Energies 2018, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies
Optimal Scheduling of an Isolated Wind–Diesel–2
Energy Storage System Considering Fast Frequency
Response and Forecast Error
Nhung Nguyen Hong 1*, Yosuke Nakanishi 2
1 Graduate School of Environment and Energy Engineering, Waseda University, Tokyo, Japan;
2 Graduate School of Environment and Energy Engineering, Waseda University, Tokyo, Japan;
* Correspondence: email@example.com; Tel.: +81-803-462-2888
Received: date; Accepted: date; Published: date
Abstract: Nowadays, the hybrid wind–diesel system is widely used on small islands. However,
the operation of these systems faces a major challenge in frequency control due to their small
inertia. Furthermore, it is also difficult to maintain the power balance when both the wind power
and load are uncertain. To solve these problems, energy storage systems (ESS) are usually
installed. This paper demonstrates the effectiveness of using ESS to provide Fast Frequency
Response (FFR) to ensure that the frequency criteria are met after the sudden loss of a generator.
An optimal day-ahead scheduling problem is implemented to simultaneously minimize the
operating cost of the system, take full advantage of the available wind power, and ensure that the
ESS has enough energy to provide FFR when the wind power and demand are uncertain. The
optimization problem is formulated in terms of two-stage chance-constrained programming and
solved using a Modified Sample Average Approximation (MSAA) algorithm, a combination of the
traditional Sample Average Approximation (SAA) algorithm and the k-means approach. The
proposed method is tested with a realistic islanded power system, and the effects of the ESS size
and its response time is analyzed. Results indicate that the proposed model should perform well
under real-world conditions.
Keywords: chance-constrained programming; day-ahead scheduling; energy storage system; fast
frequency response; wind power
Indices and Sets
31 ∈ Diesel generators
32 ∈ Time intervals (of variable duration)
33 ∈ Wind generators
34 Random vector
36 , Start-up cost of diesel generator i ($/kWh)
37 , Shutdown cost of diesel generator i ($/kWh)
38 , No-load cost of diesel generator i ($/kWh)
39 Operating cost of diesel generator i ($/kWh)
40 Charge cost of energy storage ($/kWh)
41 Discharge cost of energy storage ($/kWh)
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Maximum power output of diesel generator i (kW)
43 Minimum power output of diesel generator i (kW)
44 Minimum uptime (hours)
45 Minimum downtime (hours)
46 Power rating of energy storage system (kW)
47 Capacity of energy storage system (kWh)
48 Charging/discharging efficiency of energy storage system
49 Inertia constant of diesel generator i (s)
50 Nominal frequency (Hz)
51 Minimum frequency threshold (Hz)
52 Dead band of governor (Hz)
53 Maximum governor ramp rate of generator i (kW/s)
54 ∆ The sustain duration of Fast Frequency Response provided by energy storage system
Forecasted wind power at time t (kW)
Forecast error of wind power at time t (%)
Maximum possible wind power at time t (kW)
Forecasted demand at time t (kW)
Forecast error of demand at time t (%)
62 Actual demand at time t (kW)
64 Start-up state of diesel generator i at time t (binary)
65 Shutdown state of diesel generator i at time t (binary)
66 ON/OFF state of diesel generator i at time t (binary)
67 Power output of diesel generator i at time t (kW)
68 Reserve of diesel generator i at time t (kW)
ON/OFF state of wind turbine at time t (binary)
70 Actual wind power at time t (kW)
Charging state of energy storage system at time t (binary)
, Charge power of energy storage system at time t (kW)
, Discharge power of energy storage system at time t (kW)
Discharge power of energy storage system after a contingency event at time t (kW)
Energy stored in the energy storage system at time t (kWh)
Initial energy stored in the energy storage system at t = 0 (kWh)
Energy stored in the energy storage system at the end of the day (t = 24) (kWh)
Frequency is an important criterion in the power system's operation and is related to the
instantaneous balance between supply and demand. To ensure a stable operation of a power system,
the balance between power demand and supply must be kept at all times; the system frequency is
only allowed to vary in a tight band around the nominal value. Large frequency disturbances,
caused by events such as the sudden loss of a generator, lead to serious active power imbalances and
may lead to load shedding or partial or complete blackout. Fortunately, immediately after a
frequency disturbance, the kinetic energy stored in the spinning masses of the generators is released
into the power system to preserve the power balance, thereby reducing the rate of frequency change.
This process is called the Inertial Response (IR) of the generator. The most crucial frequency control
activity is the Primary Frequency Control Response (PFR), which is based on the characteristic of the
conventional speed governor. PFR automatically starts in the event of a large frequency deviation to
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adjust the power output of generators. Large synchronous generators can provide both IR and PFR,
thereby ensuring the frequency stability of the power system.
In recent years, the penetration of renewable energy sources (RES) such as wind and solar into
the power system is increasing rapidly. Unlike conventional power plants, which are based on
synchronous machines, RES power plants use inverter-based generators that do not have IR.
Therefore, the overall power system inertia is reduced by the increasing penetration of RES.
Furthermore, due to the stochastic nature of wind and solar irradiation, the reserved capacity for
PFR from RES is also uncertain. Both of these factors contribute to the more complicated frequency
control problem for power systems containing large fractions of wind and solar generation . In
islanded power systems, frequency control and regulation are even more challenging because the
primary resources are diesel generators (DGs) with low inertia and limited operating capability.
With the increasing penetration of renewable generation and energy storage in power systems,
the Fast Frequency Response (FFR) method has been introduced as a measure to improve frequency
stability. In the Australian electricity market, FFR is defined as “any type of rapid active power
increase or decrease by generation or load, in a timeframe of less than two seconds, to correct
supply-demand imbalances and assist with managing frequency” . There have been several
studies of FFR encompassing a wide range of technologies [3–6]. In one study , it was shown that
wind generators (WG) can provide IR for a very short duration (around 10 s). Although this method
proved to be useful for frequency regulation, the kinetic energy provided by wind turbines is highly
dependent on the wind speed; as a result, insufficient support is delivered in the case of low wind
speed. Furthermore, in a low-inertia system, the frequency can drop below the threshold of the
Under-Frequency Load Shedding (UFLS) relay within only 1 s, so the response of a WG is not
effective. For the case of photovoltaics (PV), another method involves keeping the PV power setpoint
below the total available power, at the expense of economic performance .
With a very short response time, which can be less than 250 ms depending on the technology,
energy storage systems (ESS) are able to instantly increase or decrease their power output to
counteract a system power imbalance. There have been many previous studies focusing on the
support of an ESS in frequency response such as [7–12]. The authors of [9–12] focused on the size of
the ESS, whereas [7,8] propose control strategies for an ESS to provide virtual inertia. The results
presented in these articles show the effectiveness of using an ESS for frequency response control.
An interesting research approach in frequency-constrained operation planning is to include
frequency constraints in Unit Commitment (UC) models. The authors of  proposed a UC
formulation including a frequency limit constraint based on the general-order system frequency
response model. In contrast, a first-order model for a governor–prime mover system was used in
. However, the UC model in [13,14] did not consider the uncertainty in the available wind power
so the results are less reliable for actual operation. Reference  develops a sophisticated
representation for the frequency dynamics, which includes load damping, but this work did not
consider the application of ESS in frequency response. A more recent study  developed a UC
framework, in which the ESS is considered to provide frequency response. However, the wind
power uncertainty in  is described by only three scenarios: the central forecast, the upper bound,
and the lower bound. Although this approach helps reduce the computational complexity, it may
lead to conservative solutions with higher operating cost.
In this work, we propose a frequency stability-constrained UC models and apply it for a
realistic isolated wind–diesel system on Phu Quy Island, Binh Thuan province, Vietnam. The UC
model in this work focuses on FFR. The primary goal of ESS in this model is to compensate for the
fluctuation of wind and solar generation and to help increase the energy produced from renewable
sources (rather than using curtailment). Besides, this ESS provides FFR in large frequency
disturbances, such as loss of a generator, as an ancillary service. The proposed model has practical
implications for isolated power systems with a high penetration level of renewable resources.
To account for the stochastic nature of wind power and demand, the optimal scheduling in this
work is formulated as a two-stage chance-constrained optimization problem. The constraints related
to uncertain parameters are written as probabilistic constraints with a chosen risk level [17,18]. A
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common method used to solve chance-constrained problems is the Sample Average Approximation
(SAA) algorithm, which involves Monte Carlo simulation to approximate the distribution function
of a random vector using N samples [19–23]. Although SAA is simple and convenient, all N samples
are considered to have the same probability regardless the true distribution of the random vector, so
the number of samples must be large to ensure that a feasible solution is found. If there are many
uncertain parameters, the size of the optimization problem increases and a significantly longer
computing time is required. For this reason, it is necessary to improve the algorithm, which is one of
the objectives of the present work.
The salient features of the present study include:
1. The research focuses on frequency stability-constrained UC models. The ESS, which is
employed to keep power balance and take advantage of wind power, is considered to provide
fast frequency response (FFR) in large frequency disturbances, such as loss of a generator. The
frequency dynamics is approximated using a first-order representation.
2. The proposed UC model is based on a two-stage stochastic programming framework which is
suitable for the short-term planning of power systems with uncertain sources. The model is
formulated as a chance constraint problem to allow a certain risk level in the day-ahead
3. The impact of ESS sizing and its response time on frequency nadir is analyzed.
As the computing time required for solving chance-constrained optimization is usually
significant, in this paper a Modified Sample Average Approximation (MSAA) method is presented
and applied to solve the proposed optimization problem. The combination of SAA and k-means
clustering approach is proven to be more effective than the original SAA approach.
The rest of the paper is organized as follows: Section 2 demonstrates the application of the ESS
for FFR. Section 3 presents the mathematical formulation of the proposed chance-constrained
optimization problem to determine the optimal scheduling of the power system. Section 4 presents
the MSAA algorithm. The computation results are collected and analyzed in Section 5. Section 6
concludes the paper.
2. Fast Frequency Response and the Role of the ESS
As discussed in section 1, IR plays an important role in maintaining frequency after a generator
is lost. This process slows down the change of frequency before the governors fully react to provide
PFR. We can evaluate this process using two important criteria including the
rate-of-change-of-frequency (RoCoF) and the lowest frequency known as the frequency nadir .
Consider, for example, a power system with I generators. If at time t generator j with power
output (kW) is lost, the RoCoF immediately after the contingency event is defined as
where is the system inertia (kW.s/Hz) after the loss of generation j and is a function of the
inertia of the online generators:
179 =∑ (2)
where , , and are the inertia constant, maximum capacity, and ON/OFF state of the
remaining generators, respectively; is a binary variable that is equal to 1 if generator i is online
and 0 if it is offline.
If the inertia of the system is sufficient, the frequency will stop before reaching the threshold of
the UFLS relays. However, in an islanded power system, the primary resources are DGs with low
inertia constants. Additionally, modern wind turbines are connected to the grid through an
electronic power converter, which does not contribute to the system inertia. It is easy to see that the
lower the system inertia, the faster the frequency drops. Therefore, even though DGs can adjust their
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power output quickly, it is still difficult to arrest the frequency decline before reaching the minimum
The concept of FFR is applied to solve this problem. Unlike both IR and PFR, which slowly
adjust the power based on the frequency deviation, thus arrest the deviation and restore frequency,
the objective of FFR is to immediately inject active power into the grid to correct the power
imbalance. This is implemented based on wind turbines or ESS that can change their power output
almost instantaneously. FFR can be considered as a measure to compensate for the “interval”
between IR and PFR, during which the frequency is too low and PFR is still not fully active.
Note that FFR cannot completely replace PFR - it is only a support measure while waiting for
the DGs to provide PFR. Thus, a sustained FFR duration is not necessary. Report  shows that in
ERCOT (Texas), this duration is 10 minutes while EirGrid/SONI (Ireland) only requires an 8-seconds
An important requirement for FFR is fast response time, and systems with a higher RoCoF will
require a faster response time. Assuming the system has a nominal frequency of 50 Hz and a
minimum frequency of 49 Hz, FFR must fully react within 250 ms in the case of a 4 Hz/s RoCoF. The
response time depends on not only the detection method but also the type of device used to provide
FFR. From , the times required to detect contingency and to send the control signal, as well as the
reaction time and rise time of the ESS, are summarized in Table 1. It can be seen that, with total
response times ranging from 100 ms to 200 ms, the ESS is suitable for providing FFR.
Table 1. Response times of various detection methods and types of ESS .
time + rise
Direct detection 40–60 ms ≈20 ms
RoCoF detection/PMU 40–60 ms ≈20 ms
measurement ≈100 ms ≈0
Type of ESS
Lithium Batteries 10–20 ms
Flow Batteries 10–20 ms
Lead-acid Batteries ≤100 ms
Flywheel ≤4 ms
Supercapacitor 10–20 ms
In this section, we outline the constraints on the power output of the DGs for each hour and the
response of the ESS needed to satisfy the frequency criterion, .
Using the first-order model for a governor–prime mover presented in [14,25], we arrive at an
approximate model of the system’s response after a sudden generation loss of amount and the
application of ESS for FFR, as described in Figure 1. After the governor’s dead time , which
corresponds to the frequency dead band , the power output of the DGs will change due to the
governor’s response with the system ramp rate =∑ ∑
⁄. On the other hand, a
control signal is also sent to the ESS to increase its output from
adjustment provided by the ESS is
, (Figure 1).
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To simplify the model, we make two assumptions:
The rise time of ESS is negligible
The ESS can fully compensate for the power shortage. This means that the frequency decay
stops immediately after the ESS fully responds, at which point the frequency nadir occurs.
The time evolution of the system frequency deviation can be described by:
where ∆() and ∆() describe the additional power provided by DGs (due to the response
of the governors) and ESS, respectively. ∆() and ∆() are formulated as follows:
226 ∆()= 0 <
(−) ≤ ≤
(−) ≤ (4)
∆()= 0 <
Using the model presented in Figure 1, we can find the relationship between the adjustment
power provided by the ESS and the time when the ESS can fully respond:
Considering (4)–(6), the equation (3) can be integrated between =0 and =.
Assuming that before the contingency event, the system frequency is at the nominal value
234 , we have:
235 −=− 1
= − 1
Reference  shows that the duration of is related to the governor dead-band (Hz) by
. Besides, noting that the frequency nadir should not be below the
predefined threshold , we obtain the frequency nadir’ s requirement as follow.
Substituting (6) into (7) and noting that
,, we obtain the
Constraint (8) shows that the number of DGs in operation and their power output per hour is
limited by the time taken for the ESS to fully react, and this will be used in the optimal scheduling
formulation presented in Section 3.
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Figure 1. Application of an energy storage system for fast frequency response.
3. Problem Formulation
3.1. Wind and Demand Models
In the present study, we focus on the optimal schedule of an islanded power system
considering FFR. A challenge in this problem is the uncertainty in the expected wind power and
demand. The wind power and demand forecast errors will affect the operation of the system, so the
optimal scheduling problem must be formulated as a predictive optimization with results expressed
as ranges of values that assure reliable operation of the system.
Both wind power and demand can be defined as the sum of the forecasted value and the
are assumed to follow a normal distribution with zero mean
and the standard deviation
for wind power and for demand. This means the maximum error
would be 3
for wind power and 3 for demand in correspondence to the confidence level of
3.2. The Optimal Scheduling Problem
This study implements day-ahead scheduling of the islanded power system including DGs,
WG, and ESS with frequency criteria. The operating cost of DGs is minimized when wind power is
used as much as possible. With the given forecasted wind power and demand, the operating mode
of the DGs including ON/OFF state and power output is determined every hour for a 24-hour time
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horizon. The total energy excess or deficiency due to forecasting error is fully compensated by the
ESS. Furthermore, the ESS must provide FFR if a generator is tripped. For this reason, the ESS should
be scheduled such that it not only has enough energy to discharge but also can recharge to utilize the
most energy from wind power.
The process of optimal scheduling comprises two stages, as illustrated in Figure 2. In the first
stage, the deterministic day-ahead schedule in the commitment and the dispatch of DGs is decided
and sent to the grid operator; the solid blue arrows in Figure 2 illustrate this process. The first-stage
problem is a day-ahead UC problem that is implemented at least one day before the actual operation
date. At that time, the wind and demand values are long-term forecast results with errors. Thus,
only the results of the unit commitment (on/off states) and the power output of DGs are fixed. The
ESS charge/discharge power and WG’s power output are adjusted after wind power and demand
are known with higher accuracy using a very short-term forecast.
This two-stage optimal scheduling is formulated as a two-stage chance-constrained
optimization model (Figure 3). The constraints in the first stage refer to the deterministic planning,
and the second stage ensures that the power balance and frequency criteria after a contingency event
are met with a chosen probability.
Figure 2. Schematic illustrating the optimal scheduling problem for an islanded power system.
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Figure 3. The two-stage optimization model.
3.2.1. Objective Function
The objective function to be minimized comprises the first-stage operating cost and the
expected value of the second-stage cost:
295 ∑ ∑,+,+,++
where represents a random vector including wind and demand;
,() are ESS
charge/discharge power decided corresponding to the actual values or the very short-term forecast
value of .
3.2.2. First-Stage Constraints
The first stage is characterized by the following constraints:
Active power balance constraint. The total active power output from the DGs , the wind
plant , and the storage system (
,) must equal the given forecasted load
any time t:
DG operating constraints. The power output of each DG must be in the operating range
between and , which are specified by the manufacturer. The binary variable in
constraint (12) is used to keep the DG power output equal to zero if it is shut down. Constraints
(13) and (14) describe the minimum uptime and downtime limitations of each DG.
309 ≤≤ (12)
310 −=− (13)
∑ ≤1− (14)
Constraint (14) ensures that if at hour t, the generator is offline (=0), the generator cannot
start up within the duration from − to (=0 with all values of k from − to ). In
contrast, if at hour t, the generator is online (=1) then this constraint ensures that the generator
cannot shutdown within the duration from − to (=0 with all values of k from −
Primary reserve constraints for DGs. This constraint shows that each DG can take part in the
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WG operating constraint. For a given forecasted wind power
, the power output of the WG
must satisfy constraint (16), where
is the minimum wind power output:
ESS constraints. Constraint (17) states that the charging and discharging power of the ESS must
be smaller than the actual power rating of the storage device . The process of
charging/discharging the ESS is described by constraint (18). This constraint also imposes that
the energy stored in the ESS should be smaller than its rated capacity at all times:
Frequency nadir limit. As presented in Section 2, to ensure that the frequency does not drop
below the minimum allowable level, constraint (19) must be satisfied:
Note that the system inertia and ramp rate depend on the number of the online DGs.
Thus, this constraint determines not only ESS charge/discharge power but also the number of DGs in
operation and the power output of each DG before the contingency event to ensure the frequency
Post-contingency energy storage capacity constraint. This constraint ensures that the ESS has
enough energy to sustain FFR for ∆.
3.2.3. Second-Stage Constraints
In the second stage of the optimization model, uncertainties in wind power generation and load
consumption are considered. The power output of the ESS and the WGs are re-dispatched as
,(), and (), where represents a random vector. When a generator is lost, the
ESS will discharge
() to decrease the disturbance to
,(). The second-stage constraints can be summarized as follows:
Active power balance constraint.
≥1 − (21)
WG operating constraint.
,() ≤ (25)
Frequency nadir limit.
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Post-contingency energy storage capacity constraint.
In the above constraints, constraint (21) guarantees that the ESS and WG will be adjusted so that
the probability of power imbalance is less than a risk level . Similarly, constraint (26) is the
probability formulation of the constraint (19) and ensures that the frequency criterion will be met
after a contingency event with high probability, even if the ESS is re-dispatched due to the difference
between the long-term and very short-term forecast results of .
4. The Modified Sample Average Approximation
Consider a simple two-stage chance-constrained optimization model
366 =min()+(,) (28)
368 Pr((,,)≤0)≥1− (29)
where is the first-stage variable, is the second-stage variable, and is random input data.
Many studies in the literature show that this model can be solved by the SAA method [19–23].
In this method, Monte Carlo simulation is used to approximate the distribution function of the
random vector ξ by N samples. The optimization formulation (28) then becomes
where (,)(,,) is an indicator function that is equal to one if (,,)≤0 and zero
It is assumed that the N samples have the same probability (1/). This assumption helps to
simplify the formulation of the optimization; however, a large number of samples are required to
guarantee accuracy, which means the CPU time required to solve it increases accordingly.
In the present study, a modified approach to the SAA is proposed, by using a k-means
clustering approach to reform the samples. Instead of using all N samples, the k-means clustering
divides the samples into M clusters. The probability of each cluster is the sum of the probabilities of
the constituent samples. Next, M centroids of the clusters are used as the SAA algorithm input
samples, with the probability of each centroid being equal to the probability of the cluster that it
represents. Figure 4 illustrates a small example: 1000 samples generated from the standard normal
distribution N(0,1) are replaced by 10 centroids.
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Figure 4. Example of using k-means clustering to reform samples.
For M centroids and their corresponding probabilities, (28) and (29) are reformulated as
where is the probability of each centroid ( = 1,2, … ).
Now let ̅ and , respectively, be the optimal solution and value of the optimal problem in
(32)–(33) and check whether this solution is feasible or not. Using Monte Carlo simulation to
generate a new set of ’ samples where ’ is much larger than , we find the value of the
probability constraint (31) with solution ̅ is
The (1–ɛ)-confidence lower bound on (̅) is then computed using
where Φis the inverse normal distribution function. ̅ is a feasible solution of the original
problem only if (̅)≥1−. Repeat this process K times according to the flow chart illustrated in
Figure 5 and find the maximum value
and minimum value of the optimal value . If the
optimality gap given by (
⁄×100% is smaller than a predetermined threshold, the
algorithm terminates, and we obtain the optimal solution of the original problem.
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Figure 5. Flow chart of the MSAA algorithm.
5. Results and Discussion
5.1. Study System
The data used in this study is based on an actual power system on Phu Quy Island, Binh Thuan
province, Vietnam. This system includes six 500 kW DGs and two WGs, each with a rated power of
1.8 MW. The purpose of the ESS installation is to utilize as much wind power as possible and to
mitigate the uncertainty in demand and wind speed. The optimal size of an ESS taking into account
the relevant wind scenarios and the annual load growth factor was analyzed in our previous work
k = 1
Generate N=1000 samples
Solve SAA problem
using (33) and (34)
k = k + 1
̅, ̅ ,
Generate N' samples
Find maximum value and
minimum value of
s = s + 1
s = 1
Set upp er bound
Set lower bound
Find Mcentroids by k-
means clustering Nsampes
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. The ESS helps balance the fluctuation in wind power, increase the wind penetration level, and
also provide FFR as a supplementary service. We assume that the wind power variable costs are zero
and the ESS efficiency factor for the charging and discharging process is 90%. The risk level of the
probability constraints in the second stage is 5%, which means these constraints should be met with
a probability of greater than 95%. The other parameters are presented in Table 2.
Table 2. Input data.
= 170 kW
= 49.2 Hz
= 0.02 Hz
FFR’s sustain duration
= 5 minutes
The scenario of wind and demand considered in this problem is described in Figure 6. The
maximum possible instantaneous penetration of wind power is approximately 45% during the first
hour and highest in the fifth hour (49%). However, this ratio is only 15% when the load is highest at
the 19th hour. Forecast errors are modeled using a normal distribution with zero mean. The
standard deviations are assumed to be 0.05 for both wind power and demand, which means that the
maximum forecasting error in the values each hour can be considered to be 15%.
Figure 6. Forecasted wind power and demand.
In the present study, the following cases are considered to evaluate the effectiveness of using
ESS to provide FFR:
Case 1a. This case simulates the original UC problem without frequency constraints. The ESS is
used to take advantage of wind power and compensate for the uncertainty in wind speed and
Case 1b. The original UC problem with frequency constraints. The ESS is not used to provide
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Case 2. UC problem with frequency constraints. The ESS is used to provide FFR.
Besides comparing the frequency criteria after a contingency event in these cases, the
effects of other parameters such as ESS size are also considered. The optimization problem is solved
using the MSAA approach presented in Section 4 with CPLEX version 12.6 and the YALMIP toolbox
5.2. Optimization Results
5.2.1. Case 1a: Original Optimal Scheduling Model Without Frequency Criteria
With an ESS rating of 400 kW/800 kWh and the other input data given in Section 5.1, the
optimal daily schedule for DGs in Case 1 is presented in Figure 7a. Figure 7b shows the scheduled
operation of DGs, wind generators and the ESS. Note that the DGs operating schedules are first stage
variables. The ESS charging/discharging power and wind power are second stage variables,
considering the uncertain nature of wind speed and load forecast error. Therefore, they are
represented using box plots. It can be seen that, although the possible wind capacity and demand are
uncertain, the available wind power is still fully utilized most of the time; this is undoubtedly due to
the involvement of the ESS in the grid.
The RoCoF immediately after a contingency event in which the DG having the largest power
output is lost is presented in Figure 8. During the period from the 17th to the 20th hour, demand is at
its highest, whereas available wind power is quite low, so four DGs are online. This means that the
stored kinetic energy in this period is higher than that in the rest of the day. However, the RoCoF is
still approximately 10 Hz/s. Although DGs can increase their power output very quickly, even from
a cold start condition (10–15 s), the frequency declines rapidly to below the minimum threshold. This
can be explained by the fact that the inertia constant of the DGs being small (H = 0.8).
Figure 7. Case 1a (frequency criteria not considered): (a) optimal daily schedule for the DGs; (b) box
plots of wind power and ESS power, DG power and forecasted demand.
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Figure 8. RoCoF after a contingency event in Case 1a.
5.2.2. Case 1b: Optimal Scheduling Problem Considering Frequency Criteria
Accounting for the frequency criteria in the optimal scheduling problem, as many DGs as
possible are kept online while keeping their power output at a low level (Figure 9). For example,
during peak load hours, there are six DGs in operation even though only four DGs are needed in
Case 1a. This operating strategy helps increase the inertia of the system and reduce the RoCoF. The
reduction of RoCoF can be seen by comparing the results in Figure 8 and Figure 10. However, the
frequency nadir at almost all hours is still much smaller than the minimum threshold
472 (49.2 Hz). Moreover, increasing the power output of the DGs leads to higher operating costs
Figure 9. Case 1b results (frequency criteria considered): (a) optimal daily schedule for the DGs;
(b) box plots of wind power, ESS power, DG power and forecasted demand.
Figure 10. RoCoF and frequency nadir for Case 1b.
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Table 3. Comparison of optimal costs between Cases 1 and 2.
Case 1a Case 1b Case 2
The optimal cost (USD) 11103 11376 11120
5.2.3. Case 2: Optimal Scheduling Problem Considering Frequency Criteria with FFR Provided by
As in Cases 1a and 1b, an ESS rated at 400 kW/800 kWh is used in this case. However, the ESS is
not only used to maintain the power balance but also to provide FFR. The optimal schedule of the
DGs and box plots of the ESS and wind power for this case are presented in Figure 11.
As discussed in Section 2, if the DG with the largest power output is lost, the frequency
deviation will activate the ESS response, and here we assume that the total response time for FFR is
100 ms. Based on the state of the ESS before the contingency event, the charge/discharge power of
the ESS post-contingency at each hour can be in a range, as shown by the box plot in Figure 12. The
frequency nadir values for each hour are obtained using Equation (6), as shown in Figure 13.
Figure 11. Case 2 results: (a) optimal daily schedule for the DGs; (b) box plots of wind power and ESS
power, DG power and forecasted demand.
Figure 12. Box plot of the ESS charge/discharge power after a contingency event for Case 2.
Energies 2018, 11, x FOR PEER REVIEW 18 of 22
Figure 13. Frequency nadir with ESS providing FFR for Case 2.
To evaluate the effect of the ESS on FFR, we compare the RoCoF immediately after the
contingency event for Cases 2 and 1b. Figure 14 shows that the RoCoF in Case 2 is higher than in
Case 1b in a few hours. However, the frequency nadir in Case 2 is ensured, while it is violated in
several hours in Case 1b.
Note that constraints (18) and (25) in the optimization formulation limit the power output of
each DG. This explains why the number of hours with six DGs in operation in Case 2 is more than
that in Case 1b. In contrast, the ESS support in Case 2 helps to ensure the frequency criteria after
contingency events, even when the number of online DGs is less than in Case 1, thus ensuring
maximum utilization of the available wind power. This can be seen by comparing the box plots of
wind power in Figure 9b and Figure 11b. It is interesting to note that, when the ESS is able to provide
FFR, the UC solution will reduce DGs uptime and increase the wind power/ESS output, which in
turn reduces the operating cost. The optimal cost of Case 2 is smaller than that of Case 1b and is not
significantly higher than the non-constrained optimal cost (Table 3).
Figure 14. Comparison of RoCoF between Cases 1b and 2.
5.2.4. Impact of ESS Size and Response Time
In this section, we consider the effect of the ESS size and the total response time from the
moment the contingency event occurs until the ESS fully responds. Table 4 shows the smallest
possible value of the frequency nadir for two total response times, 100 ms and 200 ms, and several
ESS sizes, which are defined by the power rating and the full charge/discharge duration
⁄ (from 0.5 h to 4 h). It can be seen that the ESS size must be larger than 200 kW/400
kWh to ensure the problem has a feasible solution. It should also be noted that the forecast errors are
assumed to be ±15%, so a too small ESS will not be able to compensate for the mismatch between the
predicted and actual values of the load and wind power. However, even if the optimization problem
Energies 2018, 11, x FOR PEER REVIEW 19 of 22
has a feasible solution, depending on the size of the ESS, there will still be a nonzero probability that
the frequency nadir is lower than the minimum threshold (these values are shown in red in Table 4).
The reason for this is that the frequency nadir constraint (26) is formulated as a probabilistic
constraint with a risk level of 5%. On the other hand, constraint (26) shows that a longer response
time requires a lower power output from each DG or more DGs in operation to provide enough
kinetic energy; consequently, increasing the ESS power rating is necessary. It can be seen from Table
4 that when the response time is 200 ms, the power rating of the ESS must be larger than 600 kW to
maintain the frequency nadir above 49.2 Hz, whereas an ESS with rated power 400 kW is acceptable
if the response time is 100 ms.
Table 4. Summary of frequency nadir for different ESS sizes and response times.
Response time ESS Parameters Lowest possible value of frequency nadir (Hz)
0.5 h x 48.91 49.22 49.22 49.22
1 h x 49.09 49.22 49.22 49.22
2 h x 49.21 49.23 49.22 49.22
3 h 49.08 49.22 49.22 49.23 49.22
4 h 49.09 49.14 49.35 49.22 49.22
0.5 h x 48.67 48.69 48.69 49.27
1 h x 48.67 48.97 49.20 49.27
2 h x 48.99 49.01 49.25 49.22
3 h 48.77 48.85 49.15 49.22 49.30
4 h 48.79 48.89 49.27 49.26 49.27
5.2.5. Comparison Between the MSAA, SAA and the Robust chance-constrained algorithm
Case 2 was solved by the traditional SAA algorithm and the MSAA algorithm to compare their
computational efficiency. To solve Case 2, the SAA algorithm must be repeated at least 50 times per
loop (K = 50) and needs at least 100 samples per loop (N = 100). On the other hand, the original set of
1000 samples can be replaced with five centroids, and the MSAA algorithm needs 50 iterations to
obtain the results. Interestingly, five centroids in the MSAA algorithm are equivalent to five samples
in the SAA algorithm; thus, it is easy to see that the computing time required for the MSAA
algorithm is much smaller than that of the SAA algorithm (Table 5).
The performance of the proposed MSAA is also compared with that of the Robust
chance-constrained formulation, which is also a popular approach. In this comparison, a two-stage
Energies 2018, 11, x FOR PEER REVIEW 20 of 22
robust chance-constrained model, solved by column-and-constraint generation algorithm (CCG)
[28–30], is implemented. The constraints related to power balance and frequency criteria are also
formulated as probability constraints with the same risk level. Besides, the results obtained with a
two-stage robust model is also shown. The results in Table 5 clearly demonstrate the compromise
between CPU time and economic performance: although the required CPU time for MSAA is longer
than the robust method, the optimal cost obtained by MSAA is significantly lower than both robust
models – with or without chance constraints.
Table 5. Comparison of computing time between the MSAA, SAA and the robust method.
Method N M N’ K S CPU time
SAA 100 1000 50 1 2950 11154
MSAA 1000 5 1000 50 1 853 11120
constrained UC 8 12196
Robust UC 8 14435
In this paper, an optimal day-ahead scheduling problem concerning the application of ESS for
FFR is considered and analyzed in detail. The optimization problem is formulated within a
two-stage chance-constrained framework, in which the load and the maximum possible wind power
are uncertain. In this model, power balance and frequency criteria constraints are formulated as
probability constraints with a certain risk level. Based on the first-order model of frequency
dynamic, the relationships between the power output of each DG, the ESS charge/discharge power,
and the response time are studied. The impact of the size and response time of the ESS on the
frequency nadir after the sudden loss of a DG is also analyzed. It is also noteworthy that an MSAA
approach was proposed in the present study to solve a chance-constrained problem, and the
effectiveness of this method was demonstrated.
The results obtained in two cases—with and without FFR provided by ESS—demonstrate the
effectiveness of FFR in arresting frequency deviations after a contingency event. The proposed
method ensures that the minimum frequency threshold is not violated, even when the actual values
of wind power and demand are different from the predicted values incorporating the predetermined
maximum errors. The results also show that a slower FFR will lead to a larger ESS to ensure
The proposed approach can be extended to consider multiple contingencies such as line outages
or load interruptions as well as equipment failures. The model can also be readily adapted to include
other uncertain factors, such as solar power generation or electricity prices. These topics are left for
Acknowledgments: This research was supported by Japan Science and Technology Agency as part of the
e-ASIA Joint Research Program (e-ASIA JRP).
Author Contributions: Nhung Nguyen Hong conceived the methodology, developed the theory, and
performed the computations under the guidance of Prof. Yosuke Nakanishi. The results were discussed by all
authors, and the final manuscript was written with contributions from all authors.
Conflicts of Interest: The authors declare no conflict of interest.
1. NERC | 2013 Special Reliability Assessment: Performance of Variable Resources During and After
System Disturbance | Performance of Distributed Energy Resources During and After System Disturbance
Energies 2018, 11, x FOR PEER REVIEW 21 of 22
Voltage and Frequency Ride-Through Requirements-DRAFT; 2013. Available online:
https://www.nerc.com/comm/PC/Integration of Variable Generation Task Force
I1/IVGTF17_PC_FinalDraft_December_clean.pdf (accessed on 03 January 2019).
2. FAST FREQUENCY RESPONSE IN THE NEM; 2017. Available online: www.aemo.com.au (accessed on
03 January 2019).
3. Lew, D.; Piwko, R.; Hannett, L.; Achilles, S.; Macdowell, J.; Richwine, M.; Wilson, D.; Adamiak, M.
Technology Capabilities for Fast Frequency Response. 2017. Available online: www.aemo.com.au
(accessed on 03 January 2019).
4. RoCoF Alternative Solutions Technology Assessment High Level Assessment of Frequency Measurement
and FFR Type Technologies and the Relation with the Present Status for the Reliable Detection of High
Rocof Events in a Adequate Time Frame. 2015. Available online: www.dnvgl.com (accessed on 03 January
5. Brogan, P.V.; Best, R.J.; Morrow, D.J.; McKinley, K.; Kubik, M.L. Effect of BESS Response on Frequency
and RoCoF During Underfrequency Transients. IEEE Trans. Power Syst. 2019, 34, 575–583,
6. Tielens, P.; Van Hertem, D. Receding Horizon Control of Wind Power to Provide Frequency Regulation.
IEEE Trans. Power Syst. 2017, 32, 2663–2672, doi:10.1109/TPWRS.2016.2626118.
7. Delille, G.; Francois, B.; Malarange, G. Dynamic Frequency Control Support by Energy Storage to Reduce
the Impact of Wind and Solar Generation on Isolated Power System’s Inertia. IEEE Trans. Sustain. Energy
2012, 3, 931–939, doi:10.1109/TSTE.2012.2205025.
8. Cheng, M.; Sami, S.S.; Wu, J. Benefits of Using Virtual Energy Storage System for Power System Frequency
Response. Appl. Energy 2017, 194, 376–385, doi:10.1016/j.apenergy.2016.06.113.
9. Knap, V.; Chaudhary, S.K.; Stroe, D.-I.; Swierczynski, M.; Craciun, B.-I.; Teodorescu, R. Sizing of an
Energy Storage System for Grid Inertial Response and Primary Frequency Reserve. IEEE Trans. Power Syst.
2016, 31, 3447–3456, doi:10.1109/TPWRS.2015.2503565.
10. Mercier, P.; Cherkaoui, R.; Oudalov, A. Optimizing a Battery Energy Storage System for Frequency
Control Application in an Isolated Power System. IEEE Trans. Power Syst. 2009, 24, 1469–1477,
11. Oudalov, A.; Chartouni, D.; Ohler, C. Optimizing a Battery Energy Storage System for Primary Frequency
Control. IEEE Trans. Power Syst. 2007, 22, 1259–1266, doi:10.1109/TPWRS.2007.901459.
12. Aghamohammadi, M.R.; Abdolahinia, H. A New Approach for Optimal Sizing of Battery Energy Storage
System for Primary Frequency Control of Islanded Microgrid. Int. J. Electr. Power Energy Syst. 2014, 54, 325–
13. Ahmadi, H.; Ghasemi, H. Security-Constrained Unit Commitment with Linearized System Frequency
Limit Constraints. IEEE Trans. Power Syst. 2014, 29, 1536–1545, doi:10.1109/TPWRS.2014.2297997.
14. Chavez, H.; Baldick, R.; Sharma, S. Governor Rate-Constrained OPF for Primary Frequency Control
Adequacy. IEEE Trans. Power Syst. 2014, 29, 1473–1480, doi:10.1109/TPWRS.2014.2298838.
15. Teng, F.; Trovato, V.; Strbac, G. Stochastic Scheduling with Inertia-Dependent Fast Frequency Response
Requirements. IEEE Trans. Power Syst. 2016, 31, 1557–1566, doi:10.1109/TPWRS.2015.2434837.
16. Wen, Y.; Li, W.; Huang, G.; Liu, X. Frequency Dynamics Constrained Unit Commitment with Battery
Energy Storage. IEEE Trans. Power Syst. 2016, 31, 5115–5125, doi:10.1109/TPWRS.2016.2521882.
17. Zheng, Q.P.; Wang, J.; Liu, A.L. Stochastic Optimization for Unit Commitment—A Review. IEEE Trans.
Power Syst. 2015, 30, 1913–1924, doi:10.1109/TPWRS.2014.2355204.
18. Tahanan, M.; van Ackooij, W.; Frangioni, A.; Lacalandra, F. Large-scale Unit Commitment under
uncertainty. 4OR 2015, 13, 115–171, doi:10.1007/s10288-014-0279-y.
19. Pagnoncelli, B.K.; Ahmed, · S; Shapiro, · A; Ahmed, S.; Shapiro, A. Sample Average Approximation
Method for Chance Constrained Programming: Theory and Applications. J. Optim. Theory Appl. 2009, 142,
20. Ahmed, S.; Shapiro, A.; Stewart, H.M. Solving Chance-Constrained Stochastic Programs via Sampling and
Integer Programming. INFORMS 2008, 261–269, doi:10.1287/educ.1080.0048.
21. Wang, Q.; Guan, Y.; Wang, J. A Chance-Constrained Two-Stage Stochastic Program for Unit Commitment
with Uncertain Wind Power Output. IEEE Trans. Power Syst. 2012, 27, 206–215,
Energies 2018, 11, x FOR PEER REVIEW 22 of 22
22. Zhao, C.; Wang, Q.; Wang, J.; Guan, Y. Expected Value and Chance Constrained Stochastic Unit
Commitment Ensuring Wind Power Utilization. IEEE Trans. Power Syst. 2014, 29, 2696–2705,
23. Pozo, D.; Contreras, J. A Chance-Constrained Unit Commitment With an $n-K$ Security Criterion and
Significant Wind Generation. IEEE Trans. Power Syst. 2013, 28, 2842–2851,
24. Riesz, J.; Palermo, J. INTERNATIONAL REVIEW OF FREQUENCY CONTROL ADAPTATION
Australian Energy Market Operator; 2016. Available online: www.aemo.com.au (accessed on 03 January
25. Egido, I.; Fernandez-Bernal, F.; Centeno, P.; Rouco, L. Maximum Frequency Deviation Calculation in
Small Isolated Power Systems. IEEE Trans. Power Syst. 2009, 24, 1731–1738,
26. Nguyen-Hong, N.; Nguyen-Duc, H.; Nakanishi, Y. Optimal Sizing of Energy Storage Devices in Isolated
Wind-Diesel Systems Considering Load Growth Uncertainty. IEEE Trans. Ind. Appl. 2018, 54, 1983–1991,
27. Lofberg, J. YALMIP: a toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004
IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508); IEEE; pp. 284–289.
28. An, Y.; Zeng, B. Exploring the Modeling Capacity of Two-Stage Robust Optimization: Variants of Robust
Unit Commitment Model. IEEE Trans. Power Syst. 2015, 30, 109–122, doi: 10.1109/TPWRS.2014.2320880.
29. Zeng, B. Solving Two-stage Robust Optimization Problems by A Constraint-and-Column Generation
Method; 2011. Available online: http://www.optimization-online.org/DB_FILE/2011/06/3065.pdf (accessed
on 19 February 2019).
30. Zhao, L.; Zeng, B. An Exact Algorithm for Two-stage Robust Optimization with Mixed Integer Recourse
Problems; 2012. Available online: https://pdfs.semanticscholar.org/6b15/8459656b321a0791cc9df4a09af
161c8f5e2.pdf (accessed on 19 February 2019)
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