AbstractThis paper presents a robust optimization framework that integrates three energy sources for superior smart grid economical and environmental operations. The renewable hydro and wind energy sources as well as nonrenewable coal-fired steam energy source are considered to be connected via an asynchronous link. The robust control framework treats the hydro and steam power plants as controllable energy sources while it treats the wind turbine as an exogenous energy source. The proposed robust system framework ensures optimal smart grid power generation for acceptable load demand tracking or for ecological benefits under wind power and model uncertainties. Simulation results demonstrate the efficiency of the proposed framework for superior smart grid power source integration under uncertain operation conditions.

Dhansri”, Naren Reddy

Yilmaz, “Muhittin

Elsevier - Publisher Connector 

“A Smart Grid Robust Optimization Framework” 

 Procedia Computer Science  12 ( 2012 )  301 – 306 
1877-0509 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Missouri University of Science and Technology. 
doi: 10.1016/j.procs.2012.09.075 
Complex Adaptive Systems, Publication 2
Cihan H. Dagli, Editor in Chief
Conference Organized by Missouri University of Science and Technology
2012- Washington D.C.
"A Smart Grid Robust Optimization Framework" 
"Muhittin Yilmaz, Naren Reddy Dhansri" 
"Texas A&M University-Kingsville (TAMUK), Kingsville, TX, USA" 
Abstract
This paper presents a robust optimization framework that integrates three energy sources for superior smart 
grid economical and environmental operations. The renewable hydro and wind energy sources as well as 
nonrenewable coal-fired steam energy source are considered to be connected via an asynchronous link. The robust 
control framework treats the hydro and steam power plants as controllable energy sources while it treats the wind 
turbine as an exogenous energy source. The proposed robust system framework ensures optimal smart grid power 
generation for acceptable load demand tracking or for ecological benefits under wind power and model 
uncertainties. Simulation results demonstrate the efficiency of the proposed framework for superior smart grid 
power source integration under uncertain operation conditions.
"Keywords: Smart grid; robust control, uncertain renewable power source integration" 
1. Introduction 
A smart grid is a contemporary power generation and distribution configuration that optimizes the 
renewable and nonrenewable energy generation according to consumer patterns to maintain optimal economical and 
ecological operations. However, the uncertainty associated with renewable energy sources, preferred policy choices 
for renewable sources and fluctuations in the load demand require efficient control algorithms for satisfactory 
overall system performance. The Automatic Generation Control (AGC) approach utilizes the power consumption 
data, which was made easier to obtain via smart meters, to adjust the power generation levels in smart grids by using 
different control techniques [1-8] for different area power systems, i.e., an indicator for renewable and 
nonrenewable source power generation distribution. Higher complexity associated with the increasing number of 
renewable and non renewable and possibly uncertain power sources can be handled by using robust system theory 
principles for efficient and cost-effective controller implementations.
The robust systems theory [9] provides a worst-case scenario to handle model or input uncertainties in 
smart grid implementations. Early formulations focused on H problem [10, 11] where a norm-bounded 
unstructured uncertainty term was used to describe all uncertainty components, with contemporary solution tools 
from convex formulations [12] and traditional state-space solutions [13]. However, contemporary robust system 
approaches such as P -controller offers potential exploitation of the system uncertainty, i.e., structured uncertainty 
cases, at the expense of higher complexity solution algorithms [14].  
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302   Muhittin Yilmaz and Naren Reddy Dhansri /  Procedia Computer Science  12 ( 2012 )  301 – 306 
This paper presents a robust optimization framework for a three area smart grid operation to ensure 
economical and ecological power generation and to explore dynamic smart grid characteristics with nonlinear and 
uncertain power sources. The three area power sources, namely, a hydro power plant, a coal fired steam power plant 
and a wind turbine, are considered for the smart grid, with proper electronic interfaces. Historical power 
consumption data is used for the desired load demand reference input, with a potential to extend for smart meter data 
integration in real-time operations. The paper consists of the following: section 1 introduces the research subject,
section 2 presents the robust system concepts while section 3 covers the robust optimization framework, section 4 
details the power plant uncertainty derivations, section 5 explains the simulation results, and section 6 concludes. 
2. Robust System Concepts
Robust system theories can be used to design a controller that ensures robust performance under model,
signal and performance uncertainties. Since robust system theories require Linear Time Invariant (LTI) models of a 
plant/process to derive a controller, nonlinear or complex models of the plant/process are approximated by an LTI 
linear model, with a bounded model uncertainty that is characterized by a combination of an uncertainty weighting 
function, which is a known linear, time-invariant transfer function and an associated norm-bounded uncertainty 
term. The multiplicative uncertainty description can be used to define the actual smart grid power plants such as    
, where G(s) denotes the known and fixed nominal plant, W(s) denotes the 
XQFHUWDLQW\ZHLJKWLQJIXQFWLRQFRQWDLQLQJDOOWKHNQRZQFKDUDFWHULVWLFVRIWKHXQFHUWDLQW\DQGǻVGHQRWHVWKHQRUP
bounded uncertainty. Also, smart grid operations require robust performance that implies a stable closed loop 
system and acceptable performance for all possible actual plant models. The performance and uncertainty 
specifications are implemented by using proper weighting functions. The resultant feedback control system is 
converted into a linear fractional transformation and both H and ȝ FRQWURO IRUPXODWLRQV DUH XVHG WR REWDLQ
controllers for unstructured and structured uncertainty cases, respectively. 
3. Optimization Framework
Highly complex nature of distributed power generation, varying consumer demand and cost requirements,
power system model as well as demand uncertainties and external disturbances present a challenging optimization 
problem to ensure power availability with an emphasis on renewable energy sources. However, robust system 
frameworks utilize Multi-Input Multi-Output (MIMO) feedback control system configurations that are very suitable 
for smart grid system uncertainty descriptions, load demand tracking, and renewable wind energy focus. As the 
multiplicative model uncertainties are derived from each power generation plant, the system performance 
specifications are described by using proper weighting functions for minimum steady-state error, satisfactory 
transient responses, reasonable closed loop bandwidth, and low gain at high frequencies. 
(a)                                                         (b)
Fig. 1 A Smart Grid a) Control System with Model and Performance Uncertainties and b) its LFT Representation. 
The smart grid closed loop system is given in Fig. 1-a, with the associated variable and system descriptions 
in Table 1. Also, the corresponding LFT representation is shown in Fig. 1-b, where P(s) is the augmented plant 
containing all known or fixed quantities, C(s) is the robust controller, ǻVLVWKHV\VWHPXQFHUWDLQW\PDWUL[VXFKWKDW
, and the system variables are deducted from the control system in Fig. 1-a. 
303 Muhittin Yilmaz and Naren Reddy Dhansri /  Procedia Computer Science  12 ( 2012 )  301 – 306 
Table 1. The Smart Grid Robust Control System Parameters and Their Descriptions.
G1(s) Nominal steam power plant.
G2(s) Nominal hydro power plant.
C(s) The robust controller.
E(s) The Error input to the controller.
W1(s), W2(s) Uncertainty weighting functions for steam and hydro power plants, respectively.
Wp1(s), The Performance weighting functions for steam and hydro power plants, respectively.
ǻp1Vǻp2(s) The performance uncertainty terms for steam and hydro power plants, respectively.
ǻ1Vǻ2(s) The uncertainty magnitude of steam and hydro power plants, respectively.
Wi(s) Demand uncertainty weighting function.
Wp(s) Performance weighting function for the grid.
ǻ i(s) The model demand uncertainty.
ǻp(s) The grid performance uncertainty.
Y(s) The generated power.
R(s) The power demand reference input.
D(s) The power generated by the wind energy system.
U1(s), U2(s) The control inputs to the steam and hydro power plants, respectively.
The augmented plant P(s) in Fig 1-b can be obtained by using the closed loop system relationships in Fig. 1-a and 
can be written as
The H control framework deals with unstructured uncertainty matrices, i.e., individual uncertainties are 
correlated with all other uncertainty components, and yields an easy problem to solve while producing conservative 
results. OQWKHRWKHUKDQGWKHȝFRQWUROIUDPHZRUNXVHVVWUXFWXUHGXQFHUWDLQW\ matrix descriptions, e.g., ǻV = diag 
(ǻ1(s) , ǻ2(s) , ǻ3(s) , ǻ4(s) , ǻ5(s) , ǻ6(s)) is assumed in this study to imply that each uncertainty only effects its 
own dynamics on performance and stability objectives, for more complex formulations with expected least 
conservative results. Consequently, the robust performance formulation states that, for a given control system in Fig. 
1-a and corresponding LFT in Fig. 1-b, and the uncertainty matrix ,
design a controller C(s) 
such thaWȘ >Ș1Ș2Șd, , , ] and z = [z1, z2, zd, , , ]
                                                                                                    (1)
4. Uncertainty Development
The wind turbine system power is considered to be an exogenous input whose levels are determined by the 
historical wind speed. Thus, the relevant uncertainty is assumed to be integrated into the generated wind power.
Hydro Power Plant Model Uncertainty: Robust system frameworks require plant transfer functions at worst-
case operating points, with the difference among the plant transfer functions being considered as the plant 
uncertainty. Due to historical demand curve characteristics and three power generation source capacities, it is 
assumed that the hydro power plant gate position control input is only varied between 0.6 per unit (pu) and 1 pu. At 
1pu gate position, the resultant hydro power plant transfer function is given as
. At 0.6pu gate position, the hydro power plant transfer 
function is now given as . Considering the plant operations
at low frequencies, i.e., , the nominal hydro power plant Gn(s) can be obtained as a first order 
304   Muhittin Yilmaz and Naren Reddy Dhansri /  Procedia Computer Science  12 ( 2012 )  301 – 306 
transfer function by considering a gate position of 0.8pu, i.e., the nominal model is assumed to correspond to the 
arithmetic average of worst case gate positions, resulting in . Using the 
multiplicative uncertainty description to define the two worst-case and largest deviations from the nominal hydro 
power plant model (Pn(s)), i.e., the deviation of Pn(s) to P1(s) and the deviation of Pn(s) to P2(s) such that 
and are satisfied, where W1(s), W2(s) denote the
hydro power plant unFHUWDLQW\ZHLJKWLQJIXQFWLRQVDQGǻ1Vǻ2(s)  denote the uncertainties for the two worst-case 
plant descriptions whose corresponding frequency responses are shown in Fig. 2-a with the final multiplicative 
uncertainty weighting function is determined to be after a number of attempts. 
(a)                                                                                    (b)
Fig. 2 The Uncertainty Weighting Functions for a) the Hydro Power Plant, and b) Coal-Fired Steam Power Plant. 
Coal Fired Steam Power Plant Model Uncertainty: The assumption of a fixed-gate level simplifies the control design 
while it varies within certain limits under practical conditions. After utilizing the boiler-steam turbine transfer 
functions for different valve openness levels for different operating points, the overall steam plant for a fully open 
valve is given as ,
while the steam plant for a 90% open valve is given as 
and the steam plant 
model for an 80% open valve is given as 
, Following the hydro 
plant worst case differences and associated uncertainty weighting function calculations, the equivalent worst-case 
plant uncertainty weighting functions are shown in Fig. 2-b, resulting in the multiplicative steam plant uncertainty
weighting function as .
Demand Curve Uncertainty: As there are a number of demand curve estimation algorithms, they are 
vulnerable at higher frequencies, and associated robust controllers are subject to inferior behavior. Thus, the demand 
curve uncertainty can be characterized based on frequency dependence characteristics, i.e., as error in forecasting 
algorithms is expected to be low at lower frequencies and increases with frequency, the uncertainty magnitude also 
increases with frequencies, resulting in a typical uncertainty characteristics in Fig. 3, i.e., demand curve uncertainty 
is implied to be 10% at high frequencies and 1% at low frequencies. 
Fig. 3 The Demand Curve Uncertainty Weighting Function.
305 Muhittin Yilmaz and Naren Reddy Dhansri /  Procedia Computer Science  12 ( 2012 )  301 – 306 
5. Numerical Results
The proposed smart grid optimization framework in LFT form, as shown in Fig. 1-b, was implemented in 
Matlab Robust Control Toolbox [15] and the controller design efficiency was evaluated. Since the augmented 
nominal plant does not satisfy the H-infinity condition, i.e., the state-space D matrix is zero, the nominal plant was 
perturbed with an arbitrarily small D matrix that was only insignificantly effective at very high frequency range. The 
smart grid system performance weighting function was determined to achieve the desired transient and steady-state 
performance specifications. It should be noted that the performance weighting function selection is subjective and 
there is no exact relationship between specifications and implied frequency characteristics while the general
guidelines are typically used to ensure proper operations. If an unstable system is obtained, one of the reasons could 
be ambitious performance specifications and associated weighting functions. 
Two fictitious performance weighting functions, one for the steam power plant input and other for the 
hydro power plant input, are selected, as shown in Fig. 4-a-b, to reflect the effects of saturation nonlinearity in the 
practical control system implementation. Since the steam power plant has a larger settling time compared to the
hydro power plant, it can be utilized more at lower frequencies and less at higher frequencies. Also, the hydro power 
plant has a smaller settling time and can be utilized more at higher frequencies and less at lower frequencies.
(a)                                                                      (b)
Fig. 4 The Performance Uncertainty Weighting Function Frequency Responses for a) the Steam Power Plant, and b) the hydro Power Plant. 
The Matlab Robust Control Toolbox was used to design a controller for robust performance. The closed-
loop step response of the smart grid nominal model with the controller is shown in Fig. 5-a while the smart grid 
response for an historical demand curve is also shown in Fig. 5-b, clearly verifying the desired operation. Moreover, 
the step responses in Fig. 5-c for different plant models validate the effectiveness of the robust system frameworks.
                      (a)                           (b)                                                                         (c)
Fig. 5 The Smart Grid Closed-Loop a) Nominal System Step Response, b) System Response for a 24-Hour Historical Demand Curve, and c) 
System Responses for Worst Case Operations.
A ȝ-controller was also designed to improve the system performance after modifying the performance 
weighting functions and assuming a diagonal uncertainty matrix. It should be noted that the controller performance 
closely depends on the particular uncertainty structure. The Matlab Robust Control Toolbox was used to design a 
306   Muhittin Yilmaz and Naren Reddy Dhansri /  Procedia Computer Science  12 ( 2012 )  301 – 306 
controller, satisfying thHȝQRUPless than 1 after a couple of D-K iterations, i.e., D-Scaling was performed to find 
the highest controller upper bound, and uncertainty bounds are determined for the highest controller bound in the 
subsequent step in iterations XQWLOȝLVVDWLVIied. The closed-loop step responses of the smart grid nominal model 
with the H DQGȝFRQWUROOHUV are shown in Fig. 6.
Fig. 6. The Closed Loop Step Responses of the Nominal Plant with H-,QILQLW\DQGȝ&RQWUROOHUV
6. Conclusions
A real time demand-based automatic generation control and optimization framework has been developed for real-
time smart grid operations by integrating conventional and renewable energy sources under varying consumer 
demand and cost requirements. Three energy sources were manipulated to emphasize on the renewable energy source 
and to handle the system uncertainties by using robust system theories. 
The simulation results demonstrate the effectiveness of the designed robust controllers under bounded power 
plant model uncertainties and exogenous wind input excitation. The proposed framework offers a new worst-case 
deterministic optimization algorithm for smart grid automatic generation control, maximizing economical benefits 
while ensuring a good power quality.
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“A Smart Grid Robust Optimization Framework”

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