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Stochastic Optimization of Renewable-Based Microgrid Operation Incorporating Battery Operating Cost Tu A. Nguyen and M. L. Crow, Fellow, IEEE

Abstract—Integration of renewable energy resources in microgrids has been increasing in recent decades. Due to the randomness in renewable resources such as solar and wind, the power generated can deviate from forecasted values. This variation may cause increased operating costs for committing costly reserve units or penalty costs for shedding load. In addition, it is often desired to charge/discharge and coordinate the energy storage units in an efficient and economical way. To address these problems, a novel battery operation cost model is proposed which considers a battery as an equivalent fuel-run generator to enable it to be incorporated into a unit commitment problem. A probabilistic constrained approach is used to incorporate the uncertainties of the renewable sources and load demands into the unit commitment (UC) and economic dispatch problems. Index Terms—Energy storage, microgrids, renewable energy, unit commitment.

I. INTRODUCTION

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URING the past decades, the electric power industry has undergone significant changes in response to the rising concerns of global climate change and volatile fossil fuel prices. For more efficient, reliable, and environmentally friendly energy production, it is critical to increase the deployment of distributed generation, especially from renewable energy resources (RE), as well as distributed energy storage (ES). This trend has evolved into the concept of a “microgrid” which can be described as a cluster of distributed energy resources, energy storage and local loads, managed by an intelligent energy management system [1], [2]. Similar to bulk power grid operation, microgrid operation can be determined by unit commitment (UC) and economic dispatch (ED). The UC is performed from one day to one week ahead providing the start-up and shut-down schedule for each generation and storage unit, which can minimize the operating cost of the microgrid. After the UC is determined, then ED is performed from few minutes to one hour in advance to economically allocate the demand to the on-line units while considering all unit and system constraints [3]. Although operation optimization for bulk power systems has been well studied in the literature, the traditional UC methods

Manuscript received January 07, 2015; revised May 04, 2015 and May 20, 2015; accepted July 06, 2015. This work has been supported in part by the Department of Energy SunShot program under DE-0006341 and in part by the National Science Foundation FREEDM ERC program. Paper no. TPWRS-000042015. The authors are with the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65401 USA. Digital Object Identifier 10.1109/TPWRS.2015.2455491

cannot be applied directly to microgrids with high penetration of RE and ES. Specifically, due to the stochastic nature of the renewable resources such as solar and wind, the mismatch in forecast and realized power may result in extra operating costs for committing costly reserve units or penalty cost for curtailing demand. In addition, to better utilize the renewable energy in the microgrid it is necessary to charge/discharge and coordinate multiple energy storage units in an efficient and economical way. To address these problems, the stochastic model of renewable energy and load demands as well as the working characteristics and operating cost of the ES devices should be incorporated into the UC and ED. Several approaches related to stochastic optimization of operation for renewable-based microgrids have been conducted. In [4], the day-ahead scheduling of a microgrid is developed as a two-stage stochastic problem in which the first stage identifies the optimal dispatch for the distributed units while the second stage considers the variability and uncertainty of photovoltaic (PV) and wind energy generation. The probabilistic UC in [5] is similarly formulated as a two-stage stochastic programming problem in order to incorporate the uncertainty in load and PV forecast. Forecast errors are modeled by normal distribution. A two-stage stochastic programming is also used in [6] and [7]. Most of the existing studies are based on scenario-based stochastic programming [4]–[9]. This approach is based on the replication of deterministic models across scenarios which are generated by Monte Carlo simulations. The computational burden in this approach increases exponentially with the number of investigated scenarios [2]. Scenario reduction using different techniques might ease the problem of computational overhead; however, this approach may overlook low-probability but high-impact scenarios. Therefore in this paper, we propose a probabilistic constrained approach to incorporate the uncertainties into the UC and ED for microgrids. In this approach, the hard constraint of exact power balance is relaxed by introducing a probabilistic constraint which contains renewable powers and load demands as random variables. The power balance constraint is enforced with high probability while the penalty for the constraint violation is applied in the cost function. The advantage of this method over the scenario-based method is that all possibilities of load demands and renewable generations are covered without the need to consider a large number of scenarios. Furthermore a stochastic dynamic programming is used to solve the UC. Most previous studies assume the energy storage system to be modeled with a constant efficiency and zero operating cost [4]–[16]. We propose a cost model for the battery that explicitly models the operating cost as a function of efficiency, life cycle, and state of health (SOH). This new cost model enables

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the battery to be treated as an equivalent fossil fuel generator in the economic dispatch problem. This cost model also makes economic dispatch for multiple batteries possible in the microgrid system without introducing additional objective functions to maximize their efficiencies and their life spans. Specifically, the following contributions are made: • A novel battery operation cost model is proposed accounting for charge/discharge efficiencies as well as cycling life time of the batteries. The model enables the battery to be treated as an equivalent fossil fuel generator in the UC and ED. • A probabilistic constrained approach is proposed to incorporate the uncertainties of the renewable sources and load demands in microgrids into the UC and ED problems. The UC is solved by stochastic dynamic programming. II. BATTERY OPERATION COST MODEL For a small-scale fossil fuel generator in a microgrid, the operating cost is typically the fuel cost. The cost can be characterized as a function of its output power [3], [10]:

price. In this paper, is be defined as the cost to have 1 kWh of storage capacity available: (4) is the total lifetime cycling capacity of a battery. where By convention, an electrochemical battery, such as lead acid or lithium-ion, is often considered to be at the end of its life when its has degraded to 80% of its rated energy capacity [17]. Assuming that a battery will be discharged to its rated depth of discharge every cycle, the average capacity degradation rate is in which is the battery rated capacity and is the rated life time. As opposed to a lead acid or a lithium-ion battery, a vanadium redox battery (VRB) has negligible capacity degradation from repeated deep discharges and recharges. The cycle life of a VRB mainly depends on the life expectancy of its proton exchange membrane and its pumps. A VRB can last over 10 000 cycles until its membrane degrades or the pumps fail. Therefore, the total lifetime usable capacity of a battery can be estimated as follows: • Lead acid and lithium-ion battery:

(1) (in $/gal) is the fuel price, (in gal/h) is where fuel consumption and is output power of generator . As opposed to a generator, a battery consumes no fuel to operate. This makes it a challenge to evaluate the operating cost of a battery. However, in terms of the energy conversion process, a battery and a generator are analogous. In a generator, energy is stored in fuel form and generated into electricity via combustion process. Similarly, in a battery, electricity is charged and discharged via an electrochemical process. In general, charging a battery is analogous to refilling fuel for a generator; thus, the input electricity (kWh) can be considered as the “fuel” for the battery. The input electricity cost is denoted as to emphasize this analogy. Therefore, the operating cost of a battery can price be determined in the same form as (1) by deriving and consumption of the battery. In this paper, lead-acid, lithium-ion, and vanadium redox batteries are considered. A.

Price for Battery

For a generator, the price of fossil fuel two components:

is composed of

(5) (6) • Vanadium redox battery: (7) where is the depth of discharge. The operating cost model of the battery is built based on the similarity with fuel cost model of a generator, therefore there is little added complexity over standard approaches. The kWhf price for a battery does not change as frequently as fuel prices. The price includes the replacement cost, the rated capacity, and life cycle which are determined at the time of purchase and do not need to be updated. B.

Consumption for Batteries

consumption of a battery during discharge is deThe fined as the energy usage for supplying a load during a unit time: (8)

(2) represents the cost for fuel and represents in which availability cost. The availability cost includes fuel transportation cost and other service costs such as cost for on-site storage facility. Depending on the location of the generator, can be due to transportation and other service much larger than costs. Similarly, the price for a battery can be determined: (3) where is the price of energy used to charge the battery represents the availability cost of battery capacity. In and a microgrid, if renewable energy is used to charge the battery, can be zero; therefore, is the main portion of the

is the battery output power and is the power in which loss during discharge. consumption of a battery during charge is defined The as the energy loss for charging the battery during a unit time: (9) is the battery charge power, is in which the power loss during charge. Depending on battery type, and can be characterized as functions of and , respectively. In this section, and are characterized for lead-acid, lithium-ion, and vanadium redox batteries. 1) Lead Acid and Lithium-Ion Battery: The power loss in lead-acid or li-ion batteries is mainly caused by the heat loss

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during charge or discharge. The heat is generated by ohmic resistances of the electrodes and electrolytes, and also by polarization effects [18]. The power loss is proportional to the voltage drop (polarization) caused by the current:

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TABLE I VRB LOSS MODEL COEFFICIENTS

(10) For lead-acid and li-ion batteries, the voltage drop can be determined based on the empirical method proposed in [19]: • During discharge: (11)

From (16), (17), the consumption during charge and discharge for the VRB can be determined: • During discharge:

• During charge: (12) (18) where is the internal ohmic resistance, is a constant which can be calculated from manufacturer's data, and is the rated capacity of the battery. From (11), (12), the consumption during charge and discharge for lead acid and lithium-ion batteries can be determined: • During discharge:

• During charge:

(19) III. STOCHASTIC UNIT COMMITMENT FOR MICROGRIDS A. Problem Formulation

(13)

In this paper, the stochastic unit commitment is formulated to minimize the expected operation cost of a microgrid over a time horizon . The objective function is therefore

• During charge: (20) (21) in which (14) is the state of charge and is the rated voltage of where battery. 2) Vanadium Redox Battery: The power loss of a vanadium redox battery during charge and discharge includes two components: power for pumping the electrolytes and stack loss power due to internal resistance and electrochemical process. Based on the empirical approach proposed in [13], the open circuit voltage and stack voltage can be characterized as functions of and charge/discharge power: • Open circuit voltage: (15) • During discharge: (16) • During charge: (17) All model coefficients are given in Table I where are the rated voltage and rated current of the VRB.

and

(22)

(23)

(24) (25) where • is the time horizon, is the time step; • is the total expected operation cost in period ; is the total transition cost which accounts for the start-up and shut-down cost of the generators in period . • and are the number of generators and batteries, respectively; and denote generator and battery , respectively; • • are respectively the total operation cost during period of the generators, the discharging batteries



and the charging batteries; is the cost due to power mismatch; • are respectively the binary status during period of the generator and battery ; due to the fact that a battery could not charge and discharge at the same time. • is the fuel price for generator ; is the price for battery ; is the fuel consumption of generator ; • and are respectively the consumptions during discharging and charging of battery ; • are respectively the dispatched power during period to generator , discharging battery and charging battery ; is the power mismatch during period . To better define the problem, the following conventions are introduced: • Charging power is considered as negative generation. • Renewable sources (PV and Wind turbines generators) are not dispatchable and considered as a negative load. The net load at period is defined as

Since are random, is considered as a random variable. • The batteries are charged only when . during period is the differ• The power mismatch ence between total generation and the net load defined as follows: — If

R6 Once a generator is brought online, it should remain online for a minimum set time; when a generator is powered off, it should remain off a minimum time before it can be restarted. In a small system such as a microgrid, grid power should not be used to charge the energy storage because of the relatively low ESS round-trip efficiency. Therefore, ESS should not charge other ESS, nor should grid power be used for energy storage. Only renewable energy should be used to charge the ESS; this is reflected in constraint R2. The above constraints are formulated as follows:

if if if

is online

if

is offline

is discharging is charging

is the state of charge of battery in period and where can be updated: if discharging if charging.

with — If

(26)

will happen By enforcing constraint R1, of the time when and of the time when . From (26) and (27), R1 can be rewritten as

(27)

The parameter governs the level of probability of importing/ exporting electricity to the microgrid from the grid. If , then the probability that the microgrid produces sufficient power internally is zero and it must import the required power from the grid. At the other extreme, if , then the probability that sufficient power is always produced within the microgrid is 1. This is not a realistic condition, therefore is constrained to be strictly less than 1.0 (but may ideally be quite close to 1.0). Similarly, if , then all net renewable energy will be used for charging the energy storage. For example, if , there is 50% probability that the excess power from the renewable energy will be exported to the grid. Choosing a larger will reduce the probability that renewable energy is used to charge the energy storage, enabling more renewable energy to be exported to the grid. Therefore, depending on energy management policies, and can be flexibly selected and potentially vary by time period (or remain constant). The amount of imported/exported power is governed by the choice of the and parameters. Choosing a smaller will increase the probability that the system will import power from

is the total generation.

with is the total charge. • The microgrid is grid connected. The electricity price and buy back price are considered as deterministic. The constraints of the problem are defined based on the energy management strategies and physical limits of the devices in the microgrid. The following constraints are considered: R1 The power mismatch is greater than zero with a predefined probability. R2 One battery should not be discharged to charge another battery; Generators should not be used to charge the batteries. R3 Each storage device cannot be charged (or discharged) beyond the maximum (or minimum) SOC. R4 The charge (or discharge) rate for each storage device should not exceed the maximum (or minimum) rates. R5 Each generator should exceed its minimum output setpoint when online.


the grid to supply its load, while choosing a larger will increase the probability that the system will export excess renewable power to the grid. The selection of and provide a level of freedom to decide whether or not and how much to import/export power from/to the grid, depending on desired energy management policies. B. Uncertainties in Forecasting Error of Load Demands and Renewable Sources To realize the cost function (21) and constraint R1, the cumulative distribution function (CDF) and mean value of need to be specified. In practice, the predicted values of load, PV, and wind generation at time period can be obtained beforehand based on forecast. Therefore, the realization of actual load, PV, wind generation and net load can be expressed as [5], [8], [9] (28) (29)

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if if The system will generate (or charge) more or less power by and . The expected operation choosing different values of cost in (21) can be expressed: (33) in which

(34)

(30) (31)

(35)

where are the forecasting errors which depend on the forecasting method and forecasting horizon. To model the uncertainties of load and renewable source forecast, are considered to be random variables. Although the wind power forecast error can be more precisely described with a Weibull, Cauchy [20] or mixed Normal-Laplace distribution [21], it can be approximated with a zero mean Normal distribution [22]. Furthermore, the net load error , which is the sum of all errors, can be approximated with a zero mean Normal distribution [22] due to the fact that the load demands and PV generation forecast errors are very close to a Normal distribution [5], [8], [9], [22]. The standard deviation of can be calculated as follows:

where is the price of the electricity which is exported (imported) to the grid.

(32) As a result, the following expected values and probabilities can be calculated:

C. Stochastic Dynamic Programming This UC problem can be categorized as a sequential decision-making problem for which dynamic programming (DP) is well known. Dynamic programming is a method to find the shortest route to the destination by breaking it down to a sequence of steps over time; at each step DP finds the possible optimum sequences (routes) based on the possible optimum subsequences in the previous steps and finally find the optimum sequence at last step. The main advantage of DP is it can maintain solution feasibility by its ability to find the optimum subsequence while searching for the optimum sequence. The primary drawback to DP is it can be computationally burdensome. For example, in an -unit system, there are combinations periods the total number of combinaat each period and for tions is [3]. For a large scale system the computation required to traverse this space can be overwhelming. However, in microgrid applications, the small number of units and large number of constraints significantly decrease the search space, therefore DP can be an appropriate choice of algorithm [13], [23]. Due to the uncertainties associated with the stochastic problem, the cost at each stage is generally a random variable [24]. Therefore, in the stochastic DP technique, the problem is formulated to minimize the expected cost. For applying DP, the states space at stage is defined as follows: (36)

(37) where is the CDF of Standard Normal distribution Therefore constraint R1 can be realized as

.

is the set of feasible states in stage ; is the number where is the binary status of unit which can be of states of ; a generator, a discharging battery or a charging battery. is a



Fig. 2. Typical microgrid. Fig. 1. Forward DP algorithm. TABLE II DIESEL GENERATOR DATA

valid state if it satisfies constraints R2, R3, R6 and the following conditions: if otherwise.

TABLE III BATTERIES DATA

In this paper, the forward DP algorithm is used (shown in Fig. 1). The algorithm to compute the minimum cost to arrive in stage is at state

(38) where is the minimum cost to arrive to state ; is the operating cost for state and is the to state . transition cost from state The operating cost can be found by performing an economic dispatch (ED) to minimize the cost function (33) with constraints R1, R4, and R5. In this paper, a steepest descent algorithm is used to solve the ED. IV. CASE STUDY AND RESULTS A case study is developed to test the proposed approach. Fig. 2 shows a typical microgrid which is connected to the low voltage side of a distribution transformer to power residential loads. The microgrid includes a 50 kW diesel generator, 2 20 kW wind turbines, a 50 kW PV array, a 10 kW/40 kWh Vanadium Redox battery, and an 12 kW/30 kWh AGM lead-acid battery. The total load is 50 kW at peak. The cost for the AGM battery is estimated at $8000. The replacement cost for the VRB is estimated at $20 000 [25]. The use of diesel generators in this case study is application specific. While most urban grids do not rely on auxiliary generation, many rural, developing nation, and military microgrids are in areas that rely on auxiliary diesel units. This case study approach would be valid in these applications.

Fig. 3. Load and renewable power forecast.

The data of generator and batteries, extracted from manufacturers' data [26]–[28], and their initial states are given in Table II and Table III. Day-ahead forecast values of total load demands, PV and wind generations are given in Fig. 3 based on per-unit data from [29] and [30]. The standard deviations for load, PV, and wind power forecast errors are respectively 3.12%, 12.5%, and 13.58% [31], [32]. The parameters and are chosen to be 0.9 and 0.1,


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TABLE IV NET LOAD FORECAST ERROR STANDARD DEVIATION

Fig. 5. Stochastic UC solution.

Fig. 4. Deterministic UC solution.

respectively. The high value of indicates a high probability of meeting all loads internally. The low value of indicates a low probability of exporting renewable energy to the grid (i.e. the preference is to use excess generation to charge the energy storage units). The standard deviations of the net load forecast error at each hour are calculated from (32) and given in Table IV. The results for the deterministic UC is shown in Fig. 4. By incorporating the operation cost functions of the batteries, the ED tends to dispatch power to the batteries which have a longer cycle life, lower replacement cost, and higher efficiency. In this case, the VRB has lower , however the AGM battery has higher efficiency, therefore their dispatched powers, as shown in the results, are close. Compared to the diesel generator, the batteries have lower operating cost due to lower “fuel” price and higher efficiency. However, the batteries are limited by their maximum depth of discharge. For that reason, the batteries can only discharge for few hours at night, as observed in the results. The results for the stochastic UC is shown in Fig. 5 in comparison with the deterministic case. The impact of the choice of and can be seen by comparing the stochastic and deterministic cases. • When the load is high and renewable generation is low (approximately hours 15 through 24), the stochastic algorithm over-commits the diesel generator compared to the deterministic case as indicated by . Since is chosen to be large (0.9), this indicates that the load must be met internally with high probability. Since renewable energy is not available during these hours, this requires that the diesel generator must be available to accommodate any potential variability in the load. • When the load is high and the renewable generation is high (approximately hours 9 through 14), the stochastic algorithm over-commits charging the energy storage units. Since is chosen to be low (0.1), this indicates a low probability of sending excess generation to the grid, thereby increasing the likelihood of charging the batteries.

• When the load is low and there is no renewable generation, the stochastic commitment algorithm closely tracks that of the deterministic case (hours 0 through 8). Through the choice of and , the amount of allowable risk in the system can be adjusted. In this example, the values of both and were held constant throughout the 24 hour period, but in practice, these values may be varied to account for anticipated variability in load or renewable generation. Furthermore, note that although not explicitly expressed, the two energy storage units were committed in accordance with their individual operating profiles as detailed in Section II to maximize their life spans. This case study provides the framework for assessing the impact of the proposed approach. The primary attribute of the proposed method is that it better describes the actual performance of the energy storage system performance and adjusts the allocation of resources in response. Specifically, most energy management and resource allocation approaches do not explicitly consider lifecycle degradation due to deep discharge nor do they consider efficiency as a function of output (or input) power. In this case study, the VRB and AGM lead-acid batteries fared similarly; even though the VRB was considerably more expensive to install, its better lifecycle attributes and efficiency characteristics produced similar long-term economic profiles to the AGM lead-acid batteries. Furthermore, both batteries were more cost effective than the diesel generator. Thus, it makes better economic sense to increase the size of the energy storage system with respect to the diesel generator. Furthermore, note that the results from the stochastic analysis indicate a heavier reliance on the diesel generator with respect to the deterministic case, since with increased uncertainty the diesel generator becomes a more reliable resource. This can be counteracted by decreasing the size of . V. CONCLUSIONS In this paper, a novel battery operating cost model has been proposed. The model considers cycle life and charge/discharge efficiencies for the batteries. The model makes economic dispatch for multiple batteries possible in the microgrid system without introducing additional objective functions to maximize their efficiencies and their cycle lives. In addition, a probabilistic constrained approach has been proposed to consider the uncertainties in load and renewable power forecast errors. Stochastic dynamic programming is applied in this method



to find the optimal day-ahead scheduling for a typical microgrid with a diesel generator, PV arrays, wind turbines, VRB, and AGM batteries. Results show the proposed approach can maintain the system optimal operation with a high probability without investigating a vast number of scenarios. Future work in this area will include the consideration of electricity prices in selecting and to maximize the microgrid's profit. REFERENCES [1] , N. Hatziargyriou, Ed., Microgrids: Architectures and Control. New York, NY, USA: Wiley, 2014. [2] H. Liang and W. Zhuang, “Stochastic modeling and optimization in a microgrid: A survey,” Energies, vol. 7, no. 4, pp. 2027–2050, 2014. [3] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, Control. New York, NY, USA: Wiley, 1996, vol. 2. [4] W. Su, J. Wang, and J. Roh, “Stochastic energy scheduling in microgrids with intermittent renewable energy resources,” IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 1876–1883, Jul. 2014. [5] X. Peng and P. Jirutitijaroen, “A probabilistic unit commitment problem with photovoltaic generation system,” in Proc. TENCON 2009—2009 IEEE Region 10 Conf., Jan. 2009, pp. 1–6. [6] M. Lubin, C. G. Petra, M. Anitescu, and V. Zavala, “Scalable stochastic optimization of complex energy systems,” in Proc. 2011 Int. Conf. IEEE High Performance Computing, Networking, Storage and Analysis (SC), 2011, pp. 1–10. [7] G. Martinez, N. Gatsis, and G. B. Giannakis, “Stochastic programming for energy planning in microgrids with renewables,” in Proc. 2013 IEEE 5th Int. Workshop Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2013, pp. 472–475. [8] S. Mohammadi, S. Soleymani, and B. Mozafari, “Scenario-based stochastic operation management of microgrid including wind, photovoltaic, micro-turbine, fuel cell and energy storage devices,” Int. J. Elect. Power Energy Syst., vol. 54, no. 0, pp. 525–535, 2014. [9] T. Niknam, R. Azizipanah-Abarghooee, and M. R. Narimani, “An efficient scenario-based stochastic programming framework for multi-objective optimal micro-grid operation,” Appl. Energy, vol. 99, no. 0, pp. 455–470, 2012. [10] M. Mazidi, A. Zakariazadeh, S. Jadid, and P. Siano, “Integrated scheduling of renewable generation and demand response programs in a microgrid,” Energy Convers. Manage., vol. 86, no. 0, pp. 1118–1127, 2014. [11] H. Bilil, G. Aniba, and M. Maaroufi, “Probabilistic economic emission dispatch optimization of multi-sources power system,” Energy Procedia, vol. 50, no. 0, pp. 789–796, 2014. [12] A. A. ElDesouky, “Security and stochastic economic dispatch of power system including wind and solar resources with environmental consideration,” Int. J. Renew. Energy Res. (IJRER), vol. 3, no. 4, 2013. [13] T. Nguyen, X. Qiu, J. Guggenberger, II, M. Crow, and A. Elmore, “Performance characterization for photovoltaic-vanadium redox battery microgrid systems,” IEEE Trans. Sustain. Energy, vol. 5, no. 4, pp. 1379–1388, Oct. 2014. [14] Y. Tan, Y. Cao, C. Li, Y. Li, L. Yu, Z. Zhang, and S. Tang, “Microgrid stochastic economic load dispatch based on two-point estimate method and improved particle swarm optimization,” Int. Trans. Elect. Energy Syst., 2014. [15] J. M. Lujano-Rojas, G. J. Osorio, and J. P. Catalao, “A probabilistic approach to solve economic dispatch problem in systems with intermittent power sources,” in Proc. 2014 IEEE PES T&D Conf. Expo., 2014, pp. 1–5. [16] Y. Zhang, N. Gatsis, and G. B. Giannakis, “Risk-constrained energy management with multiple wind farms,” in Proc. 2013 IEEE PES Innovative Smart Grid Technologies (ISGT), 2013, pp. 1–6. [17] T. Markel, K. Smith, and A. Pesaran, “Improving petroleum displacement potential of phevs using enhanced charging scenarios,” in EVS-24 NREL/CP-540-45730, 2009. [18] H. A. Kiehne, Battery Technology Handbook. Boca Raton, FL, USA: CRC Press, 2003, vol. 60. [19] O. Tremblay and L.-A. Dessaint, “Experimental validation of a battery dynamic model for ev applications,” World Elect. Vehicle J., vol. 3, no. 1, pp. 1–10, 2009.

[20] B. Hodge and M. Milligan, “Wind power forecasting error distributions over multiple timescales,” in Proc. 2011 IEEE Power and Energy Soc. General Meeting, 2011, pp. 1–8. [21] J. Wu, B. Zhang, H. Li, Z. Li, Y. Chen, and X. Miao, “Statistical distribution for wind power forecast error and its application to determine optimal size of energy storage system,” Int. J. Elect. Power Energy Syst., vol. 55, pp. 100–107, 2014. [22] R. Doherty and M. O'Malley, “A new approach to quantify reserve demand in systems with significant installed wind capacity,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 587–595, May 2005. [23] L. M. Costa and G. Kariniotakis, “A stochastic dynamic programming model for optimal use of local energy resources in a market environment,” in Proc. 2007 IEEE Lausanne Power Tech, 2007, pp. 449–454. [24] D. P. Bertsekas, Dynamic Programming: Deterministic and Stochastic Models, D. P. Bertsekas, Ed. Englewood Cliffs, NJ, USA: PrenticeHall, 1987. [25] G. Kear, A. A. Shah, and F. C. Walsh, “Development of the all-vanadium redox flow battery for energy storage: A review of technological, financial and policy aspects,” Int. J. Energy Res., vol. 36, no. 11, pp. 1105–1120, 2012. [26] MQ Power [Online]. Available: http://www.powertechengines.com/ MQP-DataSheets/MQP050IZ_Rev_0.pdf [27] Prudent Energy [Online]. Available: http://www.pdenergy.com/pdfs/ Prudent_Energy_Product_Brochure_2011.pdf [28] Sunxtender [Online]. Available: http://www.sunxtender.com/solarbattery.php?id=11 [29] D. Hung, N. Mithulananthan, and K. Lee, “Determining PV penetration for distribution systems with time-varying load models,” IEEE Trans. Power Syst., to be published. [30] A. A. Moghaddam, A. Seifi, and T. Niknam, “Multi-operation management of a typical micro-grids using particle swarm optimization: A comparative study,” Renew. Sustain. Energy Rev., vol. 16, no. 2, pp. 1268–1281, 2012. [31] N. Lu, R. Diao, R. Hafen, N. Samaan, and Y. Makarov, “A comparison of forecast error generators for modeling wind and load uncertainty,” in Proc. 2013 IEEE Power and Energy Soc. General Meeting (PES), 2013, pp. 1–5. [32] F. Kreikebaum, R. Moghe, A. Prasai, and D. Divan, “Evaluating the application of energy storage and day-ahead solar forecasting to firm the output of a photovoltaic plant,” in Proc. 2011 IEEE Energy Conversion Congr. Expo. (ECCE), 2011, pp. 3556–3561. [33] A. Zakariazadeh and S. Jadid, “Energy and reserve scheduling of microgrid using multi-objective optimization,” in Proc. 22nd Int. Conf. Exhib. Electricity Distribution (CIRED 2013), 2013, pp. 1–4.

Tu A. Nguyen received the B.S. degree in Power Systems from Hanoi University of Science and Technology, Hanoi, Vietnam, in 2007 and the Ph.D. degree from Missouri University of Science and Technology, Rolla, MO, USA, in 2015. He worked as a Power Transformer Test Engineer in ABB's High Voltage Test Department in Vietnam from 2008 to 2009. His research interests include microgrid system modeling/analysis and power electronics applications in microgrid systems.

M. L. Crow (S'83–M'90–SM'94–F'10) received the B.S.E. degree from the University of Michigan, Ann Arbor, MI, USA, and the Ph.D. degree from the University of Illinois, Urbana/Champaign, IL, USA. She is the F. Finley Distinguished Professor of Electrical Engineering at the Missouri University of Science & Technology, Rolla, MO, USA. Her research interests include computational methods for dynamic security assessment and the application of power electronics in bulk power systems. Prof. Crow is a Registered Professional Engineer in the State of Missouri.