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Optimal Scheduling of an Isolated Wind-Diesel-Energy Storage System Considering Fast Frequency Response and Forecast Error

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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 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.
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Energies 2018, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies
Article
1
Optimal Scheduling of an Isolated Wind–Diesel–2
Energy Storage System Considering Fast Frequency
3
Response and Forecast Error
4
Nhung Nguyen Hong 1*, Yosuke Nakanishi 2
5
1 Graduate School of Environment and Energy Engineering, Waseda University, Tokyo, Japan;
6
hong.nhung.nguyen1025@gmail.com
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2 Graduate School of Environment and Energy Engineering, Waseda University, Tokyo, Japan;
8
nakanishi-yosuke@waseda.jp
9
* Correspondence: hong.nhung.nguyen1025@gmail.com; Tel.: +81-803-462-2888
10
Received: date; Accepted: date; Published: date
11
Abstract: Nowadays, the hybrid wind–diesel system is widely used on small islands. However,
12
the operation of these systems faces a major challenge in frequency control due to their small
13
inertia. Furthermore, it is also difficult to maintain the power balance when both the wind power
14
and load are uncertain. To solve these problems, energy storage systems (ESS) are usually
15
installed. This paper demonstrates the effectiveness of using ESS to provide Fast Frequency
16
Response (FFR) to ensure that the frequency criteria are met after the sudden loss of a generator.
17
An optimal day-ahead scheduling problem is implemented to simultaneously minimize the
18
operating cost of the system, take full advantage of the available wind power, and ensure that the
19
ESS has enough energy to provide FFR when the wind power and demand are uncertain. The
20
optimization problem is formulated in terms of two-stage chance-constrained programming and
21
solved using a Modified Sample Average Approximation (MSAA) algorithm, a combination of the
22
traditional Sample Average Approximation (SAA) algorithm and the k-means approach. The
23
proposed method is tested with a realistic islanded power system, and the effects of the ESS size
24
and its response time is analyzed. Results indicate that the proposed model should perform well
25
under real-world conditions.
26
Keywords: chance-constrained programming; day-ahead scheduling; energy storage system; fast
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frequency response; wind power
28
29
Nomenclature
30
Indices and Sets
31 Diesel generators
32 Time intervals (of variable duration)
33 Wind generators
34 Random vector
35
Constants
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)
42
Energies 2018, 11, x FOR PEER REVIEW 2 of 22
 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
55
(minutes)
56
Semi-constants
57 
Forecasted wind power at time t (kW)
58 
Forecast error of wind power at time t (%)
59 
Maximum possible wind power at time t (kW)
60 
Forecasted demand at time t (kW)
61 
Forecast error of demand at time t (%)
62 Actual demand at time t (kW)
63
Variables
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)
69
ON/OFF state of wind turbine at time t (binary)
70 Actual wind power at time t (kW)
71 
Charging state of energy storage system at time t (binary)
72 
, Charge power of energy storage system at time t (kW)
73 
, Discharge power of energy storage system at time t (kW)
74 
Discharge power of energy storage system after a contingency event at time t (kW)
75 
Energy stored in the energy storage system at time t (kWh)
76 
 Initial energy stored in the energy storage system at t = 0 (kWh)
77 
 Energy stored in the energy storage system at the end of the day (t = 24) (kWh)
78
1. Introduction
79
Frequency is an important criterion in the power system's operation and is related to the
80
instantaneous balance between supply and demand. To ensure a stable operation of a power system,
81
the balance between power demand and supply must be kept at all times; the system frequency is
82
only allowed to vary in a tight band around the nominal value. Large frequency disturbances,
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caused by events such as the sudden loss of a generator, lead to serious active power imbalances and
84
may lead to load shedding or partial or complete blackout. Fortunately, immediately after a
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frequency disturbance, the kinetic energy stored in the spinning masses of the generators is released
86
into the power system to preserve the power balance, thereby reducing the rate of frequency change.
87
This process is called the Inertial Response (IR) of the generator. The most crucial frequency control
88
activity is the Primary Frequency Control Response (PFR), which is based on the characteristic of the
89
conventional speed governor. PFR automatically starts in the event of a large frequency deviation to
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Energies 2018, 11, x FOR PEER REVIEW 3 of 22
adjust the power output of generators. Large synchronous generators can provide both IR and PFR,
91
thereby ensuring the frequency stability of the power system.
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In recent years, the penetration of renewable energy sources (RES) such as wind and solar into
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the power system is increasing rapidly. Unlike conventional power plants, which are based on
94
synchronous machines, RES power plants use inverter-based generators that do not have IR.
95
Therefore, the overall power system inertia is reduced by the increasing penetration of RES.
96
Furthermore, due to the stochastic nature of wind and solar irradiation, the reserved capacity for
97
PFR from RES is also uncertain. Both of these factors contribute to the more complicated frequency
98
control problem for power systems containing large fractions of wind and solar generation [1]. In
99
islanded power systems, frequency control and regulation are even more challenging because the
100
primary resources are diesel generators (DGs) with low inertia and limited operating capability.
101
With the increasing penetration of renewable generation and energy storage in power systems,
102
the Fast Frequency Response (FFR) method has been introduced as a measure to improve frequency
103
stability. In the Australian electricity market, FFR is defined as “any type of rapid active power
104
increase or decrease by generation or load, in a timeframe of less than two seconds, to correct
105
supply-demand imbalances and assist with managing frequency” [2]. There have been several
106
studies of FFR encompassing a wide range of technologies [3–6]. In one study [6], it was shown that
107
wind generators (WG) can provide IR for a very short duration (around 10 s). Although this method
108
proved to be useful for frequency regulation, the kinetic energy provided by wind turbines is highly
109
dependent on the wind speed; as a result, insufficient support is delivered in the case of low wind
110
speed. Furthermore, in a low-inertia system, the frequency can drop below the threshold of the
111
Under-Frequency Load Shedding (UFLS) relay within only 1 s, so the response of a WG is not
112
effective. For the case of photovoltaics (PV), another method involves keeping the PV power setpoint
113
below the total available power, at the expense of economic performance [2].
114
With a very short response time, which can be less than 250 ms depending on the technology,
115
energy storage systems (ESS) are able to instantly increase or decrease their power output to
116
counteract a system power imbalance. There have been many previous studies focusing on the
117
support of an ESS in frequency response such as [7–12]. The authors of [9–12] focused on the size of
118
the ESS, whereas [7,8] propose control strategies for an ESS to provide virtual inertia. The results
119
presented in these articles show the effectiveness of using an ESS for frequency response control.
120
An interesting research approach in frequency-constrained operation planning is to include
121
frequency constraints in Unit Commitment (UC) models. The authors of [13] proposed a UC
122
formulation including a frequency limit constraint based on the general-order system frequency
123
response model. In contrast, a first-order model for a governor–prime mover system was used in
124
[14]. However, the UC model in [13,14] did not consider the uncertainty in the available wind power
125
so the results are less reliable for actual operation. Reference [15] develops a sophisticated
126
representation for the frequency dynamics, which includes load damping, but this work did not
127
consider the application of ESS in frequency response. A more recent study [16] developed a UC
128
framework, in which the ESS is considered to provide frequency response. However, the wind
129
power uncertainty in [16] is described by only three scenarios: the central forecast, the upper bound,
130
and the lower bound. Although this approach helps reduce the computational complexity, it may
131
lead to conservative solutions with higher operating cost.
132
In this work, we propose a frequency stability-constrained UC models and apply it for a
133
realistic isolated wind–diesel system on Phu Quy Island, Binh Thuan province, Vietnam. The UC
134
model in this work focuses on FFR. The primary goal of ESS in this model is to compensate for the
135
fluctuation of wind and solar generation and to help increase the energy produced from renewable
136
sources (rather than using curtailment). Besides, this ESS provides FFR in large frequency
137
disturbances, such as loss of a generator, as an ancillary service. The proposed model has practical
138
implications for isolated power systems with a high penetration level of renewable resources.
139
To account for the stochastic nature of wind power and demand, the optimal scheduling in this
140
work is formulated as a two-stage chance-constrained optimization problem. The constraints related
141
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
143
(SAA) algorithm, which involves Monte Carlo simulation to approximate the distribution function
144
of a random vector using N samples [19–23]. Although SAA is simple and convenient, all N samples
145
are considered to have the same probability regardless the true distribution of the random vector, so
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the number of samples must be large to ensure that a feasible solution is found. If there are many
147
uncertain parameters, the size of the optimization problem increases and a significantly longer
148
computing time is required. For this reason, it is necessary to improve the algorithm, which is one of
149
the objectives of the present work.
150
The salient features of the present study include:
151
1. The research focuses on frequency stability-constrained UC models. The ESS, which is
152
employed to keep power balance and take advantage of wind power, is considered to provide
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fast frequency response (FFR) in large frequency disturbances, such as loss of a generator. The
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frequency dynamics is approximated using a first-order representation.
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2. The proposed UC model is based on a two-stage stochastic programming framework which is
156
suitable for the short-term planning of power systems with uncertain sources. The model is
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formulated as a chance constraint problem to allow a certain risk level in the day-ahead
158
scheduling.
159
3. The impact of ESS sizing and its response time on frequency nadir is analyzed.
160
As the computing time required for solving chance-constrained optimization is usually
161
significant, in this paper a Modified Sample Average Approximation (MSAA) method is presented
162
and applied to solve the proposed optimization problem. The combination of SAA and k-means
163
clustering approach is proven to be more effective than the original SAA approach.
164
The rest of the paper is organized as follows: Section 2 demonstrates the application of the ESS
165
for FFR. Section 3 presents the mathematical formulation of the proposed chance-constrained
166
optimization problem to determine the optimal scheduling of the power system. Section 4 presents
167
the MSAA algorithm. The computation results are collected and analyzed in Section 5. Section 6
168
concludes the paper.
169
2. Fast Frequency Response and the Role of the ESS
170
As discussed in section 1, IR plays an important role in maintaining frequency after a generator
171
is lost. This process slows down the change of frequency before the governors fully react to provide
172
PFR. We can evaluate this process using two important criteria including the
173
rate-of-change-of-frequency (RoCoF) and the lowest frequency known as the frequency nadir.
174
Consider, for example, a power system with I generators. If at time t generator j with power
175
output (kW) is lost, the RoCoF immediately after the contingency event is defined as
176 =∆
 =
(1)
177
where is the system inertia (kW.s/Hz) after the loss of generation j and is a function of the
178
inertia of the online generators:
179 = (2)
180
where , , and are the inertia constant, maximum capacity, and ON/OFF state of the
181
remaining generators, respectively; is a binary variable that is equal to 1 if generator i is online
182
and 0 if it is offline.
183
If the inertia of the system is sufficient, the frequency will stop before reaching the threshold of
184
the UFLS relays. However, in an islanded power system, the primary resources are DGs with low
185
inertia constants. Additionally, modern wind turbines are connected to the grid through an
186
electronic power converter, which does not contribute to the system inertia. It is easy to see that the
187
lower the system inertia, the faster the frequency drops. Therefore, even though DGs can adjust their
188
Energies 2018, 11, x FOR PEER REVIEW 5 of 22
power output quickly, it is still difficult to arrest the frequency decline before reaching the minimum
189
allowable frequency.
190
The concept of FFR is applied to solve this problem. Unlike both IR and PFR, which slowly
191
adjust the power based on the frequency deviation, thus arrest the deviation and restore frequency,
192
the objective of FFR is to immediately inject active power into the grid to correct the power
193
imbalance. This is implemented based on wind turbines or ESS that can change their power output
194
almost instantaneously. FFR can be considered as a measure to compensate for the “interval”
195
between IR and PFR, during which the frequency is too low and PFR is still not fully active.
196
Note that FFR cannot completely replace PFR - it is only a support measure while waiting for
197
the DGs to provide PFR. Thus, a sustained FFR duration is not necessary. Report [24] shows that in
198
ERCOT (Texas), this duration is 10 minutes while EirGrid/SONI (Ireland) only requires an 8-seconds
199
FFR.
200
An important requirement for FFR is fast response time, and systems with a higher RoCoF will
201
require a faster response time. Assuming the system has a nominal frequency of 50 Hz and a
202
minimum frequency of 49 Hz, FFR must fully react within 250 ms in the case of a 4 Hz/s RoCoF. The
203
response time depends on not only the detection method but also the type of device used to provide
204
FFR. From [3], the times required to detect contingency and to send the control signal, as well as the
205
reaction time and rise time of the ESS, are summarized in Table 1. It can be seen that, with total
206
response times ranging from 100 ms to 200 ms, the ESS is suitable for providing FFR.
207
Table 1. Response times of various detection methods and types of ESS [3].
208
Contingency event
detection time
Control
signal time
ESS reaction
time + rise
time
Detection
Method
Direct detection 40–60 ms ≈20 ms
RoCoF detection/PMU 40–60 ms ≈20 ms
Local RoCoF/Frequency
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
209
response of the ESS needed to satisfy the frequency criterion, .
210
Using the first-order model for a governor–prime mover presented in [14,25], we arrive at an
211
approximate model of the system’s response after a sudden generation loss of amount and the
212
application of ESS for FFR, as described in Figure 1. After the governor’s dead time , which
213
corresponds to the frequency dead band , the power output of the DGs will change due to the
214
governor’s response with the system ramp rate = 
. On the other hand, a
215
control signal is also sent to the ESS to increase its output from 
,or 
, to 
. The
216
adjustment provided by the ESS is 
=

,
, (Figure 1).
217
218
Energies 2018, 11, x FOR PEER REVIEW 6 of 22
To simplify the model, we make two assumptions:
219
The rise time of ESS is negligible
220
The ESS can fully compensate for the power shortage. This means that the frequency decay
221
stops immediately after the ESS fully responds, at which point the frequency nadir occurs.
222
The time evolution of the system frequency deviation can be described by:
223 ∆()
 =∆()+∆() (3)
224
where ∆() and ∆() describe the additional power provided by DGs (due to the response
225
of the governors) and ESS, respectively. ∆() and ∆() are formulated as follows:
226 ∆()=0 <
()  ≤ 
()   (4)
227
∆()=0  < 

  (5)
228
Using the model presented in Figure 1, we can find the relationship between the adjustment
229
power provided by the ESS and the time when the ESS can fully respond:
230 
=() (6)
231
Considering (4)–(6), the equation (3) can be integrated between =0 and =.
232  ∆()

=1
 −
+1
 ()

233
Assuming that before the contingency event, the system frequency is at the nominal value
234 , we have:
235 =1
+1
1
2()()
236
= 1
+1
1
2

237
= − 1

2
239
238
Reference [14] shows that the duration of is related to the governor dead-band  (Hz) by
240
equation  =
. Besides, noting that the frequency nadir should not be below the
241
predefined threshold , we obtain the frequency nadir’ s requirement as follow.
242  =
 (7)
243
Substituting (6) into (7) and noting that 
=

,
,, we obtain the
244
following constraint:
245 2()2()()


,
,=() (8)
246
Constraint (8) shows that the number of DGs in operation and their power output per hour is
247
limited by the time taken for the ESS to fully react, and this will be used in the optimal scheduling
248
formulation presented in Section 3.
249
250
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251
Figure 1. Application of an energy storage system for fast frequency response.
252
3. Problem Formulation
253
3.1. Wind and Demand Models
254
In the present study, we focus on the optimal schedule of an islanded power system
255
considering FFR. A challenge in this problem is the uncertainty in the expected wind power and
256
demand. The wind power and demand forecast errors will affect the operation of the system, so the
257
optimal scheduling problem must be formulated as a predictive optimization with results expressed
258
as ranges of values that assure reliable operation of the system.
259
Both wind power and demand can be defined as the sum of the forecasted value and the
260
forecasting error:
261

=
+

=
+

(9)
262
The errors 
and 
are assumed to follow a normal distribution with zero mean
263
and the standard deviation
for wind power and for demand. This means the maximum error
264
would be 3
for wind power and 3 for demand in correspondence to the confidence level of
265
99.7%.
266
3.2. The Optimal Scheduling Problem
267
This study implements day-ahead scheduling of the islanded power system including DGs,
268
WG, and ESS with frequency criteria. The operating cost of DGs is minimized when wind power is
269
used as much as possible. With the given forecasted wind power and demand, the operating mode
270
of the DGs including ON/OFF state and power output is determined every hour for a 24-hour time
271
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horizon. The total energy excess or deficiency due to forecasting error is fully compensated by the
272
ESS. Furthermore, the ESS must provide FFR if a generator is tripped. For this reason, the ESS should
273
be scheduled such that it not only has enough energy to discharge but also can recharge to utilize the
274
most energy from wind power.
275
The process of optimal scheduling comprises two stages, as illustrated in Figure 2. In the first
276
stage, the deterministic day-ahead schedule in the commitment and the dispatch of DGs is decided
277
and sent to the grid operator; the solid blue arrows in Figure 2 illustrate this process. The first-stage
278
problem is a day-ahead UC problem that is implemented at least one day before the actual operation
279
date. At that time, the wind and demand values are long-term forecast results with errors. Thus,
280
only the results of the unit commitment (on/off states) and the power output of DGs are fixed. The
281
ESS charge/discharge power and WG’s power output are adjusted after wind power and demand
282
are known with higher accuracy using a very short-term forecast.
283
This two-stage optimal scheduling is formulated as a two-stage chance-constrained
284
optimization model (Figure 3). The constraints in the first stage refer to the deterministic planning,
285
and the second stage ensures that the power balance and frequency criteria after a contingency event
286
are met with a chosen probability.
287
288
289
Figure 2. Schematic illustrating the optimal scheduling problem for an islanded power system.
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291
Figure 3. The two-stage optimization model.
292
3.2.1. Objective Function
293
The objective function to be minimized comprises the first-stage operating cost and the
294
expected value of the second-stage cost:
295 ∑ ∑,+,+,++
,()
,()

 (10)
296
where represents a random vector including wind and demand; 
,() and 
,() are ESS
297
charge/discharge power decided corresponding to the actual values or the very short-term forecast
298
value of .
299
3.2.2. First-Stage Constraints
300
The first stage is characterized by the following constraints:
301
Active power balance constraint. The total active power output from the DGs , the wind
302
plant , and the storage system (
, and 
,) must equal the given forecasted load 
at
303
any time t:
304
++
,
, =
(11)
305
DG operating constraints. The power output of each DG must be in the operating range
306
between  and , which are specified by the manufacturer. The binary variable in
307
constraint (12) is used to keep the DG power output equal to zero if it is shut down. Constraints
308
(13) and (14) describe the minimum uptime  and downtime  limitations of each DG.
309  (12)
310 = (13)
311

 1 (14)
312
Constraint (14) ensures that if at hour t, the generator is offline (=0), the generator cannot
313
start up within the duration from  to (=0 with all values of k from  to ). In
314
contrast, if at hour t, the generator is online (=1) then this constraint ensures that the generator
315
cannot shutdown within the duration from  to (=0 with all values of k from 
316
to ).
317
Primary reserve constraints for DGs. This constraint shows that each DG can take part in the
318
operating reserve.
319
0
+  (15)
320
Energies 2018, 11, x FOR PEER REVIEW 10 of 22
WG operating constraint. For a given forecasted wind power 
, the power output of the WG
321
must satisfy constraint (16), where 
is the minimum wind power output:
322


(16)
323
324
ESS constraints. Constraint (17) states that the charging and discharging power of the ESS must
325
be smaller than the actual power rating of the storage device . The process of
326
charging/discharging the ESS is described by constraint (18). This constraint also imposes that
327
the energy stored in the ESS should be smaller than its rated capacity at all times:
328
0
,

0
, (1−
) (17)
329
0
=
+
,
,  ≤

=
 (18)
330
Frequency nadir limit. As presented in Section 2, to ensure that the frequency does not drop
331
below the minimum allowable level, constraint (19) must be satisfied:
332
2()2()()


,
,=() (19)
333
Note that the system inertia and ramp rate depend on the number of the online DGs.
334
Thus, this constraint determines not only ESS charge/discharge power but also the number of DGs in
335
operation and the power output of each DG before the contingency event to ensure the frequency
336
nadir criteria.
337
Post-contingency energy storage capacity constraint. This constraint ensures that the ESS has
338
enough energy to sustain FFR for ∆.
339 0

∆ (20)
340
3.2.3. Second-Stage Constraints
341
In the second stage of the optimization model, uncertainties in wind power generation and load
342
consumption are considered. The power output of the ESS and the WGs are re-dispatched as
343 
,(), 
,(), and (), where represents a random vector. When a generator is lost, the
344
ESS will discharge 
() to decrease the disturbance to 
()=
()
345 
,()
,(). The second-stage constraints can be summarized as follows:
346
Active power balance constraint.
347 Pr
+()+
,()
,()=
+

 ≥1 −  (21)
348
WG operating constraint.
349
()
()()
()
+

(22)
350
ESS constraints.
351 0
,()
() (23)
352 0
,()1
() (24)
353 0
()=
()+
,()
,() ≤  (25)
354
Frequency nadir limit.
355
Pr2()+
()
,()
,()()1

()
,()
,()=()
356
(26)
357
Energies 2018, 11, x FOR PEER REVIEW 11 of 22
Post-contingency energy storage capacity constraint.
358 0
()
()∆ (27)
359
In the above constraints, constraint (21) guarantees that the ESS and WG will be adjusted so that
360
the probability of power imbalance is less than a risk level . Similarly, constraint (26) is the
361
probability formulation of the constraint (19) and ensures that the frequency criterion will be met
362
after a contingency event with high probability, even if the ESS is re-dispatched due to the difference
363
between the long-term and very short-term forecast results of .
364
4. The Modified Sample Average Approximation
365
Consider a simple two-stage chance-constrained optimization model
366 =min()+(,) (28)
367
subject to
368 Pr((,,)0)1 (29)
369
where is the first-stage variable, is the second-stage variable, and is random input data.
370
Many studies in the literature show that this model can be solved by the SAA method [19–23].
371
In this method, Monte Carlo simulation is used to approximate the distribution function of the
372
random vector ξ by N samples. The optimization formulation (28) then becomes
373 =min()+
(,)
 (30)
374
subject to
375
(,)(,,)1
 (31)
376
where (,)(,,) is an indicator function that is equal to one if (,,)0 and zero
377
otherwise.
378
It is assumed that the N samples have the same probability (1/). This assumption helps to
379
simplify the formulation of the optimization; however, a large number of samples are required to
380
guarantee accuracy, which means the CPU time required to solve it increases accordingly.
381
In the present study, a modified approach to the SAA is proposed, by using a k-means
382
clustering approach to reform the samples. Instead of using all N samples, the k-means clustering
383
divides the samples into M clusters. The probability of each cluster is the sum of the probabilities of
384
the constituent samples. Next, M centroids of the clusters are used as the SAA algorithm input
385
samples, with the probability of each centroid being equal to the probability of the cluster that it
386
represents. Figure 4 illustrates a small example: 1000 samples generated from the standard normal
387
distribution N(0,1) are replaced by 10 centroids.
388
Energies 2018, 11, x FOR PEER REVIEW 12 of 22
389
Figure 4. Example of using k-means clustering to reform samples.
390
For M centroids and their corresponding probabilities, (28) and (29) are reformulated as
391 =min{()+(,)
 } (32)
392
subject to
393
(,)(,,)1
 (33)
394
where is the probability of each centroid ( = 1,2, … ).
395
Now let ̅ and , respectively, be the optimal solution and value of the optimal problem in
396
(32)–(33) and check whether this solution is feasible or not. Using Monte Carlo simulation to
397
generate a new set of ’ samples where is much larger than , we find the value of the
398
probability constraint (31) with solution ̅ is
399 (̅)=Pr((̅,,)0)=
(,)(̅,,)
 (34)
400
The (1–ɛ)-confidence lower bound on (̅) is then computed using
401 (̅)=(̅)Φ(1)(̅)(̅)
(35)
402
where Φis the inverse normal distribution function. ̅ is a feasible solution of the original
403
problem only if (̅)1. Repeat this process K times according to the flow chart illustrated in
404
Figure 5 and find the maximum value
and minimum value of the optimal value . If the
405
optimality gap given by (
)
×100% is smaller than a predetermined threshold, the
406
algorithm terminates, and we obtain the optimal solution of the original problem.
407
Energies 2018, 11, x FOR PEER REVIEW 13 of 22
408
Figure 5. Flow chart of the MSAA algorithm.
409
5. Results and Discussion
410
5.1. Study System
411
The data used in this study is based on an actual power system on Phu Quy Island, Binh Thuan
412
province, Vietnam. This system includes six 500 kW DGs and two WGs, each with a rated power of
413
1.8 MW. The purpose of the ESS installation is to utilize as much wind power as possible and to
414
mitigate the uncertainty in demand and wind speed. The optimal size of an ESS taking into account
415
the relevant wind scenarios and the annual load growth factor was analyzed in our previous work
416
k = 1
Generate N=1000 samples
of ξ
Solve SAA problem
to find
and
Estimate (̅)and
using (33) and (34)
1
k = k + 1
No
Set
=
<No
Yes
̅,̅ , 
k= 1,...,K
Generate N' samples
of ξ(N'>>M)
Initialization
Find maximum value and
minimum value  of
<0.1%
s = s + 1
s = 1
<
Set upp er bound
=
Set lower bound
=−∞
<
Set =
Set =
Yes
Yes
=
No
Yes
STOP
Yes
Find Mcentroids by k-
means clustering Nsampes
Energies 2018, 11, x FOR PEER REVIEW 14 of 22
[26]. The ESS helps balance the fluctuation in wind power, increase the wind penetration level, and
417
also provide FFR as a supplementary service. We assume that the wind power variable costs are zero
418
and the ESS efficiency factor for the charging and discharging process is 90%. The risk level of the
419
probability constraints in the second stage is 5%, which means these constraints should be met with
420
a probability of greater than 95%. The other parameters are presented in Table 2.
421
Table 2. Input data.
422
Diesel generator
=500 kW

= 170 kW
=0.8 s
ESS
=90%


=10%
=90%
Frequency criteria
=50 Hz

= 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
423
maximum possible instantaneous penetration of wind power is approximately 45% during the first
424
hour and highest in the fifth hour (49%). However, this ratio is only 15% when the load is highest at
425
the 19th hour. Forecast errors are modeled using a normal distribution with zero mean. The
426
standard deviations are assumed to be 0.05 for both wind power and demand, which means that the
427
maximum forecasting error in the values each hour can be considered to be 15%.
428
429
Figure 6. Forecasted wind power and demand.
430
In the present study, the following cases are considered to evaluate the effectiveness of using
431
ESS to provide FFR:
432
Case 1a. This case simulates the original UC problem without frequency constraints. The ESS is
433
used to take advantage of wind power and compensate for the uncertainty in wind speed and
434
demand.
435
Case 1b. The original UC problem with frequency constraints. The ESS is not used to provide
436
FFR.
437
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Case 2. UC problem with frequency constraints. The ESS is used to provide FFR.
438
Besides comparing the frequency criteria  after a contingency event in these cases, the
439
effects of other parameters such as ESS size are also considered. The optimization problem is solved
440
using the MSAA approach presented in Section 4 with CPLEX version 12.6 and the YALMIP toolbox
441
[27].
442
5.2. Optimization Results
443
5.2.1. Case 1a: Original Optimal Scheduling Model Without Frequency Criteria
444
With an ESS rating of 400 kW/800 kWh and the other input data given in Section 5.1, the
445
optimal daily schedule for DGs in Case 1 is presented in Figure 7a. Figure 7b shows the scheduled
446
operation of DGs, wind generators and the ESS. Note that the DGs operating schedules are first stage
447
variables. The ESS charging/discharging power and wind power are second stage variables,
448
considering the uncertain nature of wind speed and load forecast error. Therefore, they are
449
represented using box plots. It can be seen that, although the possible wind capacity and demand are
450
uncertain, the available wind power is still fully utilized most of the time; this is undoubtedly due to
451
the involvement of the ESS in the grid.
452
The RoCoF immediately after a contingency event in which the DG having the largest power
453
output is lost is presented in Figure 8. During the period from the 17th to the 20th hour, demand is at
454
its highest, whereas available wind power is quite low, so four DGs are online. This means that the
455
stored kinetic energy in this period is higher than that in the rest of the day. However, the RoCoF is
456
still approximately 10 Hz/s. Although DGs can increase their power output very quickly, even from
457
a cold start condition (10–15 s), the frequency declines rapidly to below the minimum threshold. This
458
can be explained by the fact that the inertia constant of the DGs being small (H = 0.8).
459
460
461
Figure 7. Case 1a (frequency criteria not considered): (a) optimal daily schedule for the DGs; (b) box
462
plots of wind power and ESS power, DG power and forecasted demand.
463
Energies 2018, 11, x FOR PEER REVIEW 16 of 22
464
Figure 8. RoCoF after a contingency event in Case 1a.
465
5.2.2. Case 1b: Optimal Scheduling Problem Considering Frequency Criteria
466
Accounting for the frequency criteria in the optimal scheduling problem, as many DGs as
467
possible are kept online while keeping their power output at a low level (Figure 9). For example,
468
during peak load hours, there are six DGs in operation even though only four DGs are needed in
469
Case 1a. This operating strategy helps increase the inertia of the system and reduce the RoCoF. The
470
reduction of RoCoF can be seen by comparing the results in Figure 8 and Figure 10. However, the
471
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
473
(Table 3).
474
475
Figure 9. Case 1b results (frequency criteria considered): (a) optimal daily schedule for the DGs;
476
(b) box plots of wind power, ESS power, DG power and forecasted demand.
477
478
Figure 10. RoCoF and frequency nadir for Case 1b.
479
Energies 2018, 11, x FOR PEER REVIEW 17 of 22
Table 3. Comparison of optimal costs between Cases 1 and 2.
480
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
481
ESS
482
As in Cases 1a and 1b, an ESS rated at 400 kW/800 kWh is used in this case. However, the ESS is
483
not only used to maintain the power balance but also to provide FFR. The optimal schedule of the
484
DGs and box plots of the ESS and wind power for this case are presented in Figure 11.
485
As discussed in Section 2, if the DG with the largest power output is lost, the frequency
486
deviation will activate the ESS response, and here we assume that the total response time for FFR is
487
100 ms. Based on the state of the ESS before the contingency event, the charge/discharge power of
488
the ESS post-contingency at each hour can be in a range, as shown by the box plot in Figure 12. The
489
frequency nadir values for each hour are obtained using Equation (6), as shown in Figure 13.
490
491
Figure 11. Case 2 results: (a) optimal daily schedule for the DGs; (b) box plots of wind power and ESS
492
power, DG power and forecasted demand.
493
494
Figure 12. Box plot of the ESS charge/discharge power after a contingency event for Case 2.
495
Energies 2018, 11, x FOR PEER REVIEW 18 of 22
496
Figure 13. Frequency nadir with ESS providing FFR for Case 2.
497
To evaluate the effect of the ESS on FFR, we compare the RoCoF immediately after the
498
contingency event for Cases 2 and 1b. Figure 14 shows that the RoCoF in Case 2 is higher than in
499
Case 1b in a few hours. However, the frequency nadir in Case 2 is ensured, while it is violated in
500
several hours in Case 1b.
501
Note that constraints (18) and (25) in the optimization formulation limit the power output of
502
each DG. This explains why the number of hours with six DGs in operation in Case 2 is more than
503
that in Case 1b. In contrast, the ESS support in Case 2 helps to ensure the frequency criteria after
504
contingency events, even when the number of online DGs is less than in Case 1, thus ensuring
505
maximum utilization of the available wind power. This can be seen by comparing the box plots of
506
wind power in Figure 9b and Figure 11b. It is interesting to note that, when the ESS is able to provide
507
FFR, the UC solution will reduce DGs uptime and increase the wind power/ESS output, which in
508
turn reduces the operating cost. The optimal cost of Case 2 is smaller than that of Case 1b and is not
509
significantly higher than the non-constrained optimal cost (Table 3).
510
511
Figure 14. Comparison of RoCoF between Cases 1b and 2.
512
5.2.4. Impact of ESS Size and Response Time
513
In this section, we consider the effect of the ESS size and the total response time from the
514
moment the contingency event occurs until the ESS fully responds. Table 4 shows the smallest
515
possible value of the frequency nadir for two total response times, 100 ms and 200 ms, and several
516
ESS sizes, which are defined by the power rating  and the full charge/discharge duration
517  
(from 0.5 h to 4 h). It can be seen that the ESS size must be larger than 200 kW/400
518
kWh to ensure the problem has a feasible solution. It should also be noted that the forecast errors are
519
assumed to be ±15%, so a too small ESS will not be able to compensate for the mismatch between the
520
predicted and actual values of the load and wind power. However, even if the optimization problem
521
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
522
the frequency nadir is lower than the minimum threshold (these values are shown in red in Table 4).
523
The reason for this is that the frequency nadir constraint (26) is formulated as a probabilistic
524
constraint with a risk level of 5%. On the other hand, constraint (26) shows that a longer response
525
time requires a lower power output from each DG or more DGs in operation to provide enough
526
kinetic energy; consequently, increasing the ESS power rating is necessary. It can be seen from Table
527
4 that when the response time is 200 ms, the power rating of the ESS must be larger than 600 kW to
528
maintain the frequency nadir above 49.2 Hz, whereas an ESS with rated power 400 kW is acceptable
529
if the response time is 100 ms.
530
Table 4. Summary of frequency nadir for different ESS sizes and response times.
531
Response time ESS Parameters Lowest possible value of frequency nadir (Hz)
100 ms

200

400

600

800

1000

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
200 ms

200

400

600

800

1000

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
x: Infeasible
5.2.5. Comparison Between the MSAA, SAA and the Robust chance-constrained algorithm
532
Case 2 was solved by the traditional SAA algorithm and the MSAA algorithm to compare their
533
computational efficiency. To solve Case 2, the SAA algorithm must be repeated at least 50 times per
534
loop (K = 50) and needs at least 100 samples per loop (N = 100). On the other hand, the original set of
535
1000 samples can be replaced with five centroids, and the MSAA algorithm needs 50 iterations to
536
obtain the results. Interestingly, five centroids in the MSAA algorithm are equivalent to five samples
537
in the SAA algorithm; thus, it is easy to see that the computing time required for the MSAA
538
algorithm is much smaller than that of the SAA algorithm (Table 5).
539
The performance of the proposed MSAA is also compared with that of the Robust
540
chance-constrained formulation, which is also a popular approach. In this comparison, a two-stage
541
Energies 2018, 11, x FOR PEER REVIEW 20 of 22
robust chance-constrained model, solved by column-and-constraint generation algorithm (CCG)
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[28–30], is implemented. The constraints related to power balance and frequency criteria are also
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formulated as probability constraints with the same risk level. Besides, the results obtained with a
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two-stage robust model is also shown. The results in Table 5 clearly demonstrate the compromise
545
between CPU time and economic performance: although the required CPU time for MSAA is longer
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than the robust method, the optimal cost obtained by MSAA is significantly lower than both robust
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models – with or without chance constraints.
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Table 5. Comparison of computing time between the MSAA, SAA and the robust method.
549
Method N M N’ K S CPU time
(seconds)
Optimal cost
(USD)
SAA 100 1000 50 1 2950 11154
MSAA 1000 5 1000 50 1 853 11120
Robust chance
constrained UC 8 12196
Robust UC 8 14435
6. Conclusions
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In this paper, an optimal day-ahead scheduling problem concerning the application of ESS for
551
FFR is considered and analyzed in detail. The optimization problem is formulated within a
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two-stage chance-constrained framework, in which the load and the maximum possible wind power
553
are uncertain. In this model, power balance and frequency criteria constraints are formulated as
554
probability constraints with a certain risk level. Based on the first-order model of frequency
555
dynamic, the relationships between the power output of each DG, the ESS charge/discharge power,
556
and the response time are studied. The impact of the size and response time of the ESS on the
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frequency nadir after the sudden loss of a DG is also analyzed. It is also noteworthy that an MSAA
558
approach was proposed in the present study to solve a chance-constrained problem, and the
559
effectiveness of this method was demonstrated.
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The results obtained in two cases—with and without FFR provided by ESS—demonstrate the
561
effectiveness of FFR in arresting frequency deviations after a contingency event. The proposed
562
method ensures that the minimum frequency threshold is not violated, even when the actual values
563
of wind power and demand are different from the predicted values incorporating the predetermined
564
maximum errors. The results also show that a slower FFR will lead to a larger ESS to ensure
565
frequency criteria.
566
The proposed approach can be extended to consider multiple contingencies such as line outages
567
or load interruptions as well as equipment failures. The model can also be readily adapted to include
568
other uncertain factors, such as solar power generation or electricity prices. These topics are left for
569
future work.
570
Acknowledgments: This research was supported by Japan Science and Technology Agency as part of the
571
e-ASIA Joint Research Program (e-ASIA JRP).
572
Author Contributions: Nhung Nguyen Hong conceived the methodology, developed the theory, and
573
performed the computations under the guidance of Prof. Yosuke Nakanishi. The results were discussed by all
574
authors, and the final manuscript was written with contributions from all authors.
575
Conflicts of Interest: The authors declare no conflict of interest.
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Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chance-constrained problems. First, checking feasibility of a given candidate solution exactly is, in general, impossible because this requires evaluating quantiles of random functions. Second, the feasible region induced by chance constraints is, in general, nonconvex, leading to severe optimization challenges. In this tutorial, we discuss an approach based on solving approximating problems using Monte Carlo samples of the random data. This scheme can be used to yield both feasible solutions and statistical optimality bounds with high confidence using modest sample sizes. The approximating problem is itself a chance-constrained problem, albeit with a finite distribution of modest support, and is an NP-hard combinatorial optimization problem. We adopt integer-programming-based methods for its solution. In particular, we discuss a family valid inequalities for a integer programming formulations for a special but large class of chance-constrained problems that have demonstrated significant computational advantages.