1650 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 6, NO. 4, OCTOBER 2015
A Multistate Markov Model for Dimensioning Solar Powered
Cellular Base Stations
Vinay Chamola and Biplab Sikdar, Senior Member, IEEE
Abstract—The dimensioning of photovoltaic (PV) panel and
battery sizes is one of the major issues regarding the design of
solar powered cellular base stations (BSs). This letter proposes a
multistate Markov model for the hourly harvested solar energy to
determine the cost optimal PV panel and battery dimensions for a
given tolerable outage probability at a cellular BS.
Index Terms—Green communications, solar energy.
SOLAR POWERED, offgrid cellular base stations (BSs)
provide a communication infrastructure in places without
reliable grid power. This letter presents a Markov model for
hourly solar energy and applies it to dimensioning offgrid cel-
lular BSs. Existing Markov models for solar energy lack the
day-level weather correlations that are critical for dimensioning
high-reliability systems , . Thus, we propose a model that
combines hourly and daily transitions in the weather conditions.
II. BACKGROUND DETAILS
This letter considers a long-term valuation (LTE) cellular BS
whose power consumption at time tis given by 
where Ntrx is the number of transceivers, P0is the power con-
sumption at no load (zero trafﬁc), Δpis a BS speciﬁc constant,
Pmax is the output of the power ampliﬁer at the maximum trafﬁc,
and Kis the normalized trafﬁc at the given time.
To model the trafﬁc, Poisson distributed call arrivals with
time-of-day dependent rates, and exponentially distributed call
durations with mean 2 min are used . Kis obtained by nor-
malizing the instantaneous trafﬁc by the maximum number of
calls that the BS can support at any time. We assume that lead
acid batteries are used. The battery lifetime is calculated by
counting the charge/discharge cycles for each range of depth
of discharge (DoD) for a year and is given by 
where Ziis the number of cycles with DoD in region i, and
CTFiis the cycles to failure corresponding to region i.Given
nPV photovoltaic (PV) panels each with dc rating Epanel, and nb
Manuscript received August 26, 2014; revised April 09, 2015 and May 25,
2015; accepted May 29, 2015. Date of publication August 05, 2015; date of
current version September 16, 2015. Paper no. PESL-00132-2014.
The authors are with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore (e-mail: vinay.
Digital Object Identiﬁer 10.1109/TSTE.2015.2454434
Fig. 1. (a) Transition between good and bad days. (b) Hourly transition in a
good day. For clarity, only the transitions from state G(i,1) are marked.
batteries, each with capacity Ebat, the overall PV panel dc rating
is PVw=nPV Epanel, and the battery bank capacity is Bcap =
nbEbat. This letter uses solar irradiance data made available by
National Renewable Energy Laboratory (NREL), USA .
III. MODEL DESCRIPTION
To develop the solar energy model, for any site, solar irra-
diance data of 10 years are fed into NREL’s System Advisor
Model tool  to calculate the hourly energy generated by a
PV panel with 1-kW dc rating. This data is then parsed on a
monthly basis. The solar energy output for each day in a given
month is computed and the days are sorted based on this energy.
β% of the days with the lowest energy are termed “bad,” and
the rest, “good” days. The probability of transition from one
day type to another is calculated from the data. This is modeled
as a Markov process [Fig. 1(a)] with transition matrix
pbg pbb (3)
where pgg (pbb, respectively) is the transition probability from
good to good (bad to bad), and pgb =1−pgg (pbg =1−pbb ,
respectively) is the transition probability from good to bad (bad
to good) day.
Within a day, the harvested solar energy varies with time. We
model these variations on a hourly basis as a Markov process.
For each day type (good/bad), the minimum and maximum PV
panel output for each hour of the day are calculated. The region
between the minimum and maximum values is divided uni-
formly into four regions, as shown in Fig. 2. Each of these
regions, along with the day type, represents a “state” of the
harvested solar energy. The state at time tis denoted by
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1652 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 6, NO. 4, OCTOBER 2015
This letter proposed a multistate Markov model for charac-
terizing the hourly solar irradiation. The model was used for
dimensioning solar powered cellular BSs in terms of the cost
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