电动汽车优化充电策略：基于马尔可夫决策过程的方法

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随着电动汽车和可再生能源的结合日益成为未来零化石燃料交通的重要推动力，优化电动汽车的充电策略变得至关重要。本文主要探讨如何通过应用马尔科夫决策过程（Markov Decision Processes, MDP）来设计一个有效的电动车充电算法。作者Emil B. Iversen、Juan M. Morales和Henrik Madsen来自丹麦技术大学计算机学院，他们的研究旨在解决电动汽车充电调度问题，考虑到车辆的实际使用情况、用户的风险偏好以及电力价格的不确定性。在该研究中，他们提出的算法核心在于将车辆的使用模式与充电决策相结合。充电政策不仅考虑车辆的日常行驶需求，比如行驶里程、电池容量和充电效率，还纳入了用户的风险态度。例如，如果用户倾向于避免电力价格波动带来的成本，那么充电策略可能会倾向于在电价较低时进行充电。反之，如果用户更关注即时便利性，可能更倾向于在回家后立即充电。 MDP提供了一个数学框架，使得可以根据不同的状态（如当前电池电量、剩余行程和电价）动态地做出最优决策。这种方法借鉴了随机动态规划（Stochastic Dynamic Programming）的思想，通过建立一个状态转移概率模型，预测不同充电策略下的长期效益。通过迭代计算，算法可以找到在所有可能的状态转移下，使总期望效用最大的充电策略。这个模型的灵活性体现在它能够适应各种特定的车辆特性，无论是电池类型、充电设施还是用户的个人习惯，都可以作为输入参数进行定制化调整。这使得研究结果具有很高的实用价值，对于电动车运营商、电网调度员和消费者来说，都能从中获益，共同推动电动汽车的高效管理和可持续发展。文章于2013年10月5日首次接收，经过修订后于2014年2月4日接受，最终于同年3月12日在线发表。关键词包括“电动汽车”、“驾驶模式”、“最优充电”、“马尔可夫过程”和“随机动态规划”。这项研究为电动汽车的绿色能源管理提供了一种科学且个性化的解决方案，有助于减少对化石燃料的依赖并促进清洁能源市场的整合。

to be estimated for each time step. Needless to say, the number of

parameters to be estimated increases as the number of states

grows. We refer to [17] for further details on techniques to reduce

the number of parameters to be estimated for each time step for

models with more than two states. Another problem is linked to

the number of observations available to properly carry out the esti-

mation, i.e. if

k¼1

ðs

Þ¼0 for some s

, then

ðs

Þ is undeﬁned.

To deal with the large number of parameters as well as unde-

ﬁned transition probability estimates, B-splines are applied to cap-

ture the diurnal variation in the driving pattern through a

generalized linear model. The procedure of applying a generalized

linear model is implemented in the statistical software package R

as the function glm (). For a thorough introduction to B-splines

see [20] and for a general treatment of generalized linear models

see [21]. Next we elaborate on how the ﬁtting of the Markov chain

model works in our particular case.

Each day, at a speciﬁc minute, a transition from state j to state k

either occurs or does not occur. Thus for every s on the diurnal cy-

cle we can consider the number of transitions to be binomially dis-

tributed, i.e. n

ðsÞBðz

ðsÞ; p

ðsÞÞ, where the number of Bernoulli

trials at s, given by z

ðsÞ¼

k¼1

ðsÞ, is known and the probability

of success, p

ðsÞ, is unknown. The data can now be analyzed using

a logistic regression, which is a generalized linear model [21]. The

explanatory variables in this model are taken to be the basis func-

tions for the B-spline. The logit transformation of the odds of the

unknown binomial probabilities are modeled as linear combina-

tions of the basis functions. We model Y

ðsÞ¼n

ðsÞ=z

ðsÞ and in

particular, we are interested in E Y

ðsÞ



¼ p

ðsÞ.

As the basis functions for the B-spline are uniquely determined

by the knot vector

, deciding the knot position and the amount of

knots is important to obtain a good ﬁt for the model. Here we pro-

ceed as follows: First a number of knots are placed on the interval

0; 1440

½, with one at each endpoint and equal spacing between

them. Denote this initial vector of knots by

init

. The model is then

ﬁtted using the basis functions as explanatory variables. Next, the

ﬁt of the model between the knots is evaluated via the likelihood

function and an additional knot is placed in the center of the inter-

val with the lowest likelihood value. The new knot vector is then

given by

. We repeat this procedure until the desired number

of knots is reached. To determine the appropriate number of knots

and avoid over-parametrization, on the basis of a likelihood ratio

principle, we test that adding a new knot does signiﬁcantly im-

prove the ﬁt.

2.2. Hidden Markov models

Standard Markov models can only include states that are explic-

itly recorded in the data. Thus, if the data only provides informa-

tion on whether the vehicle is either driving or not driving, the

standard Markov model is restricted to having two states: driving

or not driving. Standard Markov models also result, by default, in

the time spent in each state being exponentially distributed,

although it may be with time-varying intensity. Accordingly, in a

standard Markov model, the time until a transition from the cur-

rent state to another does not depend on the amount of time al-

ready spent in the current state. In the case of a vehicle, this

implies that the probability of ending a trip does not depend on

the duration of the trip so far. This seems unrealistic for a model

capturing the actual use of a vehicle.

To overcome these limitations, we can use a hidden Markov

model, which allows estimation of additional states that are not di-

rectly observed in the data. In fact, we can estimate these states so

that the waiting time in each state matches that which is actually

observed in the data. Adding a hidden state is done by introducing

a new state in the underlying Markov chain. The new state,

however, is indistinguishable from any of the previously observed

states. This allows for the waiting time in each observable state to

be the sum of exponential variables, which is a more versatile class

of distributions. It is worth insisting that the use of hidden Markov

models is justiﬁed here to address insufﬁcient state information in

our data, which only include whether the vehicle is driving or not

driving. Indeed, the same results could be obtained using the

underlying Markov chain without hidden states, provided that

the hidden states could be observed. In practice, though, more de-

tailed driving data (e.g. including driving speed and/or location of

the vehicle) could be available once the actual implementation is

made on a vehicle, which in turn would avert the need for a hidden

Markov model. For a detailed introduction to hidden Markov mod-

els, see [22], where techniques and scripts for estimating parame-

ters are also provided.

The hidden Markov model consists of two parts. First, an under-

lying unobserved Markov process, X

: t ¼ 1; 2; ...

, which de-

scribes the actual state of the vehicle. This part corresponds to

the Markov model with no hidden states as described previously.

The second part of the model is a state-dependent process,

: t ¼ 1; 2; ...

, such that when X

is known, the distribution of

depends only on the current state X

. A hidden Markov model

is thus deﬁned by the state-dependent transition probabilities,

ðtÞ, as deﬁned for the standard Markov chain and the state-

dependent distributions given by (in the discrete case):

ðtÞ¼P Z

¼ zj X

¼ kðÞ: ð6Þ

Collecting the d

ðtÞ’s in the matrix Dðz

Þ, the likelihood of the

hidden Markov model is given by:

¼ dDðz

ÞPð2ÞDðz

Þ; ...; PðTÞDðz

Þ; ð7Þ

where d is the initial distribution of X

. We can now maximize the

likelihood of observations to ﬁnd the estimates of the transition

probabilities between the different hidden states.

2.3. Fitting the Data

The data at our disposal is from the utilization of a single vehicle

in Denmark in the period spanning the six months from

23-10-2002 to 24-04-2003, with a total of 183 days. The data is

GPS-based and follows speciﬁc cars. One car has been chosen and

the model is intended to describe the use of this vehicle accord-

ingly. The data set only contains information on whether the

vehicle was driving or not driving at any given time. No other infor-

mation was provided in order to protect the privacy of the vehicle

owner. The data is divided into two periods, a training period for

ﬁtting the model from 23-10-2002 to 23-01-2003, and a test period

from 24-01-2003 to 24-04-2003 for evaluating the performance of

the model. The data set consists of a total of 749 trips. The time

resolution is in minutes.

We shall consider a model with one not driving state and two

(hidden) driving states. In other words, one can observe whether

the vehicle is driving, but cannot identify which type of driving state

the vehicle is in. Besides, the hidden driving states are not directly

interpretable from the data. In practice, they could correspond to

driving in different environments (urban/rural) or at different

speeds. Be as it may, the inclusion of the hidden structure allows

for the probability of ending the current trip to depend on the time

since departure, as the vehicle may pass through different driving

states before ending the trip. We then compute the transition

probability between the hidden states in such a way that the

resulting probability distribution of the trip duration follows the

one reﬂected in the data. Furthermore, to ﬁt the model to the data,

we assume that only the transition probability from the not driving

state depends on the time of day. This is done to reduce the

E.B. Iversen et al. / Applied Energy 123 (2014) 1–12

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