Model Predictive Economic/Environmental Dispatch of Power Systems ...
Model Predictive Economic/Environmental Dispatch of Power Systems with Intermittent Resources Le Xie, Student Member, IEEE, and Marija D. Ili´c, Fellow, IEEE
Abstract—This paper presents potential benefits of applying model predictive control (MPC) to solving the multi-objective economic/environmental dispatch problem in electric power systems with many intermittent resources. Based on the predictive model of the available output in the next short time period (e.g. 5 minutes) from the intermittent resources, this paper introduces a look-ahead optimal control algorithm for dispatching the available generation resources with the objective of minimizing objective function comprising both generation and environmental costs. This method is compared with (1) the static economic dispatch which treats intermittent resources as uncertain negative loads, and (2) the MPC dispatch with single objective function of minimizing the total generation cost. We show that the proposed MPC approach could lower the generation costs by directly dispatching the generator output from the renewable resources in order to compensate temporal load variations over pre-defined time horizon. Furthermore, the multi-objective economic/environmental cost function provides a formulation to study the tradeoff of efficiency and environmental impact in future energy systems. Simulation is implemented in a 12-bus power system comprising five generators to illustrate potential benefits from this look-ahead dispatch of both intermittent and more conventional power plants. The proposed method is directly applicable to managing power systems with large presence of wind and photovoltaic resources.
Keywords: Energy and Environment, Intermittent Resources, Wind and Solar Energy, Model Predictive Control, Economic Dispatch. I. I NTRODUCTION This paper is motivated by the observation that increasing penetration of intermittent energy resources, wind and photovoltaic power, for example, has raised many challenging questions to the power system operation. Due to the increasing price and environmental concern of fossil fuel, many countries and regions around the globe have mandates for electric utilities to significantly increase the portion of electricity generated from renewable resources [1], e.g. California’s 33% by the year 2020 [2]. These mandates require a substantial increase of renewable energy production compared with the average of 1 to 5 percent in most regions until a few years ago. Given the face that many renewable energy resources are intermittent, how to efficiently and reliably integrate the Le Xie is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 USA e-mail: [email protected]. Marija D. Ili´c is a professor at Carnegie Mellon University, Pittsburgh, PA, with a joint appointment in the Electrical and Computer Engineering and Engineering and Public Policy Departments. She is also the Honorary Chaired Professor for Control of Future Electricity Network Operations at Delft University of Technology in Delft, The Netherlands e-mail: [email protected].
978-1-4244-4241-6/09/$25.00 ©2009 IEEE
intermittent renewable resources into the existing power grid becomes a serious roadblock to their high penetration. The solutions toward harnessing intermittent renewable energy resources in an efficient, reliable, and environmentally friendly way have been actively attempted in recent years. Forecasting methods for predicting output of renewable energy resources, in particular, have been extensively studied, e.g. [5]-[10]. Once a good short-term output forecast (5 minutes to an hour) becomes established, it is important to explore new solutions to (a) operating the system reliably; and (b) operating the system in the most efficient and environmentally friendly way. This raises the question of how to utilize the information from good prediction to better solve the above two subproblems. In this paper, we explore the potential of using good prediction of intermittent renewable resources output to solve problem (b). In particular, we propose a model predictive control (MPC) approach to allocating the resources needed to supply the fluctuating load while considering the tradeoff between the environmental and economic costs. This is also known as load-following problem [15]. The added value of prediction is quantified. Problem (a) is beyond the scope of this paper. Important research efforts toward solving problem (a) can be found in literature such as [11]-[14]. Among the work toward solving problem (b), [4] and recently [17] have explored the potential of adopting MPC to controlling a single conventional power plant output to balance the demand. Little work has been found in the literature that addresses MPC dispatch in power systems with consideration of both economic and environmental objectives. A previous paper by the authors [6] an MPC algorithm was proposed to dispatch all available resources including intermittent energy to supply the fluctuating loads at minimum generation cost. In this paper, we propose an advanced MPC dispatch algorithm to incorporate both economic and environmental costs. Conventional approach is to minimizing total generation cost by treat intermittent resources as negative load [8]. Expected demand and expected intermittent resource output are given as fixed number at each step of the dispatch. A set of controllable generating units adjust their outputs to compensate for the uncontrollable time-varying outputs from intermittent resources, as well as the loads. Therefore, the capability of intermittent resources to follow the demand side fluctuation is not actively pursued under the conventional approach. By incorporating prediction model of intermittent resources in the optimization formulation, the proposed approach actively uses the intermittent resources as a method to follow the demand side fluctuation. Technology improvements to control the output from intermittent resources, such as wind and
photovoltaic power, make the proposed approach feasible in real systems [18] [19]. The multi-objective cost function provides the platform to start consider the tradeoff between generation cost and environmental impact in future energy systems. This paper is organized as follows: in Section II, the problem of economic/environmental dispatch with large presence of intermittent energy resources is formulated. In Section III the MPC method is briefly reviewed, and a new MPC-based algorithm for solving the economic/environmental dispatch problem is introduced. In Section IV a 12 bus test system with 20% renewable energy resources is used to simulate the new algorithm. A comparison of the results obtained from the conventional economic dispatch method and the proposed MPC economic/environmental dispatch algorithm is discussed. Quantifiable savings are obtained using the proposed MPCbased dispatch algorithm. In Section V conclusions are given and future research is discussed.
II. E CONOMIC /E NVIRONMENTAL D ISPATCH IN P OWER S YSTEMS : P ROBLEM F ORMULATION The objective of power system operation is to supply power from generators to the end users reliably and efficiently, under both normal conditions and expected contingencies. With increasing environmental concern in power plants outputs, dispatch procedure is shaped towards a multi-objective optimization in which both economic and environmental costs are factored [8]. In this paper, the problem of interest is to compute generation outputs necessary to supply the expected demand over the time in economically efficient and environmentally friendly way. This problem is different from the conventional single-objective economic dispatch problem, and is formulated follows1 :
Given G :the set of all available generators; Gr :the set of intermittent energy generators; ˆ L(k) :forecast total demand at time step k ; Ci (PGi ) :operating cost function of generator i producing at level PGi , i ∈ G; ECi (PGi ) :environmental cost function of generator i producing at level PGi , i ∈ G; :minimum output level of generator i ∈ G; PGmin i max PGi :maximum output level of generator i ∈ G; PˆGj :predicted available generation output at time step k, j ∈ Gr ; Ri :ramp rate of generator i, i ∈ G; K :number of samples in the optimization period;
Solve : min
K
PGi (k)
s.t.
k=1
PGi (k)
i PGmin i
(Ci (PGi (k) ) + ECi (PGi (k) )), i ∈ G
i
ˆ = L(k), i ∈ G, k = 1, 2, · · · , K;
≤ PGi (k) ≤ PGmax , k = 1, 2, · · · , K; i
|PGi (k + 1) − PGi (k)| ≤ Ri , i ∈ G;
(1) (2) (3) (4)
The above formulation provides a means of incorporating both the economic and environmental concerns. By assigning different relative weight to environmental cost term, insights on tradeoff between economic performance and environmental cost can be studied. Typically the cost functions for generators and the environmental cost (emission cost, primarily) are assumed to be convex (e.g. linear or quadratic). Therefore economic/environmental dispatch can be modelled as a convex optimization problem. Many algorithmic software packages have been developed to solve convex optimization problem, e.g. the CVX package for MATLAB [26]. At present, utility control centers run such optimization engine every 5 to 15 minutes to solve the economic dispatch problem. However, in the present economic dispatch optimization routine, intermittent resources are modelled as uncertain “negative loads”. This is because in conventional power systems intermittent energy output only consists less than 3 percent of the overall generation. Therefore, for all practical purposes it is acceptable to treat the intermittent resources output as negative loads. Moreover, the optimization is static with K = 1. No environmental concern is factored into the conventional optimization either. Whereas the conventional approach does not actively utilize the output of intermittent resources as decision variables in the optimization routine, it is much less desired to use the same approach in future power systems which are likely to have substantial presence of intermittent resources. It is instead necessary to optimize the use of generator resources dynamically including direct control of intermittent resources. Also multi-objective function instead of single least cost function is desired to dispatch all available resources. As a step in that direction, an MPC-based formulation of this problem is introduced in the next Section. A look-ahead optimal dispatch algorithm is proposed to dynamically schedule all available resources. III. P ROPOSED M ODEL P REDICTIVE D ISPATCH A LGORITHM In this section the proposed model predictive dispatch algorithm is presented. First we briefly review the basic formulation of MPC. This is followed by the proposed MPC-based economic/environmental dispatch algorithm and its mathematical formulation. A. Model Predictive Control
1 For
the sake of simplicity and not losing the key message of MPC , this paper does not consider transmission losses and transmission constraints in the formulation. However, the authors are aware of the importance of including them and have started to incorporate them in the next phase of this research.
Model predictive control (MPC) is a receding horizon optimization-based control method[3][4][20]. The basic concept of MPC is that at each step, a look-ahead finite-horizon
optimal control problem is solved but only the first step of control sequences is implemented. The time-domain trajectory of state variables over the prediction horizon is described by a predictive model, with the initial state being the measured state of the actual system. After the implementation of the first step, the system waits until the next step. With the new measurement, the optimal control routine is computed again. This online optimization approach has been successfully applied to many process control problems e.g. [4]. The MPC at control step i is described as: minJ(U ), U = {u0 , u1 , u2 , · · · , uN }
(5)
s.t.xk+1 = f (xk , uk , wk ), k = 0, 1, · · · , N − 1
(6)
U
g(xk , xk+1 , uk , wk ) ≤ 0, k = 0, 1, · · · , N − 1
(7)
x0 = Z(k)
(8)
where N is the prediction horizon. The optimal solution to the above problem is denoted by U ∗ = {u∗0 , u∗1 , · · · , u∗N −1 }. Only u∗0 after this iteration of optimization is implemented. The process is repeated in time. B. MPC-based Economic Environmental Dispatch Algorithm Based on the general MPC approach described in the previous subsection, the economic/environmental dispatch problem for power systems with intermittent renewable resources and uncertain demands can be implemented as follows: (i) Set k = 0; (ii) Select a moving horizon N (e.g. 24 hours discretized at 5 minutes step size). Select a prediction model for load and available intermittent resource outputs. (iii) Solve the multi-objective economic/environmental dispatch problem for the entire horizon N. Denote the optimum outputs of all generators for the entire horizon by u∗0 , u∗1 , · · · , u∗N −1 . (iv) Apply the first step of the generator outputs u(k) = u∗0 to the power system. (v) Set k = k + 1 and update the predictive model for the next control iteration, such as available wind power outputs, contingency information, etc. Go to (ii). Fig. 1 described the MPC approach to solving economic/environmental dispatch problem. By taking into consideration the prediction of available output level of intermittent resources, this approach expands the set of generator decision variables by including intermittent resource outputs. Therefore, this approach reduces the burden of supplying the entire load by means of conventional fast-start units which are typically expensive. C. Comparison of Conventional and MPC-based Economic/ Environmental Dispatch Algorithms Conventional economic dispatch with single objective of minimizing total generation cost is posed as follows: C(PGi (k)), i ∈ G\Gr (9) min PGi (k)
s.t.
i
ˆk − (PGi (k)) = L
i PGmin i
≤ PGi (k) ≤
j PGmax i
PˆGj (k), j ∈ Gr
(10) (11)
3UHGLFWLYH0RGHODQG 03&(QYLURQPHQWDO (FRQRPLF'LVSDWFK
/Ö N 3Ö PD[ N ZLQG
PD[ 3ÖVRODU N
(OHFWULF(QHUJ\6\VWHP
XN 2XWSXWYHFWRU RI DOO JHQHUDWRUV DWWLPH VWHS N
Fig. 1.
MPC algorithm to economic dispatch problem
At each step k, (9)-(12) is solved without considering the interdependencies of consecutive steps. Due to the fact that many low-cost coal units can not ramp fast enough to fully supply the time-varying loads, the optimization procedure has to assign the fast-start units (e.g. expensive natural gas generators) to follow the fast fluctuation of loads. Due to high volatility of intermittent resources, the need for fast following expensive plants has become even more pronounced. On the other hand, in the proposed algorithm, optimization is carried over a horizon (e.g 24 hours) but only the control signal in the first step (e.g. the next 5 minutes) is implemented. In this formulation, the control signals are generation output levels for all the units including the intermittent renewable resources. The state variables are the expected total load, as well as the expected maximum output levels of intermittent resources. Therefore, outputs from intermittent resources can be chosen as a means of supplying time-varying loads. We formulate this new problem as follows:
min
PGi (k)
C(PGi (k)), k = 1, 2, · · · , 24 ∗ 12
i
s.t.
(PGi (k)) =
i
ˆ (L(k))
(12) (13)
j
ˆ + 1)) = f ((L(k))) ˆ L(k max PˆG (k + 1)) = h(PˆGmax (k)) j
j
PˆGmin (k + 1)) = h(PˆGmin (k)) j j min max ˆ ˆ PG ≤ PGi (k) ≤ PG , i ∈ {Gr } i
PGmin j
i
≤ PGj (k) ≤
PGmax ,j j
∈ {Gc }
|PGi (k) − PGi (k − 1)| ≤ Ri ∗ ΔT
(14) (15) (16) (17) (18) (19)
Equation (14) represents a simple prediction model of expected load consumption. Equation (16) represents prediction model of the maximum output for intermittent renewable resources. Functional dependencies in Equations (14) and (16) can obtained from autoregressive parametrization techniques, such as [24]. While it is well-known that long-term forecasts for intermittent resources are quite inaccurate, we only need relatively short-term accurate forecast for implementing the first step u∗0 of the control sequences. In the next section, we show the results of numerical simulations on a 12-bus system. A comparison of the results obtained using the proposed algorithm and conventional dispatch is discussed.
IV. N UMERICAL E XAMPLES
2 For the sake of simplicity, the minimum output level when units are turned on is assumed to be zero.
Fig. 2. [23]
12 bus electric energy system (modified from IEEE 14 bus system
Predicted and actual available wind power output 15000 Actual available MW
10000
Predicted available
5000 0 0 2000 1500 MW
The economic/environmental dispatch problem is simulated on a 12-bus system shown in Fig. 2. This system can be considered as a simplified topology of a regional transmission organization (e.g. New York System) which consists of five generator plants of representative types. Each generator is characterized by its marginal cost and maximum capacity, as well as the ramp rate constraints2 . Table I shows the parameters of these generators. To characterize the difference between coal power plants and natural gas power plants, the coal power plants are assumed to be capable of changing output level at an hourly basis, whereas the natural gas power plants are capable of changing output levels at a 5-minute basis. For renewable power plants, there is another constraint: power output must not exceed the predicted output level at any given time. The predicted and actual power output from wind and solar power plants are shown in Fig. 3 [21][22]. Fiveminute interval system load data in the year 2006 obtained from New York ISO website [25] is used as the load profile in this simulation. Shown in Fig. 4 are the actual system load and the net system load after subtracting outputs from the renewable resources. It can be seen that the net load is much more volatile than the original system load. Three scenarios are solved, namely, the MPC economic/ environmental dispatch, the MPC purely economic dispatch without considering the environmental impact, and the conventional economic dispatch method. Both MPC and conventional methods are run in MATLAB LP Solver because the economic/environmental cost functions and the constraints are assumed to be linear. The results of these three scenarios are shown in the following four figures. Fig. 5 shows the generation output levels corresponding to the two coal units in a typical day. The result of the proposed MPC algorithm is that the base coal unit output is higher. This is because renewable energy helps meet the fluctuating loads, therefore, the base coal unit can operate at a higher level while the high-priced natural gas unit can operate at a lower level. With the environmental cost factored into the objective function of the optimization, the coal units operate at a slightly lower level than the MPC results by doing the pure economic dispatch. Fig. 6 shows the output from the natural gas unit, confirming the above explanation. In Fig. 7 wind power output is plotted for both algorithms. Although wind output is higher in the conventional algorithm, in the proposed MPC algorithm available wind power output is used to directly cancel out fast load fluctuations and because of this, it reduces the need for fast following expensive natural gas plants. Therefore, the wind-related cost savings are higher in the proposed MPC algorithm. Moreover, with environmental concerns, the wind operates at higher level than the pure economci dispatch case. If we compare the economic/environmental cost function, the relative cost saving of using MPC algorithm over the conventional algorithm is about $9.12 Million per day for the entire New York System, which is about 7.0% reduction of total cost.
500 1000 Time Steps (5 minutes interval) Predicted and actual available solar power output
1500
Actual available Predicted available
1000 500 0 0
500 1000 Time Steps (5 minutes interval)
1500
Fig. 3. Predicted and actual wind and solar power output at 5-minute interval
V. C ONCLUSIONS In this paper we have introduced a look-ahead MPC-based approach to solving the economic/environmental dispatch problem for future power systems with substantial presence of intermittent resources. The main contributions are two-folded: (1) MPC-based dispatch algorithm actively relies on direct control of intermittent resources for compensating fast load fluctuations; and (2) multi-objective economic/environmental dispatch provides a framework to study the tradeoff between the economics and environmental impact. Therefore, the proposed algorithm results in lower total generation and environmental cost than the conventional economic dispatch. A numerical simulation on a 12-bus power system confirms this claim. This newly proposed MPC-based economic/environmental
TABLE I G ENERATOR PARAMETERS OF THE 12- BUS SYSTEM
4
1.8
x 10
1.6
Generator ID
Type
Capacity
Generation Cost
Environmental Cost
Ramp Rate
1
Gas
5000MW
350$/MWh
10$/MWh
100MW/5 min
2
Coal
9000MW
150$/MWh
15$/MWh
1000MW/hour
3
Wind
3500MW
0$/MWh
0$/MWh
150MW/5 min
4
PV
1500MW
0$/MWh
0$/MWh
100MW/5 min
5
Coal
8000MW
100$/MWh
20$/MWh
800MW/hour
Natural gas power plant outputs under three scenarios
Load versus (load minus intermittent outputs)
9000
Load Net Load minus intermittent outputs
8000 7000
1.4
MPC with environmental cost MPC without environmental cost Conventional dispatch
1
5000 4000
0.8
3000
0.6
2000
0.4 0
Fig. 4.
MW
MW
6000 1.2
1000 500
1000
Time Steps (5 minutes interval)
1500
0 0
Load versus load minus wind and solar Fig. 6.
MW
8000
MPC with environmental cost MPC without environmental cost Conventional dispatch
7000
1000
1500
MW
MW
500
Time Steps (5 minutes interval) Gen ID: 5
5000 4000 3000 2000
6000 4000
MPC with environmental cost MPC without environmental cost Conventional dispatch
2000
Fig. 5.
MPC with environmental cost MPC without environmental cost Conventional dispatch
6000
8000
0 0
Natural gas power plant output in a typical day
9000
2000 0 0
1500
10000
8000
4000
1000
Wind power plant outputs under three scenarios
Gen ID: 2 10000
6000
500
Time Steps (5 minutes interval)
500
1000
Time Steps (5 minutes interval)
1000 0 0
500
1000
Time Steps (5 minutes interval)
1500
1500
Coal power plant outputs for a typical day
dispatch algorithm provides several aspects for future research. In particular, methods for making this algorithm more robust against errors in the prediction model should be explored. Also, extensions of this algorithm to computationally efficient algorithms that can be applied in large-scale real-world systems is one important aspect to be studied. Moreover, extension to include include network transmission constraints needs to be worked. Finally, our simulation is done based on hypothetical data, much more simulations work is needed to assess the potential benefits of the newly proposed MPC-based economic dispatch for real-world electric energy systems.
Fig. 7.
Wind power plant output in a typical day
ACKNOWLEDGMENT This work was supported by the U.S. National Science Foundation ITR Project CNS0428404. The authors greatly appreciate this financial help. The authors acknowledge helpful comments from Mr. Michiel Houwing of Delft University of Technology, Delft, The Netherlands. The first author would also like to acknowledge the discussion with group members of the Electric Energy Systems Group (EESG) at Carnegie Mellon University. The authors appreciate reviewers’ comments on improving the quality of this paper. R EFERENCES [1] U.S. Department of Energy, 20% Wind Energy by 2030: Increasing Wind Energy’s Contribution to U.S. Electricity Supply, May 2008.
[2] California Public Utilites Commission, Order Instituting Rulemaking Regarding Implementation and Administration of the Renewables Portfolio Standard Program, Apr 2008. [3] E. F. Camacho and C. Bordons, Model Predictive Control, 2nd ed. New York, USA: Springer-Verlag, 2004. [4] E. Gallestey, A. Stothert, M. Antoine and S. Morton, “Model predictive control and the optimization of power plant load while considering lifetime consumption,” IEEE Transactions on Power Systems, Vol. 17, Issue 1, pp.186-191, Feb 2002. [5] P. Fairley, “Scheduling wind power,” MIT Technology Review, April 17, 2008, available online at http://www.technologyreview.com/Energy/20646/. [6] L. Xie and M.D. Ili´c, ”Model predictive dispatch in electric power systems with intermittent energy sources,” Proceedings of IEEE Conference on Systems, Man and Cybernatics, Singapore, Oct 2008. [7] G. Giebel, The State-Of-The-Art in Short-Term Prediction of Wind Power: A Literature Overview, RisøNational Laboratory, Denmark: 2003. [8] J.H. Talaq, F.EI-Hawary, and M.E. EI-Hawary, “A summary of environmental/economic dispatch algorithms,” IEEE Transactions on Power Systems, Vol. 9, No. 3, pp. 1508–1506. Aug 1994. [9] J. Apt, “The spectrum of power from wind turbines,” Journal of Power Sources, Vol. 169, Issue 2, pp.369-374, Jun 2007. [10] K. Furushima, Y. Nawata and M. Sadatomi“Prediction of photovoltaic (PV) power output considering weather effects,” Proceedings of the SOLAR, Jul 2006. [11] F. Bouffard and F. D. Galiana, “Stochastic security for operations planning with significant wind power generation,” IEEE Transactions on Power Systems, Vol. 23, Issue 2, pp.306-316, May 2008. [12] M. Milligan, K. Porter, B. Parsons and J. Caldwell, “Wind energy and power system operations: a survey of current research and regulatory actions,” The Electricity Journal, Vol. 15, Issue 2, pp.56-67, Mar 2002. [13] K. Hemmes, J. L. Zachariah-Wolf, M. Geidl and G. Andersson, “Towards multi-source multi-product energy systems,” International Journal of Hydrogen Energy, Vol. 32, Issue 10-11, pp.1332-1338, Jul-Aug 2007. [14] G. A. Koeppel, Reliability Considerations of Future Energy Systems: Multi-Carrier Systems and the Effect of Energy Storage, Ph.D Dissertation, Swiss Federal Institute of Technology, Zurich, 2007. [15] Federal Energy Regulatory Commission (FERC), Order 888, 1996. [16] B. C. Ummels, M. Gibescu, E. Pelgrum, W. L. Kling and A. J. Brand, “Impacts of wind power on thermal generation unit commitment and dispatch,” IEEE Transaction on Energy Conversion, Vol. 22, Issue 1, pp.44-51, Mar 2007. [17] K. Edlund, J. D. Bendtsen, S. Børresen and T. Mølbak, “Introducing model predictive control for improving power plant portfolio performance,” Proceedings of the 17th World Congress of The International Federation of Automatic Control, Seoul, Korea, Jul 2008. [18] R. Pena, J. C. Clare and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its application to variablespeed wind-energy generation,” IEE Proceedings on Electric Power Applications, Vol. 143, Issue 3, pp.231-241, May 1996. [19] T. Shimizu, M. Hirakata, T. Kamezawa and H. Wantanabe, “Generation control circuit for photovoltaic modules,” IEEE Transactions on Power Electronics, Vol. 16, Issue 3, pp.293-300, May 2001. [20] E. Camponogara, D. Jia, B. H. Krogh and S. Talukdar, “Distributed model predictive control,” IEEE Control Systems Magazine, Vol. 22, Issue 1, pp.44-52, Feb 2002. [21] A. Sfetsos and C. Siriopoulos, “Time series forecasting of averaged data with efficient use of information,” IEEE Transactions on Systems, Man and Cybernetics, Part A, Vol. 35, Issue 5, pp.738-745, Sep 2005. [22] J. A. Momoh, Y. Wang and F. Eddy-Posey, “Optimal power dispatch of photovotaic system with random load,” Proceedings of IEEE Power Engineering Society General Meeting, Jun 2004. [23] Power System Test Case , available online at http://www.nyiso.com/public/market_data/load_data.jsp. [24] M. D. Ili´c, L. Xie, U. A. Khan and J. M. F. Moura, “Modeling Future Cyber-Physical Energy Systems,” in Proceedings of IEEE Power and Energy Society General Meeting, Pittsburgh, PA, Jul 2008. [25] Available online at http://www.nyiso.com/public/market_data/load_data.jsp. [26] M. Grant, S. Boyd, and Y. Ye, CVX: Matlab Software for Disciplined Convex Programming, available online at http://www.stanford.edu/ boyd/cvx/.
Le Xie (S’05) received his B.E. in Electrical Engineering in 2004 from Tsinghua University, Beijing, China. He got M.Sc. in Engineering Sciences from Harvard University in June 2005. Currently he is a Ph.D candidate in the Department of Electrical and Computer Engineering at Carnegie Mellon University. His industry experience included an internship (Jun 2006-Aug 2006) at ISO-New England and an internship at Edison Mission Energy Marketing and Trading (Jun 2007-Aug 2007). His research interest includes modeling and control of large-scale complex systems, smart grids and electricity market.
Marija D. Ili´c (M’80–SM’86–F’99) is currently a Professor at Carnegie Mellon University, Pittsburgh, PA, with a joint appointment in the Electrical and Computer Engineering and Engineering and Public Policy Departments. She is also the Honorary Chaired Professor for Control of Future Electricity Network Operations at Delft University of Technology in Delft, The Netherlands. She was an assistant professor at Cornell University, Ithaca, NY, and tenured Associate Professor at the University of Illinois at Urbana-Champaign. She was then a Senior Research Scientist in Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, from 1987 to 2002. She has 30 years of experience in teaching and research in the area of electrical power system modeling and control. Her main interest is in the systems aspects of operations, planning, and economics of the electric power industry. She has co-authored several books in her field of interest. Prof. Ili´c is an IEEE Distinguished Lecturer.
Abstract—This paper presents potential benefits of applying model predictive control (MPC) to solving the multi-objective economic/environmental dispatch problem in electric power systems with many intermittent resources. Based on the predictive model of the available output in the next short time period (e.g. 5 minutes) from the intermittent resources, this paper introduces a look-ahead optimal control algorithm for dispatching the available generation resources with the objective of minimizing objective function comprising both generation and environmental costs. This method is compared with (1) the static economic dispatch which treats intermittent resources as uncertain negative loads, and (2) the MPC dispatch with single objective function of minimizing the total generation cost. We show that the proposed MPC approach could lower the generation costs by directly dispatching the generator output from the renewable resources in order to compensate temporal load variations over pre-defined time horizon. Furthermore, the multi-objective economic/environmental cost function provides a formulation to study the tradeoff of efficiency and environmental impact in future energy systems. Simulation is implemented in a 12-bus power system comprising five generators to illustrate potential benefits from this look-ahead dispatch of both intermittent and more conventional power plants. The proposed method is directly applicable to managing power systems with large presence of wind and photovoltaic resources.
Keywords: Energy and Environment, Intermittent Resources, Wind and Solar Energy, Model Predictive Control, Economic Dispatch. I. I NTRODUCTION This paper is motivated by the observation that increasing penetration of intermittent energy resources, wind and photovoltaic power, for example, has raised many challenging questions to the power system operation. Due to the increasing price and environmental concern of fossil fuel, many countries and regions around the globe have mandates for electric utilities to significantly increase the portion of electricity generated from renewable resources [1], e.g. California’s 33% by the year 2020 [2]. These mandates require a substantial increase of renewable energy production compared with the average of 1 to 5 percent in most regions until a few years ago. Given the face that many renewable energy resources are intermittent, how to efficiently and reliably integrate the Le Xie is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 USA e-mail: [email protected]. Marija D. Ili´c is a professor at Carnegie Mellon University, Pittsburgh, PA, with a joint appointment in the Electrical and Computer Engineering and Engineering and Public Policy Departments. She is also the Honorary Chaired Professor for Control of Future Electricity Network Operations at Delft University of Technology in Delft, The Netherlands e-mail: [email protected].
978-1-4244-4241-6/09/$25.00 ©2009 IEEE
intermittent renewable resources into the existing power grid becomes a serious roadblock to their high penetration. The solutions toward harnessing intermittent renewable energy resources in an efficient, reliable, and environmentally friendly way have been actively attempted in recent years. Forecasting methods for predicting output of renewable energy resources, in particular, have been extensively studied, e.g. [5]-[10]. Once a good short-term output forecast (5 minutes to an hour) becomes established, it is important to explore new solutions to (a) operating the system reliably; and (b) operating the system in the most efficient and environmentally friendly way. This raises the question of how to utilize the information from good prediction to better solve the above two subproblems. In this paper, we explore the potential of using good prediction of intermittent renewable resources output to solve problem (b). In particular, we propose a model predictive control (MPC) approach to allocating the resources needed to supply the fluctuating load while considering the tradeoff between the environmental and economic costs. This is also known as load-following problem [15]. The added value of prediction is quantified. Problem (a) is beyond the scope of this paper. Important research efforts toward solving problem (a) can be found in literature such as [11]-[14]. Among the work toward solving problem (b), [4] and recently [17] have explored the potential of adopting MPC to controlling a single conventional power plant output to balance the demand. Little work has been found in the literature that addresses MPC dispatch in power systems with consideration of both economic and environmental objectives. A previous paper by the authors [6] an MPC algorithm was proposed to dispatch all available resources including intermittent energy to supply the fluctuating loads at minimum generation cost. In this paper, we propose an advanced MPC dispatch algorithm to incorporate both economic and environmental costs. Conventional approach is to minimizing total generation cost by treat intermittent resources as negative load [8]. Expected demand and expected intermittent resource output are given as fixed number at each step of the dispatch. A set of controllable generating units adjust their outputs to compensate for the uncontrollable time-varying outputs from intermittent resources, as well as the loads. Therefore, the capability of intermittent resources to follow the demand side fluctuation is not actively pursued under the conventional approach. By incorporating prediction model of intermittent resources in the optimization formulation, the proposed approach actively uses the intermittent resources as a method to follow the demand side fluctuation. Technology improvements to control the output from intermittent resources, such as wind and
photovoltaic power, make the proposed approach feasible in real systems [18] [19]. The multi-objective cost function provides the platform to start consider the tradeoff between generation cost and environmental impact in future energy systems. This paper is organized as follows: in Section II, the problem of economic/environmental dispatch with large presence of intermittent energy resources is formulated. In Section III the MPC method is briefly reviewed, and a new MPC-based algorithm for solving the economic/environmental dispatch problem is introduced. In Section IV a 12 bus test system with 20% renewable energy resources is used to simulate the new algorithm. A comparison of the results obtained from the conventional economic dispatch method and the proposed MPC economic/environmental dispatch algorithm is discussed. Quantifiable savings are obtained using the proposed MPCbased dispatch algorithm. In Section V conclusions are given and future research is discussed.
II. E CONOMIC /E NVIRONMENTAL D ISPATCH IN P OWER S YSTEMS : P ROBLEM F ORMULATION The objective of power system operation is to supply power from generators to the end users reliably and efficiently, under both normal conditions and expected contingencies. With increasing environmental concern in power plants outputs, dispatch procedure is shaped towards a multi-objective optimization in which both economic and environmental costs are factored [8]. In this paper, the problem of interest is to compute generation outputs necessary to supply the expected demand over the time in economically efficient and environmentally friendly way. This problem is different from the conventional single-objective economic dispatch problem, and is formulated follows1 :
Given G :the set of all available generators; Gr :the set of intermittent energy generators; ˆ L(k) :forecast total demand at time step k ; Ci (PGi ) :operating cost function of generator i producing at level PGi , i ∈ G; ECi (PGi ) :environmental cost function of generator i producing at level PGi , i ∈ G; :minimum output level of generator i ∈ G; PGmin i max PGi :maximum output level of generator i ∈ G; PˆGj :predicted available generation output at time step k, j ∈ Gr ; Ri :ramp rate of generator i, i ∈ G; K :number of samples in the optimization period;
Solve : min
K
PGi (k)
s.t.
k=1
PGi (k)
i PGmin i
(Ci (PGi (k) ) + ECi (PGi (k) )), i ∈ G
i
ˆ = L(k), i ∈ G, k = 1, 2, · · · , K;
≤ PGi (k) ≤ PGmax , k = 1, 2, · · · , K; i
|PGi (k + 1) − PGi (k)| ≤ Ri , i ∈ G;
(1) (2) (3) (4)
The above formulation provides a means of incorporating both the economic and environmental concerns. By assigning different relative weight to environmental cost term, insights on tradeoff between economic performance and environmental cost can be studied. Typically the cost functions for generators and the environmental cost (emission cost, primarily) are assumed to be convex (e.g. linear or quadratic). Therefore economic/environmental dispatch can be modelled as a convex optimization problem. Many algorithmic software packages have been developed to solve convex optimization problem, e.g. the CVX package for MATLAB [26]. At present, utility control centers run such optimization engine every 5 to 15 minutes to solve the economic dispatch problem. However, in the present economic dispatch optimization routine, intermittent resources are modelled as uncertain “negative loads”. This is because in conventional power systems intermittent energy output only consists less than 3 percent of the overall generation. Therefore, for all practical purposes it is acceptable to treat the intermittent resources output as negative loads. Moreover, the optimization is static with K = 1. No environmental concern is factored into the conventional optimization either. Whereas the conventional approach does not actively utilize the output of intermittent resources as decision variables in the optimization routine, it is much less desired to use the same approach in future power systems which are likely to have substantial presence of intermittent resources. It is instead necessary to optimize the use of generator resources dynamically including direct control of intermittent resources. Also multi-objective function instead of single least cost function is desired to dispatch all available resources. As a step in that direction, an MPC-based formulation of this problem is introduced in the next Section. A look-ahead optimal dispatch algorithm is proposed to dynamically schedule all available resources. III. P ROPOSED M ODEL P REDICTIVE D ISPATCH A LGORITHM In this section the proposed model predictive dispatch algorithm is presented. First we briefly review the basic formulation of MPC. This is followed by the proposed MPC-based economic/environmental dispatch algorithm and its mathematical formulation. A. Model Predictive Control
1 For
the sake of simplicity and not losing the key message of MPC , this paper does not consider transmission losses and transmission constraints in the formulation. However, the authors are aware of the importance of including them and have started to incorporate them in the next phase of this research.
Model predictive control (MPC) is a receding horizon optimization-based control method[3][4][20]. The basic concept of MPC is that at each step, a look-ahead finite-horizon
optimal control problem is solved but only the first step of control sequences is implemented. The time-domain trajectory of state variables over the prediction horizon is described by a predictive model, with the initial state being the measured state of the actual system. After the implementation of the first step, the system waits until the next step. With the new measurement, the optimal control routine is computed again. This online optimization approach has been successfully applied to many process control problems e.g. [4]. The MPC at control step i is described as: minJ(U ), U = {u0 , u1 , u2 , · · · , uN }
(5)
s.t.xk+1 = f (xk , uk , wk ), k = 0, 1, · · · , N − 1
(6)
U
g(xk , xk+1 , uk , wk ) ≤ 0, k = 0, 1, · · · , N − 1
(7)
x0 = Z(k)
(8)
where N is the prediction horizon. The optimal solution to the above problem is denoted by U ∗ = {u∗0 , u∗1 , · · · , u∗N −1 }. Only u∗0 after this iteration of optimization is implemented. The process is repeated in time. B. MPC-based Economic Environmental Dispatch Algorithm Based on the general MPC approach described in the previous subsection, the economic/environmental dispatch problem for power systems with intermittent renewable resources and uncertain demands can be implemented as follows: (i) Set k = 0; (ii) Select a moving horizon N (e.g. 24 hours discretized at 5 minutes step size). Select a prediction model for load and available intermittent resource outputs. (iii) Solve the multi-objective economic/environmental dispatch problem for the entire horizon N. Denote the optimum outputs of all generators for the entire horizon by u∗0 , u∗1 , · · · , u∗N −1 . (iv) Apply the first step of the generator outputs u(k) = u∗0 to the power system. (v) Set k = k + 1 and update the predictive model for the next control iteration, such as available wind power outputs, contingency information, etc. Go to (ii). Fig. 1 described the MPC approach to solving economic/environmental dispatch problem. By taking into consideration the prediction of available output level of intermittent resources, this approach expands the set of generator decision variables by including intermittent resource outputs. Therefore, this approach reduces the burden of supplying the entire load by means of conventional fast-start units which are typically expensive. C. Comparison of Conventional and MPC-based Economic/ Environmental Dispatch Algorithms Conventional economic dispatch with single objective of minimizing total generation cost is posed as follows: C(PGi (k)), i ∈ G\Gr (9) min PGi (k)
s.t.
i
ˆk − (PGi (k)) = L
i PGmin i
≤ PGi (k) ≤
j PGmax i
PˆGj (k), j ∈ Gr
(10) (11)
3UHGLFWLYH0RGHODQG 03&(QYLURQPHQWDO (FRQRPLF'LVSDWFK
/Ö N 3Ö PD[ N ZLQG
PD[ 3ÖVRODU N
(OHFWULF(QHUJ\6\VWHP
XN 2XWSXWYHFWRU RI DOO JHQHUDWRUV DWWLPH VWHS N
Fig. 1.
MPC algorithm to economic dispatch problem
At each step k, (9)-(12) is solved without considering the interdependencies of consecutive steps. Due to the fact that many low-cost coal units can not ramp fast enough to fully supply the time-varying loads, the optimization procedure has to assign the fast-start units (e.g. expensive natural gas generators) to follow the fast fluctuation of loads. Due to high volatility of intermittent resources, the need for fast following expensive plants has become even more pronounced. On the other hand, in the proposed algorithm, optimization is carried over a horizon (e.g 24 hours) but only the control signal in the first step (e.g. the next 5 minutes) is implemented. In this formulation, the control signals are generation output levels for all the units including the intermittent renewable resources. The state variables are the expected total load, as well as the expected maximum output levels of intermittent resources. Therefore, outputs from intermittent resources can be chosen as a means of supplying time-varying loads. We formulate this new problem as follows:
min
PGi (k)
C(PGi (k)), k = 1, 2, · · · , 24 ∗ 12
i
s.t.
(PGi (k)) =
i
ˆ (L(k))
(12) (13)
j
ˆ + 1)) = f ((L(k))) ˆ L(k max PˆG (k + 1)) = h(PˆGmax (k)) j
j
PˆGmin (k + 1)) = h(PˆGmin (k)) j j min max ˆ ˆ PG ≤ PGi (k) ≤ PG , i ∈ {Gr } i
PGmin j
i
≤ PGj (k) ≤
PGmax ,j j
∈ {Gc }
|PGi (k) − PGi (k − 1)| ≤ Ri ∗ ΔT
(14) (15) (16) (17) (18) (19)
Equation (14) represents a simple prediction model of expected load consumption. Equation (16) represents prediction model of the maximum output for intermittent renewable resources. Functional dependencies in Equations (14) and (16) can obtained from autoregressive parametrization techniques, such as [24]. While it is well-known that long-term forecasts for intermittent resources are quite inaccurate, we only need relatively short-term accurate forecast for implementing the first step u∗0 of the control sequences. In the next section, we show the results of numerical simulations on a 12-bus system. A comparison of the results obtained using the proposed algorithm and conventional dispatch is discussed.
IV. N UMERICAL E XAMPLES
2 For the sake of simplicity, the minimum output level when units are turned on is assumed to be zero.
Fig. 2. [23]
12 bus electric energy system (modified from IEEE 14 bus system
Predicted and actual available wind power output 15000 Actual available MW
10000
Predicted available
5000 0 0 2000 1500 MW
The economic/environmental dispatch problem is simulated on a 12-bus system shown in Fig. 2. This system can be considered as a simplified topology of a regional transmission organization (e.g. New York System) which consists of five generator plants of representative types. Each generator is characterized by its marginal cost and maximum capacity, as well as the ramp rate constraints2 . Table I shows the parameters of these generators. To characterize the difference between coal power plants and natural gas power plants, the coal power plants are assumed to be capable of changing output level at an hourly basis, whereas the natural gas power plants are capable of changing output levels at a 5-minute basis. For renewable power plants, there is another constraint: power output must not exceed the predicted output level at any given time. The predicted and actual power output from wind and solar power plants are shown in Fig. 3 [21][22]. Fiveminute interval system load data in the year 2006 obtained from New York ISO website [25] is used as the load profile in this simulation. Shown in Fig. 4 are the actual system load and the net system load after subtracting outputs from the renewable resources. It can be seen that the net load is much more volatile than the original system load. Three scenarios are solved, namely, the MPC economic/ environmental dispatch, the MPC purely economic dispatch without considering the environmental impact, and the conventional economic dispatch method. Both MPC and conventional methods are run in MATLAB LP Solver because the economic/environmental cost functions and the constraints are assumed to be linear. The results of these three scenarios are shown in the following four figures. Fig. 5 shows the generation output levels corresponding to the two coal units in a typical day. The result of the proposed MPC algorithm is that the base coal unit output is higher. This is because renewable energy helps meet the fluctuating loads, therefore, the base coal unit can operate at a higher level while the high-priced natural gas unit can operate at a lower level. With the environmental cost factored into the objective function of the optimization, the coal units operate at a slightly lower level than the MPC results by doing the pure economic dispatch. Fig. 6 shows the output from the natural gas unit, confirming the above explanation. In Fig. 7 wind power output is plotted for both algorithms. Although wind output is higher in the conventional algorithm, in the proposed MPC algorithm available wind power output is used to directly cancel out fast load fluctuations and because of this, it reduces the need for fast following expensive natural gas plants. Therefore, the wind-related cost savings are higher in the proposed MPC algorithm. Moreover, with environmental concerns, the wind operates at higher level than the pure economci dispatch case. If we compare the economic/environmental cost function, the relative cost saving of using MPC algorithm over the conventional algorithm is about $9.12 Million per day for the entire New York System, which is about 7.0% reduction of total cost.
500 1000 Time Steps (5 minutes interval) Predicted and actual available solar power output
1500
Actual available Predicted available
1000 500 0 0
500 1000 Time Steps (5 minutes interval)
1500
Fig. 3. Predicted and actual wind and solar power output at 5-minute interval
V. C ONCLUSIONS In this paper we have introduced a look-ahead MPC-based approach to solving the economic/environmental dispatch problem for future power systems with substantial presence of intermittent resources. The main contributions are two-folded: (1) MPC-based dispatch algorithm actively relies on direct control of intermittent resources for compensating fast load fluctuations; and (2) multi-objective economic/environmental dispatch provides a framework to study the tradeoff between the economics and environmental impact. Therefore, the proposed algorithm results in lower total generation and environmental cost than the conventional economic dispatch. A numerical simulation on a 12-bus power system confirms this claim. This newly proposed MPC-based economic/environmental
TABLE I G ENERATOR PARAMETERS OF THE 12- BUS SYSTEM
4
1.8
x 10
1.6
Generator ID
Type
Capacity
Generation Cost
Environmental Cost
Ramp Rate
1
Gas
5000MW
350$/MWh
10$/MWh
100MW/5 min
2
Coal
9000MW
150$/MWh
15$/MWh
1000MW/hour
3
Wind
3500MW
0$/MWh
0$/MWh
150MW/5 min
4
PV
1500MW
0$/MWh
0$/MWh
100MW/5 min
5
Coal
8000MW
100$/MWh
20$/MWh
800MW/hour
Natural gas power plant outputs under three scenarios
Load versus (load minus intermittent outputs)
9000
Load Net Load minus intermittent outputs
8000 7000
1.4
MPC with environmental cost MPC without environmental cost Conventional dispatch
1
5000 4000
0.8
3000
0.6
2000
0.4 0
Fig. 4.
MW
MW
6000 1.2
1000 500
1000
Time Steps (5 minutes interval)
1500
0 0
Load versus load minus wind and solar Fig. 6.
MW
8000
MPC with environmental cost MPC without environmental cost Conventional dispatch
7000
1000
1500
MW
MW
500
Time Steps (5 minutes interval) Gen ID: 5
5000 4000 3000 2000
6000 4000
MPC with environmental cost MPC without environmental cost Conventional dispatch
2000
Fig. 5.
MPC with environmental cost MPC without environmental cost Conventional dispatch
6000
8000
0 0
Natural gas power plant output in a typical day
9000
2000 0 0
1500
10000
8000
4000
1000
Wind power plant outputs under three scenarios
Gen ID: 2 10000
6000
500
Time Steps (5 minutes interval)
500
1000
Time Steps (5 minutes interval)
1000 0 0
500
1000
Time Steps (5 minutes interval)
1500
1500
Coal power plant outputs for a typical day
dispatch algorithm provides several aspects for future research. In particular, methods for making this algorithm more robust against errors in the prediction model should be explored. Also, extensions of this algorithm to computationally efficient algorithms that can be applied in large-scale real-world systems is one important aspect to be studied. Moreover, extension to include include network transmission constraints needs to be worked. Finally, our simulation is done based on hypothetical data, much more simulations work is needed to assess the potential benefits of the newly proposed MPC-based economic dispatch for real-world electric energy systems.
Fig. 7.
Wind power plant output in a typical day
ACKNOWLEDGMENT This work was supported by the U.S. National Science Foundation ITR Project CNS0428404. The authors greatly appreciate this financial help. The authors acknowledge helpful comments from Mr. Michiel Houwing of Delft University of Technology, Delft, The Netherlands. The first author would also like to acknowledge the discussion with group members of the Electric Energy Systems Group (EESG) at Carnegie Mellon University. The authors appreciate reviewers’ comments on improving the quality of this paper. R EFERENCES [1] U.S. Department of Energy, 20% Wind Energy by 2030: Increasing Wind Energy’s Contribution to U.S. Electricity Supply, May 2008.
[2] California Public Utilites Commission, Order Instituting Rulemaking Regarding Implementation and Administration of the Renewables Portfolio Standard Program, Apr 2008. [3] E. F. Camacho and C. Bordons, Model Predictive Control, 2nd ed. New York, USA: Springer-Verlag, 2004. [4] E. Gallestey, A. Stothert, M. Antoine and S. Morton, “Model predictive control and the optimization of power plant load while considering lifetime consumption,” IEEE Transactions on Power Systems, Vol. 17, Issue 1, pp.186-191, Feb 2002. [5] P. Fairley, “Scheduling wind power,” MIT Technology Review, April 17, 2008, available online at http://www.technologyreview.com/Energy/20646/. [6] L. Xie and M.D. Ili´c, ”Model predictive dispatch in electric power systems with intermittent energy sources,” Proceedings of IEEE Conference on Systems, Man and Cybernatics, Singapore, Oct 2008. [7] G. Giebel, The State-Of-The-Art in Short-Term Prediction of Wind Power: A Literature Overview, RisøNational Laboratory, Denmark: 2003. [8] J.H. Talaq, F.EI-Hawary, and M.E. EI-Hawary, “A summary of environmental/economic dispatch algorithms,” IEEE Transactions on Power Systems, Vol. 9, No. 3, pp. 1508–1506. Aug 1994. [9] J. Apt, “The spectrum of power from wind turbines,” Journal of Power Sources, Vol. 169, Issue 2, pp.369-374, Jun 2007. [10] K. Furushima, Y. Nawata and M. Sadatomi“Prediction of photovoltaic (PV) power output considering weather effects,” Proceedings of the SOLAR, Jul 2006. [11] F. Bouffard and F. D. Galiana, “Stochastic security for operations planning with significant wind power generation,” IEEE Transactions on Power Systems, Vol. 23, Issue 2, pp.306-316, May 2008. [12] M. Milligan, K. Porter, B. Parsons and J. Caldwell, “Wind energy and power system operations: a survey of current research and regulatory actions,” The Electricity Journal, Vol. 15, Issue 2, pp.56-67, Mar 2002. [13] K. Hemmes, J. L. Zachariah-Wolf, M. Geidl and G. Andersson, “Towards multi-source multi-product energy systems,” International Journal of Hydrogen Energy, Vol. 32, Issue 10-11, pp.1332-1338, Jul-Aug 2007. [14] G. A. Koeppel, Reliability Considerations of Future Energy Systems: Multi-Carrier Systems and the Effect of Energy Storage, Ph.D Dissertation, Swiss Federal Institute of Technology, Zurich, 2007. [15] Federal Energy Regulatory Commission (FERC), Order 888, 1996. [16] B. C. Ummels, M. Gibescu, E. Pelgrum, W. L. Kling and A. J. Brand, “Impacts of wind power on thermal generation unit commitment and dispatch,” IEEE Transaction on Energy Conversion, Vol. 22, Issue 1, pp.44-51, Mar 2007. [17] K. Edlund, J. D. Bendtsen, S. Børresen and T. Mølbak, “Introducing model predictive control for improving power plant portfolio performance,” Proceedings of the 17th World Congress of The International Federation of Automatic Control, Seoul, Korea, Jul 2008. [18] R. Pena, J. C. Clare and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its application to variablespeed wind-energy generation,” IEE Proceedings on Electric Power Applications, Vol. 143, Issue 3, pp.231-241, May 1996. [19] T. Shimizu, M. Hirakata, T. Kamezawa and H. Wantanabe, “Generation control circuit for photovoltaic modules,” IEEE Transactions on Power Electronics, Vol. 16, Issue 3, pp.293-300, May 2001. [20] E. Camponogara, D. Jia, B. H. Krogh and S. Talukdar, “Distributed model predictive control,” IEEE Control Systems Magazine, Vol. 22, Issue 1, pp.44-52, Feb 2002. [21] A. Sfetsos and C. Siriopoulos, “Time series forecasting of averaged data with efficient use of information,” IEEE Transactions on Systems, Man and Cybernetics, Part A, Vol. 35, Issue 5, pp.738-745, Sep 2005. [22] J. A. Momoh, Y. Wang and F. Eddy-Posey, “Optimal power dispatch of photovotaic system with random load,” Proceedings of IEEE Power Engineering Society General Meeting, Jun 2004. [23] Power System Test Case , available online at http://www.nyiso.com/public/market_data/load_data.jsp. [24] M. D. Ili´c, L. Xie, U. A. Khan and J. M. F. Moura, “Modeling Future Cyber-Physical Energy Systems,” in Proceedings of IEEE Power and Energy Society General Meeting, Pittsburgh, PA, Jul 2008. [25] Available online at http://www.nyiso.com/public/market_data/load_data.jsp. [26] M. Grant, S. Boyd, and Y. Ye, CVX: Matlab Software for Disciplined Convex Programming, available online at http://www.stanford.edu/ boyd/cvx/.
Le Xie (S’05) received his B.E. in Electrical Engineering in 2004 from Tsinghua University, Beijing, China. He got M.Sc. in Engineering Sciences from Harvard University in June 2005. Currently he is a Ph.D candidate in the Department of Electrical and Computer Engineering at Carnegie Mellon University. His industry experience included an internship (Jun 2006-Aug 2006) at ISO-New England and an internship at Edison Mission Energy Marketing and Trading (Jun 2007-Aug 2007). His research interest includes modeling and control of large-scale complex systems, smart grids and electricity market.
Marija D. Ili´c (M’80–SM’86–F’99) is currently a Professor at Carnegie Mellon University, Pittsburgh, PA, with a joint appointment in the Electrical and Computer Engineering and Engineering and Public Policy Departments. She is also the Honorary Chaired Professor for Control of Future Electricity Network Operations at Delft University of Technology in Delft, The Netherlands. She was an assistant professor at Cornell University, Ithaca, NY, and tenured Associate Professor at the University of Illinois at Urbana-Champaign. She was then a Senior Research Scientist in Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, from 1987 to 2002. She has 30 years of experience in teaching and research in the area of electrical power system modeling and control. Her main interest is in the systems aspects of operations, planning, and economics of the electric power industry. She has co-authored several books in her field of interest. Prof. Ili´c is an IEEE Distinguished Lecturer.
