H ∞ Control of Microgrids Involving Gas Turbine Engines and Batteries
Masaaki Nagahara, Yutaka Yamamoto, Seiya Miyazaki, Takahiro Kudoh, and Naoki Hayashi

Abstract— This paper proposes a new power management
control method for microgrids based on H ∞ control theory.
Microgrid systems consist of distributed power sources such ascogeneration systems and photovoltaic systems with batteries.
In general, power generation by photovoltaic systems and power
consumption of various loads cause large fluctuations depending
on weather conditions and varied lifestyles. Moreover, efficientmanagement of battery capacity is a critical issue to prolong
the battery life. Therefore, in power management control ofmicrogrids, it is necessary to take account of a number of
aspects such as power balancing performance, maintaining
battery capacity, and robustness against power fluctuations.
For such multiobjective control problems, we apply H ∞ control
theory which offers a unified robust control method withdesirable power balancing performance and efficient battery
management. The experimental results illustrate the effectiveness of the proposed control method.

I. I NTRODUCTION
Tackling global warming is the most urgent issue on
all levels of scientific and engineering fields. The IEA(International Energy Agency) reported that about 41[%] of
current energy-related CO2 emissions comes from the power
sector [1]. With the growing attention for the realization ofsustainable development, smart grids are expected to play acentral role to decarbonize the power sector while improving network efficiency and reliability [2], [3]. Microgrids
are small-scale smart grids that can operate independently
from commercial utility grids as electrically isolated power
networks [4], [5]. In the most common configuration, amicrogrid is connected to a utility grid at a single point. The
point that connects a microgrid and a utility grid is calleda point of common coupling (PCC). The power consumed
in a microgrid is mainly supplied by distributed powersources such as photovoltaic systems, storage batteries, andcogeneration systems. In addition, a microgrid can also be
supplied power from a utility grid through a PCC whenneeded. The supplied power from a utility grid at a PCC ispredetermined according to a demand prediction. If there isa considerable difference between the predetermined power
and the actually supplied power from a utility grid, power
M. Nagahara is with Department of Applied Analysis and Complex
Dynamical Systems, Graduate School of Informatics, Kyoto University,
Japan. nagahara@i.kyoto-u.ac.jp
Y. Yamamoto is with Department of Applied Analysis and Complex
Dynamical Systems, Graduate School of Informatics, Kyoto University,
Japan. yy@i.kyoto-u.ac.jp
S. Miyazaki is with Energy Solutions Development Center, Panasonic
Corporation, Japan. miyazaki.seiya@jp.panasonic.com
T. Kudoh is with Energy Solutions Development Center, Panasonic
Corporation, Japan. kudoh.takahiro@jp.panasonic.com
N. Hayashi is with Division of Electrical, Electronic and Information
Engineering, Graduate School of Engineering, Osaka University, Japan.

n.hayashi@eei.eng.osaka-u.ac.jp

flow in the entire network including the microgrid and theutility grid can become unstable. Therefore, power balancing
control that adjusts power flow at a PCC to a predetermined
set point as close as possible is essential to stabilize the entire
network.
The main problem encountered in power balancing control
is that power generation by distributed power sources andpower consumption of various loads are difficult to control
and cause large fluctuations of power flow in microgrids.
Moreover, such fluctuations are unpredictable because of
sudden changes of weather conditions and varied lifestyles.
Therefore, in power control of microgrids, it is important to
reduce power fluctuations and achieve power balancing by
appropriately controlling batteries and cogeneration systems.
In this paper, we propose power balancing control basedon H ∞ control theory by regarding power fluctuations as
disturbances added to microgrids. H ∞ control theory has beendeveloped to provide a quantitative way of simultaneously
mitigating the effect by uncertainty of parameters and stabilizing control systems [6]. The objective of the proposed
H ∞ control method is to achieve robust power balancing evenunder unpredictable power fluctuations.
Another problem of power balancing control is how toassign power generation among distributed power sources.
For example, storage batteries can track relatively highfrequency fluctuations but they cannot provide enough power
used in the entire microgrid because of their limited capacities. On the other hand, gas turbine engines can supply
more power than batteries but they generally have a slow
response. In this paper, we take account of these operational
characteristics of batteries and gas turbine engines. Moreprecisely, we separate power fluctuations in a microgrid
into two subsignals so that gas turbine engines can control
lower frequency fluctuations and batteries can control higher
frequency fluctuations. However, tracking for high-frequency
fluctuations causes large variation of state-of-charge (SOC)of batteries and this results in shorter battery lifetime. In thispaper, we take battery capacity into account when designing
controllers for batteries to reduce such variation of SOC.
There have been proposed a number of power balancing
control methods [7]. Yoshimoto and co-workers proposed the
SOC feedback control to regulate power flow in microgrids
and keep a battery capacity within a certain range [8].
Bolognani and Zampieri considered distributed control ofreactive power in microgrids [9]. They proposed a randomized gossip-like algorithm to solve a convex optimization
problem of reactive power compensation. Kim and coworkers proposed PI (Proportional-Integral) control strategy
with two layered control architecture for power balancing

PCC (Point of Common
Coupling)[10]. Guo and Lee proposed PI-based control of regulating
power flow in microgrids with fuel-cell and micro-turbine
engines [11]. Although the PI-based control approach isdemonstrated to achieve desirable power control, it requires
repetitive parameter tuning and it is difficult to evaluate theeffect of parameter variation in a systematic way. To remedy
such a drawback, H ∞ control approach has been considered
in a control problem of regulating inverters in microgrids.
Yang and co-workers considered H ∞ control scheme that
achieves desirable tracking performance on an output voltage of threephase inverters under parameter variation [12].
However, the objective considered in [12] is different fromours in that we consider power balancing control of the entire
microgrid system instead of just voltage control of specificinverters.
The paper is organized as follows: Section II presents
a model of a microgrid that consists of photovoltaic panels, batteries, and gas turbine engines as distributed powersources. In Section III, we formulate a power balancing
problem. Then we propose power balancing control basedon H ∞ control theory in Section IV. Section V shows numerical examples for the proposed power balancing control.
In Section VI, we present some experimental results on a
microgrid testbed. Finally, we give a conclusion in Section
VII.

In this paper, we consider a microgrid that consists ofphotovoltaic panels, batteries, and gas turbine engines as
distributed power sources. The microgrid is connected to autility grid at a PCC as shown in Fig. 1. A difficulty in power
balancing control of microgrids is to smooth out fluctuations
caused by photovoltaic systems and various loads. Figs. 2
and 3 show typical examples of the photovoltaic power
generation Ppv and the load power consumption Pld over
24 hours, respectively. As observed in these figures, power
generation by photovoltaic systems and power consumptionof loads cause large fluctuations in power flow of microgrids.
To overcome such power fluctuations, we propose power balancing control based on H ∞ control theory by regarding thepower fluctuations as disturbances added to the microgrid.
In this paper, we assume that the battery Gbt and the gas
turbine engine Gge of the microgrid can be modeled as firstorder systems (see e.g., [13]):1
0.1, Tbt =,
Tbt s + 1
2π
1
10Gge (s) =, Tge =.
Tge s + 1
2π
Gbt (s) =(1)(2)

Note that the time constant of the battery is 100 times fasterthan that of the gas turbine engine. This reflects the factthat batteries can track higher frequency fluctuations but gas
turbine engines can only track lower frequency fluctuations
because of their slow responses. In this paper, we takeaccount of these operational characteristics of batteries andgas turbine engines when designing controllers in Section
IV.

Power from Utility Grid

Microgrid System
Photovoltaic
PanelsPower Balancing
ControlPower from Distributed
Power Sources

Batteries

Loads
Gas Turbine
EnginesFig. 1.

Configuration of a microgrid system.
Photovoltaics

7000
6000
5000

Ppv [W]

II. M ODEL OF M ICROGRIDS

Utility Grid

4000
3000
2000
1000
0
0

1

2

Fig. 2.

3

4
5
time [sec]

6

7

8

9

4

x 10

Photovoltaic power generation Ppv .

III. P ROBLEM S ETTINGSIn this section, we formulate a power balancing problemof the microgrid modeled in the previous section. The power
supplied to the microgrid from a utility grid at a PCC ispredetermined according to a demand prediction. In thispaper, we refer to the predetermined power supplied from autility grid as the reference power. The goal of the power
balancing problem is to appropriately regulate distributed
power sources in order to track the reference power evenunder large power fluctuations.
Let Ppv (t), Pbt (t), and Pge (t) be the power generation of thephotovoltaic system, the battery, and the gas turbine engine
at time t (≥ 0). Note that Ppv (t) and Pge (t) have always
nonnegative real values for all t ≥ 0, and
Pbt (t) ≥ 0 if batteries are discharging,
Pbt (t) < 0 if batteries are charging.

Load
8000

Flow

r1
+ −

Kge

Gge

Pge

r

7000

+ y
+

Pbt

Pload [W]

6000

Fhigh

5000

r2
+ −

Gbt
Kbt

1
s

P̃bt

4000
3000
2000

Fig. 4.

Block diagram of the proposed microgrid system.

1000
0
0

each time t (≥ 0):
1

2

Fig. 3.

3

4
5
time [sec]

6

7

8

9
4

x 10

Load power consumption Pld .

Then the supplied power from a utility grid Pnet (t) is equal
to the difference between the power consumption of loads
Pld (t) and the total power generation of the distributed powersources; that is,
Pnet (t) = Pld (t) − Pbt (t) − Pge (t) − Ppv (t),t ≥ 0.

t ≥ 0.

(3)

IV. H ∞ C ONTROL OF BATTERIES AND G AS T URBINEE NGINES
In this section, we design controllers that stabilize andbalance power flow in the microgrid. Fig. 4 shows the block
diagram of the proposed microgrid system, wherer(t) := Pld (t) − Ppv (t) − Pref (t),y(t) := Pbt (t) + Pge (t),Tge + Tbt
2
be the common time constant for the filters. In this paper,
we give Flow and Fhigh as
Tc :=If Pnet (t) > 0, distributed power sources cannot supply
enough power to loads and the necessary power Pnet (t) is
supplied from a utility grid. On the other hand, Pnet (t) < 0means that the power demand of loads is less than the
total power generation of distributed power sources and theredundant power is sold to a utility grid. Then the power
balancing problem can be summarized as follows:
Problem 1 (Power Balancing Problem): Given the reference power Pref , find the power generation of the battery Pbt
and the gas turbine engine Pge to minimize the tracking error(3) at each time t (≥ 0):e(t) = |Pref (t) − Pnet (t)| ,e(t) = |r(t) − y(t)| , t ≥ 0.
(4)
As shown in Figs. 2 and 3, the input signal r contains
various frequency components. In general, batteries cantrack high-frequency fluctuations but their storage capacity
is limited. On the other hand, gas turbine engines can supply
more power than batteries but they can track only relatively
lower frequency fluctuations. Considering such operational
characteristics of batteries and gas turbine engines, we separate the signal r into two subsignals using the low-pass filter
Flow and the high-pass filter Fhigh . Lett ≥ 0.

In Fig. 4, Kbt and Kge are controllers for the battery and thegas turbine engine, and Gbt and Gge represent plant models
for the battery (1) and the gas turbine engine (2).
Then the power balancing problem can be restated asfollows:
Problem 2: Given the input signal r, find the controllers
Kbt and Kge in order to minimize the tracking error (4) at

1
Tc s, Fhigh (s) =.
Tc s + 1
Tc s + 1
Fig. 5 shows the Bode gain plots of the low-pass filter
Flow and the high-pass filter Fhigh . Then r1 contains lower
frequency fluctuations that are regulated by the gas turbine
engine and r2 contains higher frequency fluctuations that are
regulated by the battery.
In addition to power fluctuations, battery life is of great
concern in microgrids because of the high costs of battery
replacement. In this paper, we consider a battery capacity
to reduce variation of SOC. The battery capacity P̃bt iscomputed by the integral of the charged/discharged power
Pbt . In the proposed microgrid system, P̃bt is modeled byusing an integrator 1/s as shown in Fig. 4.
The controllers Kbt and Kge are designed as H ∞ controllers. Fig. 6 shows the block diagram for designing thecontroller Kge . The weights Wge1 , Wge2 , and Wge3 are theweighting functions for tracking performance, gain margin of
the microgrid system, and robustness for power fluctuations.
For example, we need to choose Wge1 to have a higher
gain in an appropriate bandwidth to achieve better tracking
performance. In this paper, we set these weights as
1
10.005s,, Wge2 (s) =Wge1 (s) = · 10s 2π s + 1
0.1s + 1
Flow (s) =5
sWge3 (s) = 0.01 5 2π .
s
+
1
2π

Bode Plot
200
Wge1
Wge2
Wge3

150

Magnitude [dB]

100

50

0

−50

−100

−150

−200

−5

10

Fig. 5. Bode gain plots of the low-pass filter Flow (solid line) and thehigh-pass filter Fhigh (dashed line).

Wge1
Wge2
r

Flow

Fig. 6.

Kge

−

Gge

0

5

10
Frequency [rad/sec]

10

Fig. 7. Bode gain plots of the weights Wge1 (solid line), Wge2 (dashed line),and Wge3 (dashed-dotted line).

z1

Wbt1

z2

Wbt2

Wge3

z1
z2
z3

Wbt3

z3
r

Fhigh

−

Kbt

Gbt

1
s

Wbt4

z4

Block diagram for the controller Kge .

The Bode gain plots of the weights are shown in Fig. 7. Thenthe controller Kge can be obtained by solving the following
H ∞ problem.
Problem 3 (H ∞ Problem for Controller Kge ): Find
thecontroller Kge that minimizes the H ∞ norm of the systemfrom r to z = [z1 z2 z3 ] .
Note that Problem 3 is a standard H ∞ problem and we
can find the controller Kge using Robust Control Toolbox of
MATLAB.
The block diagram for designing the controller Kbt isshown in Fig. 8. We set the weights of the block diagram as
1,s
2π + 1

Wbt1 (s) = 0.5

Wbt2 (s) = 0.001,

0.5s
Wbt3 (s) = 0.01 0.52π,
2π s + 1

Wbt4 (s) = 1.

The purposes of the weights Wbt1 , Wbt2 , and Wbt3 are the sameas those of the weights Wge1 , Wge2 , and Wge3 , respectively.
The weight Wbt4 is for the battery capacity. The higher thegain of the weight Wbt4 is, the more likely we must consider
variation of the battery capacity. Fig. 9 shows the Bode gainplots of these weights. The controller Kbt can be obtained bysolving the following H ∞ problem:

Fig. 8.

Block diagram for the controller Kbt .

Problem 4 (H ∞ Problem for Controller Kbt ): Find
thecontroller Kbt that minimizes the H ∞ norm of the systemfrom r to z = [z1 z2 z3 z4 ] .
Problem 4 is also a standard H ∞ problem and solved by
Robust Control Toolbox of MATLAB.
V. S IMULATIONIn this section, we show some numerical examples of the
proposed power balancing control.
Fig. 10 shows the Bode plots of the H ∞ -optimal controllers
Kge and Kbt obtained by solving Problems 3 and 4 with
MATLAB. Note that the controller Kge has a pole at s = 0
since the weight Wge1 (s) has a pole there. On the other hand,the controller Kbt has a higher gain in a high frequency domain. This shows that low-frequency fluctuations are mainly
controlled by the gas turbine engine while high-frequency
fluctuations are controlled by the battery.
To show the effectiveness of the proposed method to thebattery life, we also design a controller by solving the H ∞control problem of the system from r to z = [z1 z2 z3 ] .
This controller does not take account of the state of capacity

Bode Plot

Pnet (with SOC control)
2.5
Pnet
Pref

0
−20

2

−40
1.5
Power [kW]Magnitude [dB]

−60
−80
−100

1

−120
0.5
−140
−160

Wbt1
Wbt2
Wbt3
Wbt4

−180
−200

−5

10

0

0

−0.5

5

10
Frequency [rad/sec]

100Fig. 9. Bode gain plots of the weights Wbt1 (solid line), Wbt2 (dashed line),
Wbt3 (dashed-dotted line), and Wbt4 (dotted line).

2

Fig. 11.

4

6Time [hour]

8

10

12

Pnet for 12 hours with SOC control
Pnet (without SOC control)

2.5

Bode Plot

Pnet
Pref

200
Kge
Kbt(1)
Kbt(2)

150

2

1.5Power [kW]Magnitude [dB]

100

50

1

0

0.5

−50

0

−100

−0.5
−5

10

0

10
Frequency [rad/sec]

0

2

5

10

Fig. 12.
Fig. 10. Bode gain plots of the controllers Kge (solid line) and Kbt (dashedline and dashed-dotted line).

(SCO) of the battery. Fig. 11 shows the reference power Prefand the net power Pnet with SOC control, while Fig. 12 showsthat without SOC control. The RMS (root mean square) valueof the error is 0.086 [kW] with SOC control and 0.077 [kW]without SOC control. The difference is just 0.09 [kW].
Fig. 13 shows the SOC [%] of the battery. The initial SOC
is 50 %. From Fig. 13, we can see that the SOC is maintained
closer to the initial SOC with SOC control than that without
SOC control.
VI. E XPERIMENTAL R ESULTS
We developed a microgrid testbed that consists of a gas
turbine engine, a lead battery, and 3 units of photovoltaic
inverters. The maximum power of the gas turbine engine is
5 [kW]. The H ∞ controller of the gas turbine engine Kge is
implemented in the testbed. The total capacity of the leadbattery that is regulated by a PI controller is 10 [kWh].

4

6Time [hour]

8

10

12

Pnet for 12 hours without SOC control

In the microgrid testbed, we used a photovoltaic simulator
that emulates the behavior of real photovoltaic inverters.
We assume that the maximum powers of the photovoltaic
inverters are 15 [kW], 10 [kW], and 4 [kW]. The power
consumption of 2 loads are also generated by a simulator.
The peak power consumption of these loads are assumed tobe 10 [kW] and 8 [kW].
Figs. 14 and 15 show the results of the experiment on the
testbed with a conventional PID controller and with the H ∞controller Kge for the gas turbine engine. We can see that the
tracking performance for the reference signal is significantly
improved by using the H ∞ controller Kge .
VII. C ONCLUSION
In this paper, we proposed power balancing control for
microgrids based on H ∞ control theory. We designed H ∞controller for gas turbine engines and batteries to smooth out
power fluctuations caused by photovoltaic systems and loads.

H−infinity control

Battery SOC

553
with SOC control
without SOC control
initial capacity

52

4.5
4

51

3.5
Power [kW]

SOC [%]

50

49

48

3
2.5
2

47

1.5

46

1
0.5

45

0

5

10
Time [hour]

15

20

0
0

Fig. 13. SOC (state of capacity) of the battery with SOC control (solidline) and without SOC control (dashed line).

10

20

30
Time [sec]

40

50

60

Fig. 15. H ∞ control for the gas turbine engine: reference signal (dashedline), power generation (solid line).

PID control
54.5
4Power [kW]

3.5
3
2.5
2
1.5
1
0.5
0
0

10

20

30
Time [sec]

40

50

60

Fig. 14. PID control for the gas turbine engine: reference signal (dashedline), power generation (solid line).

The obtained controllers can appropriately assign power
generation based on operational characteristics of batteries
and gas turbine engines. Numerical examples showed thatthe proposed method can achieve power balancing control
considering SOC of the batteries. We also implemented
the obtained H ∞ controller for a gas turbine engine on a
microgrid testbed. The experimental results showed that the
proposed H ∞ control method can achieve better tracking
performance compared to a PID-based control.
In the experiment on the testbed, we only implemented a
H ∞ controller for a gas turbine engine. The evaluation of a
H ∞ controller for a battery on the testbed is a target of our
subsequent work.
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