NI}(_{k}_{,NI}(

The penalty of all NI stations is:

The EVs not signing the agreement are regarded as normal load. They start to charge as soon as finishing the last trip in a day, and continue until the battery is full. When connected to the power facilities, these EVs charge at power _{s}

In this mode, interaction between the EVs and the power system is implemented by battery exchange stations. At a given time t, the state of the BE station connected with bus

In Equations (4) and (5), _{m}_{,BE}(

At a given time, the battery exchange demand is expected to follow a Poisson distribution:

We do not discuss the reactive demand and supply of EVs in this paper, as it is close to zero.

We consider all other types of demand to be regular load and follow a random distribution. Therefore, a single probability distribution function can be used to describe this active/reactive demand at each bus. This distribution can be either derived from measurements or simply assumed to be the normal distribution _{l}_{l}

The output of a wind farm is considered to be in proportion to the third power of the wind speed:
_{w}

The wind speed curve is given in

By using an appropriate scheduling strategy, EVs can mitigate problems caused by the difference between power generation and demand at different times of the day, with minimized operational cost. We also need to meet the demand of EV charging. Thus, the objective function is:

The expressions of _{G}_{M}_{P}

We must minimize the objective function subject to a number of constraints.

_{W}

_{G}

_{loss }

_{= }

_{EV }

_{+ }

_{L}

This constraint ensures that the dispatch schedule satisfies the EV owners’ requirements.

Equation (19) ensures that the battery exchange station will operate sustainably. It is assumed that, to achieve a sustainable operation, the battery in BE stations should be full at the start and end of each day.

We can write the objective function in Equation (11) as:
_{total} = _{obj}_{j}_{,}_{g}_{k}_{,NI}(_{m}_{,BE}(_{i}_{,w}(_{k}_{k}_{,sev}(

We can conclude from

Learning automata (LA) are adaptive decision makers that learn to choose the optimal action from a set of available actions by using noisy reinforcement feedback from their environment [

Some definitions are given as follows: _{obj}_{obj}_{obj}

The goal of FALA is to find _{obj}^{i}_{1}(_{r}_{i}^{i}_{i}_{r}^{i}_{i}

The FALA algorithm ^{r}^{r}_{1}, _{2}, … _{r}}, is the set of output actions of the automaton, and ^{th} iteration, generated based on the action probability vector _{obj}

Our FALA approach is described as follows.

Step 1: The variables are initialized. The decision variable has n-dimensions (_{1}, …, _{n}^{n}^{n}_{n}_{,M}(^{th} iteration, where:
_{i}_{,j}(_{i}^{th} interval at time _{i}

Calculate the initial optimal value in the following way: all the stochastic variables are set to their expected value, and the interactive power between EV facilities is equal to the expected power demand of EVs. Then, calculate the output of the conventional generators using the optimal power flow method. The optimal value (_{total}(0)), _{total}(0)

Step 2: A set of random variables ξ(_{i}_{,w}(_{k}_{k}_{,sev}(

Step 3: A set of control actions _{n}_{,M}(

Step 4: The constraints are checked and the value of the objective function, _{obj}

Step 5: Calculate the response of the environment using:

The response of the environment is 0 when the current control variables are “better” (constraints are satisfied and the total cost is small), the response is 1 when the current control variables are “worse”.

Step 6: Update the probability distributions of the actions. For the intervals that _{i}

For the other intervals, the probabilities are updated using:

In Equations (34) and (35),

Step 7: Update the current optimal value of objective function:

Step 8: Decide if the algorithm should terminate using:
^{i}

Flow chart of the FALA-based approach.

A test system was developed based on the standard IEEE-30 bus system [

As previously mentioned, the number of cars arriving at the NI stations at each time interval and the driving distance of a day has been estimated using the driving pattern data collected in the NHTS [

Our examples assume that the energy consumed by an EV is proportional to the distance it has driven. The predictive probability of the battery exchange demand curve, conventional load demand curve, and predicative wind speed curve are shown in

Other parameters used for these examples are: T = 1 h, C_{w} = 500 Ұ/MWh, C_{NI} = 1250 Ұ/MWh, C_{BE} = 1000 Ұ/MWh, C_{un1} = C_{un2} = C_{un3} = 3000 Ұ/MWh, v_{ci} = 3 m/s, v_{c}_{o} = 22 m/s, P_{R}= 10 MW, P^{2}_{emin} = −10 MW, P^{2}_{emax} = 10 MW, P^{8}_{emin} = −5 MW, P^{8}_{emax} = 5 MW, E_{max} = 24 MWh, U^{i}_{min} = 0.95, and U^{i}_{max} = 1.05. The parameters of the conventional generators are listed in

Driving pattern data: trip length [

Driving pattern data: driving distance [

Power demand and wind farm output prediction.

Parameters of conventional generators.

Bus Number | α
_{i} |
β
_{i} |
γ_{i} (Ұ/MW^{2}) |
||
---|---|---|---|---|---|

1 | 0.124 | 12.4 | 0 | 0 | 80 |

2 | 0.1085 | 10.85 | 0 | 0 | 80 |

22 | 0.3875 | 6.2 | 0 | 0 | 50 |

27 | 0.5171 | 20.15 | 0 | 0 | 55 |

23 | 0.155 | 18.6 | 0 | 0 | 30 |

13 | 0.155 | 18.6 | 0 | 0 | 40 |

The parameters of FALA were set as: ^{24}, _{0} = 0.01. The PSO-based (particle swarm optimization) approach is summarized as follows:

The stochastic problem was transformed into a deterministic problem using CCP (see [

The transformed problem was solved using the PSO algorithm (see Equation (6) in [_{ic}_{1} _{2}

All programs were written using MATPOWER 4.1 [^{®} Core™ i7-2600 3.40 GHz CPU, and 16 GB kst 4g/1333 RAM.

The convergence characteristics of the three algorithms are shown in

The convergence criterion of the FALA algorithm is given in Equation (35). It converged after approximately 150 iterations. In the first 30 iterations, the action probability vector changed rapidly because “_{obj}

The CALA algorithm converged after approximately 270 iterations for this ED problem. The convergence criterion of the CALA algorithm fluctuated more dramatically than that of FALA algorithm, because it had the possibility to increase (opposite to the converging direction) even though the response of the current solution was good enough.

The characteristic curve of the PSO algorithm shows the change of the current global optimal value with each iteration. The value was transformed using:

Convergence characteristics of three algorithms.

The main computation work needed for these algorithms was the power flow calculation. To calculate the objective function of one set of control variables in a deterministic environment, the computer needs to calculate the power flow 24 times. During each iteration, the FALA algorithm needs one objective function calculation; the CALA needs two, while the PSO needs 200 (the Popsize is 20, calculating the response of the environment of one pop using the CCP model requires

Algorithm complexity and computing time.

Algorithm | _{0bj} |
Iteration Times | Total Times of
_{0bj} |
Computing Time(s) |
---|---|---|---|---|

FALA | 1 | 160 | 160 | 239 |

CALA | 2 | 270 | 540 | 815 |

PSO | 200 | 70 | 14,000 | 15,872 |

The environment is stochastic and the goal of the LA algorithms is to optimize the expectation of the environment’s response. Moreover, the LA algorithms do not establish an evaluation system of a solution in the stochastic environment. To compare the results of the three algorithms, each solution was tested in the stochastic environment 30 times (

Performance of algorithm results.

Algorithm | Mean Cost (Ұ) | Mean Diff (MW) | Mean Diff When Charging/Discharging Disorderly (MW) |
---|---|---|---|

FALA | 57,939 | 62.53 | 92.02 |

CALA | 57,925 | 60.48 | 91.62 |

PSO | 57,970 | 63.91 | 94.80 |

The results in

We define

As seen in

Change of comprehensive load.

In this paper, we have developed a stochastic ED model that considers a high penetration of EVs. Both the NI and BE modes are assumed to be schedulable in our model. We have proposed a FALA-based approach to solve the ED problem and compared the result with other two intelligent algorithms. Our results show that the FALA algorithm required less time than the CALA algorithm to reach the optimal solution, while PSO needed much more time than both. FALA can only converge to an interval, the measure of which is determined by the initial values. We conclude that the decrease in computation time of the FALA algorithm is obtained by sacrificing the accuracy. However, the results show that the accuracy of the FALA algorithm is satisfactory in the stochastic environment that we established in this paper. We also conclude that orderly charging of EVs can mitigate the fluctuation of load, and decrease the peak-value difference of thermal power-generating units.

However, the ED model developed in this paper should be used with caution. Considering the computation speed, the lower model of the NI station is not optimized, and the penalty functions (Equation (1) and Equation (5)) are only an approximation of the non-satisfiability of the dispatch schedule. The models in this paper should be further developed and more efficient stochastic optimization algorithms should be investigated before the techniques are applied.

A. Nouns, Numbers, and Sets:

Electrical vehicle

Normal interaction model of EVs

Battery exchange model of EVs

_{a}

Set of EVs arriving at the NI station

_{NI}Ω

_{W}

_{G}

Set of buses connected with NI stations, wind farms and conventional generators

_{w}

Number of wind turbines in a wind farm

_{k}

_{,s}

Total number of EVs signing the dispatch agreement in the NI station

Set of control variables

Set of random variables

B. Constants:

_{s}

Maximum power of EV charging/discharging

Time interval between operations

_{power},

_{energy}

Penalty parameters of unsatisfied battery charging demand

_{BE}

Penalty parameter of the unsatisfied battery exchange demand

_{s}

Battery capacity of a single EV

_{ex}

Predictive value of the battery exchange demand

_{R}

Rated power of a wind farm

Air density

_{p}

Energy conversion efficiency of a wind farm

Radius of the wind turbine blade

_{R}

_{ci}

_{co}

Rated wind speed, cut-in wind speed and cut-out wind speed

Predicted value of the wind speed

Shape parameter of Weibull distribution

_{w}

Power factor of a wind farm

_{j}

_{j}

_{j}

Power generation cost parameters of a conventional generator

_{w}

Cost of wind power generation per MW

_{NI}

Cost of interaction power between NI EVs and the power system per MW

_{BE}

Cost of interaction power between NI EVs and the power system per MW

Lower and upper generation limits of a conventional generator

Lower and upper limits of interactive power of the battery exchange stations

Lower and upper limits of the voltage of bus

_{ev}

Battery capacity of a EV

Maximum energy storage capacity of the BE station

C. Uncertain Quantities:

_{l}

Energy left in the battery of EV

_{k}

Total energy in the batteries of the EVs that arrive in the NI station

_{k}

Number of EVs that finish the last trip in the NI station

_{m}

Stalled energy of the station

_{m}

_{,BE}

The power transmission between BE station

_{m}

_{,ex}

Number of the EVs that need battery exchange

_{m}

_{,un}

Unsatisfied battery exchange demand

_{total}

Total operational cost of the whole sampling period

_{i}

_{,w}

Wind farm output of bus

_{k}

_{,sev}

Interactive power between the power system and the NI EVs not signing the dispatch agreement in NI station

_{W}

_{G}

_{L}

Output of the wind farms, output of conventional generators and power Loss through the transmission lines

_{EV}

Total interactive power between the power system and the EV installation

_{i}

Injection power of bus

Maximum power transmission of the power line that connects bus

_{i}

Voltage of bus

_{m}

Stalled energy of the BE station

D. Decision Variables:

_{k}

_{,NI}

Interactive power between the system and the NI EVs signing the user agreement in bus

_{j}

_{,g}

Output of a conventional generator of bus

E. Functions:

_{k}

_{,NI}()

Penalty function of the NI station of bus k

_{NI}

_{,penalty}()

Penalty function of all NI stations

_{BE}

_{,penalty}()

Penalty function of all BE stations

_{G}

_{M}

_{P}

Fuel cost, maintenance cost and penalty cost of the system

_{obj}

Objective function of the optimization model

Equality and inequality constraint function

Reward function for the automation

This work was supported by the State Grid Corporation of China (PD71-13-031), the National High Technology Research and Development Program of China (863 Program Grant 2012AA050210), the National Science Foundation of China (51277027), the Fundamental Research Funds for the Central Universities, the Natural Science Foundation of Jiangsu Province of China (SBK201122387).

The listed authors contributed together to achieve this research paper. Junpeng Zhu designed the optimization algorithm, Ping Jiang contributed to economic dispatch formulation, Wei Gu contributed to the modeling of power demand and supply, Wanxing Sheng and Xiaoli Meng did the case studies, and Jun Gao assisted in typesetting and revision work.

The authors declare no conflict of interest.