Control of Smart Grid Architecture

Chapter 19

Control of Smart Grid Architecture Hrvoje Pandzˇic* and Tomislav Dragicˇevic† * †

Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia, Department of Energy Technology, Aalborg University, Aalborg, Denmark

NOMENCLATURE The nomenclature used in this chapter is listed here for a quick reference. In general, variables take lowercase letters and parameters take uppercase letters.

Sets ΩM Ωpv ΩT Ωλ

Set Set Set Set

of of of of

parts of a piecewise thermal generator cost curve indexed by m. photovoltaic output scenarios indexed by p. time periods indexed by t. market price scenarios indexed by l.

Variables ct drft drrft gpv t gpv p, t gpv,curt t gpv,curt p, t gth t

Cost of operating the thermal generator during time period t, €/MW. Reduced part of the flexible demand during time period t, MW. Retrieved amount of the reduced part of the flexible demand during time period t, MW. Output of photovoltaics during time period t, MW. Output of photovoltaics during time period t and under photovoltaics output scenario p, MW. Curtailed part of the available photovoltaics output during time period t, MW. Curtailed part of the available photovoltaic output during time period t and under photovoltaics output scenario p, MW. Output of thermal generator during time period t, MW.

Control of Power Electronic Converters and Systems. https://doi.org/10.1016/B978-0-12-816136-4.00019-1 © 2018 Elsevier Inc. All rights reserved.

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gth m, t gw t gw,curt t ht qpt qst rw t sch t sdis t soct ut vt xt yt zt αt βt

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Output of the mth segment of thermal generator during time period t, MW. Output of wind generator during time period t, MW. Curtailed part of the available wind generator output during time period t, MW. Binary variable equal to 1 if storage is charged during time period t, and 0 otherwise. Electricity purchased in the market at time period t, MW. Electricity sold in the market at time period t, MW. Auxiliary variable used for linearization in robust optimization, MW. Charging power of storage during time period t, MW. Discharging power of storage during time period t, MW. Storage state of charge at the end of time period t, MWh. Auxiliary variable used for robust optimization, MW. Auxiliary variable used for robust optimization, MW. Binary variable equal to 1 if thermal generator is online during time period t, and 0 otherwise. Binary variable equal to 1 if thermal generator is started during time period t, and 0 otherwise. Binary variable equal to 1 if thermal generator is shut down during time period t, and 0 otherwise. Binary variable equal to 1 during the demand response provision, and 0 otherwise. Binary variable equal to 1 during the demand response retrieval, and 0 otherwise.

Parameters A Bm Dft Dnf t DRf,max t DRf,max t DT DURdr DURdrr Kf Pmax

Fixed generation cost of thermal generator, €. Slope of the mth segment of the thermal generator piecewise cost curve, €/MW. Flexible demand during time period t, MW. Nonflexible demand during time period t, MW. Maximum portion of flexible demand during time period t that can be reduced. Maximum portion of flexible demand during time period t that can be retrieved. Thermal generator minimum down time in time periods. Duration of demand response in time periods. Duration of demand response recovery in time periods. Load retrieval coefficient. Maximum output of thermal generator, MW.

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Pmax m Pmin Ppv t Ppv p, t Pw t RDth RUth Sch, max Sdis, max Sth SOCmax SOCmin UT Γ tpv Γ tw Δt ηch ηdis λt πl πp

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Maximum output of the mth segment of the piecewise cost curve of thermal generator, MW. Minimum output of thermal generator, MW. Available photovoltaic output during time period t, MW. Available photovoltaic output during time period t and under photovoltaic output scenario p, MW. Available wind turbine output during time period t, MW. Ramp down limit of thermal generator, MW/h. Ramp up limit of thermal generator, MW/h. Maximum charging power of storage, MW. Maximum discharging power of storage, MW. Start-up cost of thermal generator, €. Maximum storage state of charge, MWh. Minimum storage state of charge, MWh. Thermal generator minimum up time in time periods. Budget of uncertainty of the photovoltaics output, used in robust optimization. Budget of uncertainty of the wind turbine output, used in robust optimization. Length of the time step, e.g., 1 h. Storage charging efficiency. Storage discharging efficiency. Cost of electricity in the market at time period t, €/MW. Weight of each market price scenario. Weight of each photovoltaic output scenario.

This chapter builds upon the previous chapter where the internal microgrid control is described. It is shown that a microgrid is built from a number of power electronic converters, which comprise the lowest architectural level of a microgrid. From the control standpoint, those converters are responsible for regulating the physical variables (i.e., the current and/or the voltage) depending on their functionality. This level of control is commonly referred to as the local control layer and ensures that every converter and, consequently, the microgrid are operated in a safe way. The second part of Chapter 18 also stresses the importance of integrating a coordinated control layer on top of the basic regulation. The purpose of that layer is to optimize the real-time performance of the microgrid in a long run. Its calculated outputs are forwarded to the local control layer in the form of voltage and current references, which the latter simply follows [1, 2]. From a practical point of view, coordinated control layer can be realized through either centralized, decentralized, or distributed implementation [3]. It should be noted that this research topic has been more prevalent in DC microgrid research community since the challenges of realizing the basic functionalities are relatively straightforward. Nonetheless, real-time

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optimization procedures have been proposed for AC microgrids as well [4], focusing on optimization of technical performance indicators within the microgrid, while neglecting the cost optimization, which is the main operation driver in the Smart Grid environment [5]. This chapter is therefore focused on elaborating the principles of minimizing the long-term microgrid operating cost through a hierarchical control structure, which comprises two main layers: the local control layer and the optimization layer. Fig. 19.1 shows physical and communication structure of a microgrid. The upper part shows physical microgrid elements (conventional and renewable generators, storage, and loads). The microgrid is connected to the distribution

Microgrid sources and loads Solar panels

Battery storage

DC loads

Distribution system

Wind turbine

AC loads

Data acquisition

AC generators

Coordinated control commands

Microgrid control layer System state parameters

Microgrid optimal setpoints Energy bids

Microgrid optimization layer

Energy prices Local distribution-level market

FIG. 19.1 Physical and communication structure of a microgrid. Full lines represent energy flows and dashed lines represent information and control flows.

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system. The dotted lines represent communication and control within the microgrid. The optimization layer makes decisions on optimal economic operation of the microgrid based on the current system state parameters, expected load levels, outputs of noncontrollable energy sources within the microgrid, and expected energy prices in the local distribution-level market. The outcome of this economic optimization are optimal microgrid setpoints in the coming time periods and energy bids in the market. On the other hand, the microgrid control layer sends coordinated control commands to all microgrid elements in order to follow the optimal setpoints provided by the optimization layer. At the same time, the control layer collects data on actual states of the microgrid elements (actual outputs of electricity sources; actual load levels; and energy storage charge, discharge, and state of charge values). These data are used as inputs in the microgrid control layer, as well as in the microgrid optimization layer when necessary. The optimization layer derives the long-term operational schedule based on forecasted values of the uncertain parameters: output of local renewable energy sources, for example, wind turbines and photovoltaics, local load curve, and electricity prices (in case of dynamic pricing). Since this layer usually looks 24 h ahead and considers uncertain parameters, it usually uses simple, that is, linear, models of the microgrid elements and produces the microgrid trajectory (setpoints) for the following 24 h. The optimal trajectory is a set of variable values in time. For instance, if the optimization is performed in 15-min discrete intervals, the trajectory consists of the values of variables at each discrete time step. This trajectory is then passed on to the control layer that uses a detailed representation of the microgrid elements. The control layer runs continuously in a closed loop and tries to match the imposed trajectory setpoints making the necessary adjustments in real time. The optimization level is restarted if (i) updated forecasts of uncertain parameters are available; or (ii) if the control layer starts to deviate from the imposed trajectory and is unable to find a feasible solution. As shown in Fig. 19.2, all values within the microgrid are constantly measured and fed as input data to both layers on demand. The mechanics of the interaction between the optimization layer described in this chapter and the control layer described in the previous chapter are shown in Figs. 19.3–19.5. The optimization layer passes on many values for each time period to the control layer (thermal generator output, photovoltaics output, wind turbine output, storage state of charge, flexible demand reduction and recovery). However, for clarity, in the following figures, we only show the storage state of charge trajectory. Also, we only consider the uncertainty of wind turbine output. The time step of the optimization layer is 1 h and the optimization horizon is 24 h. The optimization layer derives optimal storage state of charge trajectory, shown as black curve in Fig. 19.3 based on the central wind forecast (gray curve). Ideally, the optimization layer should consider a range of wind uncertainty represented as the light-gray area in Fig. 19.3. This range of uncertainty is

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Renewable energy sources output forecast

Current state

Load forecast

Price forecast

Optimization layer Stochastic optimal scheduling Runs every 24 h or less

Monitoring Optimal trajectory

Data acquisition Current state

Control layer Optimal adjustments of the control resources Runs continously

FIG. 19.2 Microgrid operation scheme using optimization and control layers.

Storage state of charge Wind turbine output

100%

50%

0% 0 1 2 3 4 5 6 7 8 9 10

12

16

20

24

Time periods (h) FIG. 19.3 Storage state of charge setpoints (black curve) based on the central wind forecast (gray curve) within an uncertainty range (light-gray area).

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Storage state of charge Wind turbine output 100%

50%

0% 0 1 2 3 4 5 6 7 8 9 10

12

16

20

24

Time periods (h) FIG. 19.4 Control layer cannot meet the optimization layer trajectory resulting in a lower storage state of charge (thick black dotted curve) due to less available wind power (gray dashed curve).

derived as an envelope around all possible realizations of wind forecast, while the gray curve is the most probable wind output scenario (i.e., central forecast). Fig. 19.4 shows the control layer following the trajectory from the optimization layer for the first 4 h (thick black dotted curve). However, during hour 4 the control layer struggles to follow the trajectory due to a wind forecast error. As a result, actual storage state of charge cannot reach the desired setpoint provided by the optimization layer. At this point, the optimization layer is run considering the updated wind forecast (dashed gray curve in Fig. 19.4) and current state of all microgrid elements. The simulation results in an updated storage trajectory (among other trajectories for other microgrid elements) represented with dashed black curve in Fig. 19.5. This procedure continues to repeat, which ensures optimal microgrid economic performance considering the uncertain parameters. This chapter is focused on the optimization layer described above and is organized as follows. Section 19.1 describes the microgrid optimization model used to maximize its profit. The constraints include models of all microgrid elements. Section 19.2 describes uncertainty models that can be used to tackle the unknown, but predictable nature of certain model parameters. These models include stochastic, robust, and interval optimization. Finally, a numerical example of the microgrid operation is provided in Section 19.3. This example is solved using both the deterministic and the all three modeling techniques from Section 19.2. Code written in General Algebraic Modeling System (GAMS) [6] for all four cases is provided in Appendices A–D at the end of this book.

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Storage state of charge Wind turbine output 100%

50%

0% 0 1 2 3 4 5 6 7 8 9 10 12 16 Time periods (h)

20

24

FIG. 19.5 Optimization layer is run at hour 4 due to a deviation from the black trajectory using an updated wind forecast (gray dashed curve) providing new optimal storage setpoints (black dashed curve).

19.1 MICROGRID OPTIMIZATION MODEL The long-term goal of microgrid operation is the minimization of operating costs or maximization of profit, which are assumed equivalent in this case. In other words, the goal is to satisfy all the local energy needs and maximize market performance by exercising arbitrage (selling electricity at high price and purchasing it at low price). Arbitrage can be exercised under the time-of-use tariffs, which is common in today’s distribution network, but we are interested in the case when electricity prices at the distribution level are variable. These prices are variable if the distribution-level market is implemented, so small sources of flexibility can offer their services, or an aggregator providing dynamic price signals to the microgrid, thus transferring the wholesale market dynamics to its customers. In the rest of this chapter, a microgrid under dynamic pricing is considered, and this price is referred to as the market price. We consider a general case in which a microgrid consists of the following generic elements: l l l l l l

inflexible demand; flexible demand (capable of providing demand response); thermal generator (e.g., diesel generator); photovoltaic plant; wind turbine; and energy storage (e.g., batteries).

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The objective of the microgrid operation is the maximization of its profit: " # X  p s Maximize λ t q t  qt  c t (19.1) t2ΩT

The revenue is made by selling electricity qst at price λt, while the expenses include purchased electricity qpt at price λt and cost of the local generation ct. Objective function (19.1) is subject to the following constraints.

19.1.1 Power Balance Constraint pv p T w dis f nf f f ch s gth t + gt + gt + st + drt + qt ¼ Dt + Dt + drrt + st + qt , 8t 2 Ω

(19.2)

On the left-hand side are the terms that inject electricity (thermal generator, wind turbine, photovoltaics, discharging of storage, reduced part of the flexible demand, and electricity purchased from the distribution network), while on the right-hand side are the terms that withdraw electricity (nonflexible load, flexible load, retrieved part of the reduced demand, charging of storage, and electricity sold into the local distribution network).

19.1.2 Thermal Generator Model max gth  xt , 8t 2 ΩT t P

(19.3)

min gth  xt , 8t 2 ΩT t P

(19.4)

Thermal generators might have minimum stable output (i.e., level of minimum sustainable output which a generator unit is capable of producing). This means that either the thermal unit is off or its output is higher than its minimum stable output. To model this, binary variable xt is introduced. Constraints (19.3), (19.4) work in the following way: if xt takes value 0, gth t is forced to 0 and the th unit is off; if xt equals 1, Eq. (19.3) limits gth t from above to g and Eq. (19.4) th from below to g . yt  zt ¼ xt  xt1 , 8t 2 ΩT

(19.5)

yt + zt  1, 8t 2 ΩT

(19.6)

In order to model start-up costs and minimum on and off times of a thermal generator, we need to know in which time period the thermal generator was started and in which it was shut down. This is modeled through binary variables yt (value 1 if thermal generator is started during time period t and 0 otherwise) and zt (value 1 if thermal generator is shut down during time period t and 0 otherwise). Constraint (19.5) determines if the thermal generator is started up or shut down during the time period t based on the change of its on/off status (i.e., binary variable xt) between time periods t and t  1. Constraint (19.6)

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ensures that the thermal generator cannot be started up and shut down during the same time period. X T ct ¼ Sth  yt + A  xt + Bm  gth m , 8t 2 Ω (19.7) M X

gth t ¼

m2Ω

m2Ω

T gth m, t , 8t 2 Ω

(19.8)

M

T M max gth m, t  Pm , 8t 2 Ω , m 2 Ω

(19.9)

A general thermal generator cost curve is nonlinear (thick dashed curve in Fig. 19.6). In order to preserve linearity of the optimization model, this curve is linearized by pieces (gray lines in Fig. 19.6). On top of this, if a thermal generator is started during time period t, it induces start-up cost Sth. The thermal generator output cost is thus modeled in constraint (19.7) as the sum of the start-up cost, no-load cost, and variable cost that depends on the generator output. Since the start-up cost should be only considered in the time period a thermal unit went online, constraint (19.5) sets appropriate value to binary variable yt indicating that the thermal unit went online during time period t. Constraint (19.8) sets the thermal generator output to the sum of its piecewise segments, which are limited in Eq. (19.9). Since the output cost curve is monotonically increasing, variables gth m, t will start assigning output to the first piece of the linearized cost curve, then the second, and so on, resembling the real-life operation of a thermal generator. This is because the slope of the cost curve Bm is lowest for the first part of the curve and then monotonically increased toward the last piece of the cost curve.   (19.10) xt ¼ x0 8t 2 0, Lup + Ldn t X

ytt  xt 8t 2 ½Lup ,T 

(19.11)

tt¼tUT1

Cost (€/MW) B3 Cost curve B2 A

B1 g1th

0

g2th P min

g3th P max

Power (MW)

FIG. 19.6 Piecewise linear approximation of the cost curve of a thermal generator.

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  ztt  1  xt 8t 2 Ldn , T

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(19.12)

tt¼tDT1

Constraint (19.10) sets the on/off status for the first part of the optimization horizon based on the initial status of the generator. For example, if the generator must stay on for 3 h, Lup will be 3, and Ldn will be 0. If no minimum up or down time constraints are active at the beginning of the scheduling horizon, both Lup and Ldn will be 0. Constraints (19.11), (19.12) enforce minimum up and down times for the remaining time intervals as explained in Ref. [7]. T th th gth t  gt1  RU , 8t 2 Ω

(19.13)

T th th gth t1  gt  RD , 8t 2 Ω

(19.14)

Since thermal generators cannot instantaneously change their power output, constraints (19.13), (19.14) are used to limit the amount of output a thermal generator can change in between each two consecutive time periods. Constraint (19.13) limits the up-ramping capability of thermal generator to RUth, while Eq. (19.13) limits it down-ramping capability to RDth. In general, the higher RUth and RDth, the more flexible the thermal generator, which helps counteract the intermittency of renewable generation.

19.1.3 Wind Turbine Output w,curt w gw , 8t 2 ΩT t ¼ Pt  gt

(19.15)

The available wind turbine output Pwind is mostly used to supply the microt grid, which is modeled through variable gw t , but in some cases it can be curcurt . tailed, which is captured by variable gw, t

19.1.4 Photovoltaics Output pv pv,curt gpv , 8t 2 ΩT t ¼ Pt  g t

(19.16) Ppv t

Similar to the wind turbine output, the photovoltaics output comes at no operating cost. Variable gpv t represents the portion of available photovoltaics is the curtailed portion. output used to supply the microgrid, while gpv,curt t

19.1.5 Storage Model Energy storage is characterized by its state of charge, which is changing over time, and operates within the minimum and maximum state of charge. On top of this, the storage device is limited by its charging and discharging ratings and efficiencies. Microgrids can contain different types of energy storage (e.g., flywheels, fuels cells, or any of the battery technologies). Here, a generic

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energy storage model from [8] is presented (i.e., the model is independent on the storage technology itself ). t ch soct ¼ soct1 + sch t η Δ 

sdis t  Δt , 8t 2 ΩT ηdis

(19.17)

SOCmin  soct  SOCmax , 8t 2 ΩT

(19.18)

 Sch  ht , 8t 2 ΩT sch t

(19.19)

 Sdis  ð1  ht Þ, 8t 2 ΩT sdis t

(19.20)

Energy storage state of charge is calculated using Eq. (19.17). State of charge in the current time period is equal to the state of charge in the previous time period plus the energy charged minus the energy discharged. Both charging and discharging processes are imperfect and their respective efficiencies are considered. Δt denotes the length of the time step, usually 1 h, and is required to convert the charging/discharging power to charged/discharged energy. Constraints (19.18)–(19.20) limit the energy storage state of charge, charging power, and discharging power, respectively. In order to avoid simultaneous charging and discharging, binary variable ht is introduced. This variable takes value 1 if the storage is charged and 0 if it is discharged during time period t.

19.1.6 Demand Response Model Demand response presents a portion of flexible demand that is reduced at a certain time period. This means that the load is increased at other time periods in order to satisfy the constraints of the electricity consuming process within the microgrid, or the requirements of any other flexible load. Depending on the type of the load, more or less demand that was reduced will be retrieved at other time periods. For instance, if the flexible load in question is lighting, the retrieved load at other time periods may be close to 0, because if the light in a room is dimmed due to a demand response action, it may not require brighter light at other time periods. On the other hand, devices from the heating, ventilation, and air conditioning (HVAC) group will yield higher demand retrieval, thus resulting in higher overall demand. An example of the load whose demand response actions do not change the overall demand is pool pumps [9]. A generic flexible load is modeled using the following constraints. drtf  DRft  αt , 8t 2 ΩT

(19.21)

drrtf  DRRft  βt , 8t 2 ΩT

(19.22)

αt + βt  1, 8t 2 ΩT

(19.23)

Constraint (19.21) limits the amount of flexible load that can take part in , while constraint (19.22) limits demand response at each time period to DRf,max t the amount of flexible load that can be recovered at each time period to

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DRRf,max . The role of binary variables αt and βt is to prevent simultaneous reduct tion and recovery of the flexible demand within the same time period. Note that this would happen if the return ratio of a flexible demand is below 100%. X X drtf ¼ drrtf , 8t 2 ΩT Kf  (19.24) T T t2Ω

t2Ω

In order to enforce the retrieval of the reduced flexible demand within the optimization horizon, constraint (19.24) uses the load retrieval coefficient Kf. This coefficient depends on the type of deferred load and ranges from 0 for lights to well over 1 for heating and cooling devices. Depending on the flexible load type and the desired operating paradigm, some of the following constraints can be added. If the demand recovery has to start as soon as the demand response is over: βt  αt1  αt , 8t 2 ΩT

(19.25)

Demand response duration limit is imposed using the following constraint: t +X DURdr

αtt  DURdr , 8t 2 ΩT

(19.26)

tt¼t1

The demand response recovery duration limit is imposed using the following constraint: t +X DURdrr

αtt  DURdrr , 8t 2 ΩT

(19.27)

tt¼t1

If the demand can be recovered only after it had been reduced (i.e., no precooling or preheating is allowed), the following constraint should be used: X X drtf  K f  drrtf , 8t 2 ΩT (19.28) T T t2Ω

t2Ω

Microgrid profit maximization model (19.1)–(19.28) is a mixed-integer mathematical program with specific time resolution. The resolution is usually 1 h, but it can be higher (e.g., 15 min) in order to provide more accurate results. However, very high number of time periods will significantly increase the computational burden (i.e., time necessary to obtain the optimal solution of the model). This is relevant for microgrids with large number of thermal generators, storage and demand response units, as their models contain binary variables.

19.2 UNCERTAINTY MANAGEMENT The model presented in the previous section contains uncertain parameters (i.e., market prices λt, available photovoltaics output Ppv t , available wind turbine outnf f put Pw t , nonflexible load Dt , and flexible load Dt ). Once these uncertain

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parameters are identified, they need to be modeled in an appropriate way to avoid negative consequences of the poor uncertainty management. For example, if the production of the local uncertain renewable sources is overestimated, the microgrid will not be able to meet its demand during the real-time operation and will be forced to purchase expensive balancing energy. On the other hand, if the uncertain generation is underestimated, the operating plan will not be optimal as additional revenue could have been collected by selling the excess electricity in the market. There are multiple means of considering uncertainties in microgrid operation modeling. The most commonly used techniques, described here, include stochastic optimization, robust optimization, and interval optimization.

19.2.1 Stochastic Optimization In a stochastic optimization problem, the uncertain parameter is represented by a set of stochastic scenarios, each with an assigned probability [10]. In other words, distribution of uncertainty is approximated by a set of scenarios. These scenarios can be derived using very complex models, such as numerical weather forecast for deriving wind turbine and photovolatic outputs, by much simpler models, based on historical data, often used in the scientific literature [11], or by a combination of the two. In order to derive stochastic scenarios based on historical data, first a scenario generation technique needs to be employed. In general, the more the scenarios, the better the solution. However, high number of scenarios drastically reduces computational tractability of a stochastic problem. For this reason, a scenario reduction technique is employed to reduce the number of scenarios preserving as much information possible from the original scenario set. Scenario reduction techniques aggregate similar scenarios based on particular metrics, such as their probability, hourly magnitudes, or the cost resulting from each scenario [12]. In order to address uncertainty in the model from Section 19.1, a set of scenarios needs to be assigned to each uncertain parameter. For example, in order to consider uncertainty of the available photovoltaics output, uncertain parampv eter Ppv t is replaced by its set of output forecasts Pp, t , where p is an index of the pv photovoltaic output uncertainty set Ω . Therefore, constraint (19.16) needs to be updated to include multiple available photovoltaics scenario outputs: pv pv,curt gpv , 8p 2 Ωpv , t 2 ΩT p, t ¼ Pp, t  gp, t

(19.29)

Notice that the variables directly related to the uncertain parameter depend pv,curt are different for each photoon the scenario, that is, values of gpv p, t and gp, t voltaics output scenario p. The scenario dependency reflects on constraints that include these two variables. Thus, the power balance constraint becomes: pv p pv T w dis f nf f f ch s gth p, t + gp, t + gp, t + sp, t + drp, t + qt ¼ Dt + Dt + drrp, t + sp, t + qt , 8p 2 Ω , t 2 Ω

(19.30)

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w dis f f In this updated power balance equation, variables gth p, t , gp, t , sp, t , drp, t , drrp, t , ch and sp, t have different values for each photovoltaics output scenario. Photovoltaics output scenario dependence is then propagated to thermal generator, storage, and demand response-related constraints and variables. However, it is important to notice that variables qpt and qst are not scenario dependent. This is the direct result of the nonanticipativity of the future, meaning that at the moment when microgrid offers/bids in the market, the unknown parameter (i.e., photovoltaics output) is not known. Therefore, the hourly bids and offers in the market need to be the same over all scenarios. Only after the market is cleared and time progresses will the unknown parameter unveil and controllable resources (thermal generator, storage, and flexible demand) will operate in a way to maximize the profit. In stochastic programming terminology, variables qpt and qst are the first-stage variables, often referred to as here-and-now decisions, while the remaining variables are the second-stage variables, also known as wait-and-see decisions, as their final values are determined at the second stage. Another consequence of the uncertainty modeling is the change in the formulation and the meaning of the objective function. Instead of Eq. (19.1), the new objective function considers weight π p of each photovoltaics output scenario and generates a solution that represents the expected value: " # X X  p s Maximize πp λt qt  qt  cp, t (19.31) p2Ωpv

t2ΩT

In addition, if the market prices are considered uncertain as well, the objective function becomes: " # X X X  p s πp πl λl, t qt  qt  cl, p, t (19.32) Maximize p2Ωpv

l2Ωλ

t2ΩT

P It is important toPnote that the sum of all probabilities needs to be equal to 1: p2Ωpv π p ¼ 1 and l2Ωλ π l ¼ 1. A similar modeling practice can be applied to any uncertain parameter, such f as available wind turbine output Ptw, nonflexible load Dnf t , and flexible load Dt .

19.2.2 Robust Optimization Robust optimization is a more recent approach to uncertainty representation in mathematical programming than the stochastic optimization [13]. In robust formulation, the uncertainty model is based on uncertainty set, as opposed to stochastic scenarios with assigned probabilities. Instead of the uncertainty distribution assumption, robust formulation only considers the bounds of uncertainty without any implication of uncertainty distribution. The objective function is much different than in deterministic or stochastic models, as it optimizes the worst case realization of uncertainty within the given bounds.

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s1

s2

s3

s4

Lower and upper bounds

9 8 Output (MW)

7 6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

Time periods (h) FIG. 19.7 An example of determining bounds for robust optimization model based on four stochastic scenarios s1–s4.

The construction of uncertainty set in robust optimization starts the same way as in stochastic optimization—by generating a bunch of scenarios based on historical data. After that, instead of reducing the number of scenarios, the upper and lower bounds of uncertainty are defined in a way that (almost) all scenarios are beneath the upper bound and (almost) all of them are above the lower bound. The mechanics of deriving the upper and lower bounds of a noncontrollable generator are shown in Fig. 19.7. This example contains four stochastic scenarios of a generator output over a 6-h period. In the first hour, the expected output is 3 MW for scenario s1, and 4 MW for scenarios s2, s3, and s4. Therefore, the lower and upper bounds are set to 3 and 4 MW, respectively. In the second hour, scenario s2 provides 4 MW output, while scenarios s1, s3, and s4 provide 5 MW output. As a result, the lower and upper bounds are set to 4 and 5 MW, respectively. The same reasoning is applied for the remaining hours. In case many scenarios are used, that is, hundreds, these bounds could be far apart, thus rendering the forecast model useless. In this case, the boundary scenarios should be left out of the uncertain area. For example, if bounds are set at 5th and 95th percentile, this indicates that 5% of scenarios are below the lower bound and 95% of scenarios are below the upper bound. In order to model the uncertainty of the wind turbine output using robust w optimization, Pw t is  w  modeledwas random variable pt that can take values within w w w Pt  δ , Pt + δ , where δ  0 represents deviation from the mean expected value Pw t . Constraint (19.15) is robustified in the following way: w,curt w w w w , 8t 2 ΩT gw t ¼ Pt  ut  Γ t  v t  gt

(19.33)

T w +vw uw t t  δw  rt , 8t 2 Ω

(19.34)

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19

rtw  1, 8t 2 ΩT

(19.35)

T w uw t  0, vt  0, 8t 2 Ω

(19.36)

w The wind availability constraint now includes auxiliary variables uw t , vt , and w w rt , as well as the robustness parameter Γ t that ranges from 0 to 1. The robust-

ness parameter is used to control the level of robustness of the solution. If Γw t ¼ 0, the most optimistic wind output is considered (i.e., no wind deviations). On the other hand, if Γ w t ¼ 1, the entire range of uncertainty is considered and the obtained solution is the most conservative. Any value of Γ w t in between these two bound values can be used to fine tune the level of conservatism. Also, note that Γ w t may take different values at different time periods. Additional information and mathematical proofs related to robust optimization can be found in [14], while a comprehensive model for a remote microgrid investment planning using robust optimization is available in [15].

19.2.3 Interval Optimization Interval optimization produces a schedule that maximizes microgrid profit considering the most probable (central) forecast of the uncertain parameter while guaranteeing that any realization of uncertainty within a given range around this central forecast is feasible [16]. As in the robust optimization, this range covers the area between the upper and lower bounds. As compared to the robust optimization, which endogenously determines the worst-case realization, interval optimization enforces feasibility of transitions from the upper to the lower bound between any two consecutive optimization intervals by means of deterministic constraints, as shown in Fig. 19.8. Upper and lower bounds are required Lower and upper bounds

Down and up ramp

Central forecast

9 8

Output (MW)

7 6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

Time periods (h)

FIG. 19.8 An example of determining bounds (full lines) and ramp requirements (dashed lines) in interval optimization.

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to ensure sufficient capacity to supply the local load, while the imposed transitions ensure sufficient flexibility of the microgrid assets. These assets are storage, whose flexibility is limited by its charging and discharging power capacities Sch, max and Sdis, max, and thermal generator, whose flexibility is limited by its down and up ramp limits RDth and RUth. It is important to notice that the obtained objective function values in case of interval and robust optimization should not be compared [17]. While interval optimization maximizes profit for the central forecast scenario, the robust optimization maximizes profit for the worst realization of uncertainty (i.e., the least amount of wind, highest load, worst realization of market prices, etc.). Therefore, if interval and robust optimizations are applied to the same problem and input data and solved for the same range of uncertainty, their overall profit will be significantly different (unless the worst-case scenario of the robust formulation matches the central forecast of the interval optimization). When modeling an uncertain parameter using interval optimization, three scenarios need to be used: central forecast, lower bound, and upper bound. Strictly, only the central forecast scenario is an actual scenario with its probability of occurrence, while the upper and lower bounds are here referred to as “scenarios” only for modeling purposes. In reality, these are bounds of the uncertainty area and their probability of occurrence is zero. In addition, transitions between the lower and the upper bound and vice versa are imposed in between each two consecutive time periods for variables with intertemporal constraints. As an example, let us model the uncertainty of photovoltaics output using interval optimization. Constraint (19.16) needs to consider all three scenarios (this is equivalent to the stochastic formulation (19.29)): pv pv,curt , p 2 Ωpv , t 2 ΩT gpv p, t ¼ Pp, t  gp, t

(19.37)

The power balance constraint is also identical as the one in the stochastic optimization case, as the power balance needs to hold for all three scenarios: pv p pv T w dis f nf f f ch s gth p, t + gp, t + gp, t + sp, t + drp, t + qt ¼ Dt + Dt + drrp, t + sp, t + qt , 8p 2 Ω , t 2 Ω

(19.38) gth p, t ,

As in the stochastic power balance equation, variables drp,f t , have different values for the three photovoltaics output scenarios. Again, variables qpt and qst have the same values regardless of the scenario due to the nonanticipativity requirement. Sufficient thermal generator ramping is ensured with the following constraints (in thermal generator output variable, index “ub” represents the upper bound scenario, index “cf” represents the central forecast scenario, and index “dn” represents the lower bound scenario): drrp,f t , and sch p, t

gw p, t ,

sdis p, t ,

T th th gth ub,t  glb,t1  RU , 8t 2 Ω

(19.39)

T th th gth ub,t1  glb, t  RD , 8t 2 Ω

(19.40)

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The energy storage state of charge contains intertemporal constraints as well. Besides updating variables in storage constrains (19.17)–(19.20) to condis tain the stochastic index as well (i.e., socp,t, sch p, t , and sp, t ), additional constraints are needed to ensure the possibility of switching between the states of charge from any of the three scenarios to any other scenario in between all consecutive time periods: socub,t1  socp, t 

max Sdis, t , 8p 2 Ωpv jp 6¼ ub, t 2 ΩT ηdis

socub,t1 + socp, t  Sch,max  ηch , 8p 2 Ωpv jp 6¼ ub, t 2 ΩT t soccf,t1  socp, t 

Sdis,max t , 8p 2 Ωpv jp 6¼ cf,t 2 ΩT ηdis

max soccf,t1 + socp, t  Sch,  ηch , 8p 2 Ωpv jp 6¼ cf, t 2 ΩT t

soclb,t1  socp, t 

Sdis,max t , 8p 2 Ωpv jp 6¼ lb, t 2 ΩT ηdis

soclb,t1 + socp, t  Sch,max  ηch , 8p 2 Ωpv jp 6¼ lb, t 2 ΩT t

(19.41) (19.42) (19.43) (19.44) (19.45) (19.46)

Constraints (19.41)–(19.46) consider all possible transitions of the state of charge variable and therefore ensure sufficient charging and discharging power capacity for any realization of uncertainty within the uncertainty set.

19.3 MODELING EXAMPLE As an illustrative example, a microgrid is modeled that consists of a nonflexible load, two flexible loads, two thermal generators, a wind turbine, a battery storage, and photovoltaics. The microgrid operation is optimized for a 24-h time horizon with a 1-h time step. Load, available wind output, available photovoltaics output, and electricity prices throughout the day are given in Table 19.1. Nonflexible load represents aggregated fixed loads. The first flexible load represents an industry-grade heating device whose load does not change much during the day. Only 20% of the load can be reduced at any time period and hourly load can be increased by up to 40% when returning the reduced load. Load return factor for flexible load 1 is 1.2, meaning that if 100 kWh of load was reduced at a specific time period, an additional 120 kWh needs to be consumed at other time periods. In addition, the load reduction cannot occur in two consecutive hours. The second flexible load represents a constant load in a two-shift operated industry process where machines start operating at 7 am and operate for 16 h until 11 pm. Therefore, all the increase and decrease in demand can only occur during those 16 working hours. Hourly load can be reduced by up to 20% and up to 30% of the hourly demand can be retrieved at each hour. The load return factor is 1.1.

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TABLE 19.1 Load (MW), Available Wind Output (MW), Available Photovoltaic Output (MW), and Electricity Prices (€/MW) Throughout the Day Hour

Nonflex. Load

Flex. Load 1

Flex. Load 2

Wind

PV

Prices

1

1.5

0.3

0

1.3

0

16

2

1.5

0.3

0

1.2

0

17

3

2

0.4

0

1.4

0

18

4

2

0.3

0

0.8

0

16

5

2.5

0.5

0

0.7

0

19

6

4.5

0.5

0

1.1

0

18

7

5

0.6

1

0.8

0.05

22

8

7

0.6

1

0.5

0.1

28

9

7

0.7

1

0.4

0.2

27

10

6.5

0.6

1

0.7

0.25

34

11

6.5

0.6

1

0.9

0.2

30

12

7.5

0.8

1

0.5

0.4

28

13

6

0.6

1

0.6

0.4

26

14

5.5

0.7

1

1

0.6

33

15

5.5

0.7

1

1.1

0.45

39

16

5

0.6

1

0.8

0.4

29

17

4.5

0.5

1

0.7

0.3

31

18

6

0.6

1

0.3

0.15

38

19

7

0.4

1

0.5

0

44

20

7.5

0.6

1

0.2

0

48

21

7

0.5

1

0.4

0

38

22

5

0.4

1

0.4

0

31

23

3

0.3

0

0.5

0

28

24

2.5

0.3

0

0.6

0

27

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TABLE 19.2 Thermal Generator Data Generator 1

Generator 2

Capacity (MW)

3.3

4.2

Min. output (MW)

0.4

1.5

Up ramp (MW/h)

2.5

6

Down ramp (MW/h)

2.5

6

Min. up time (h)

2

3

Min. down time (h)

2

3

Start-up cost (€)

750

600

Fixed generation cost (€)

11

17

Cost curve capacity (segment 1) (MW)

1.1

1.4

Cost curve slope (segment 1) (€/MW)

14

19

Cost curve capacity (segment 2) (MW)

1.1

1.4

Cost curve slope (segment 2) (€/MW)

15

20

Cost curve capacity (segment 3) (MW)

1.1

1.4

Cost curve slope (segment 3) (€/MW)

16

21

Thermal generator data are provided in Table 19.2. The microgrid contains two thermal generators rated at 3.3 and 4.2 MW. The first generator has lower minimum output, 0.4 MW, but needs 2 h to reach its full capacity after being offline. The second generator has high minimum stable output, 1.5 MW, but high ramp rates (it can start up and reach its rated output within 1 h). Minimum up and down times of generators 1 and 2 are 2 and 3 h, respectively. Generator 1 has slightly higher start up cost, but lower fixed generation cost and lower piecewise generation costs. Generation cost curve of both generators is divided into 3 equal parts (3  1.1 MW for generator 1 and 3  1.4 MW for generator 2). Both generators have been off for 3 h before the optimization horizon (i.e., both can be started up at any point during the day). The battery storage energy capacity is 3 MWh, while charging and discharging rates are 2 MW. Minimum state of charge is 20% of the energy capacity and both charging and discharging efficiencies are 92%. Prior to the optimization horizon, the storage state of charge was at 50% of its capacity and the final state of charge should not be below this level.

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19.3.1 Results of the Deterministic Simulation The overall profit in the deterministic case is €3795.1, which is the sum of the local thermal generation cost, €4264.2, and market profit, €469.1. Generator actions are shown in Fig. 19.9. Generator 1 starts operating at 5 am, in order to support the increased load and compensate for the reduced wind output. Due to the up ramp limit, its output reaches only 2.5 MW at hour 5. From hour 6 onward, it operates at its full capacity. On the other hand, generator 2 starts up at hour 8 and it reaches its maximum output within this hour because its up ramp constraint is not binding. This generator stays in operation at its capacity until the end of the day as well. Demand response actions are shown in Fig. 19.10. Flexible load 1, which is a cooling/heating device, performs precooling/preheating during the night hours to take advantage of the high wind turbine output and low market prices. The demand is reduced during the high net load hours (net load is defined as the actual load minus the output of the noncontrollable resources such as wind and solar) respecting constraint (19.26) that forbids demand response during 2 consecutive hours. Overall, 0.86 MW of flexible load 1 was curtailed and 1.032 MW was retrieved. Flexible load 2 operated in a way that reduces the net load in the high-price hours (hours 15 and 18–21 are the ones with the highest market prices). Overall, 1.5 MW of the flexible load 2 was reduced and 1.65 MW was returned. Energy storage operation throughout the day is shown in Fig. 19.11. The storage is charged from 50% to 100% state of charge in the first hour and stays fully charged until the end of hour 9. Discharge at the maximum rate in hour 10 is a result of the high-price spike (34€/MW) in the market. The afternoon

4.5

Generator output (MW)

4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Time periods (h) Generator 1

Generator 2

FIG. 19.9 Generator operation throughout the day for the deterministic case.

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0.4

Demand response (MW)

0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 –0.1 –0.2 –0.3

Time periods (h) Flexible load 1

Flexible load 2

FIG. 19.10 Demand response actions throughout the day for the deterministic case.

3.5

State of charge (MWh)

3 2.5 2 1.5 1 0.5 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Time periods (h) SoC

Max. and min. SoC

FIG. 19.11 Storage state of charge throughout the day for the deterministic case.

minimum market price in hour 13 is used to fully charge the storage. The afternoon spike price at hour 15 (39€/MW) causes storage to discharge at the maximum rate and again charge in the following hours at lower prices. Storage is again discharged to the minimum state of charge during hour 20 (i.e., the time period with the highest market price 48€/MW). Finally, the storage is charged back to the initial state of charge in the last hour at a low price.

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6

60

5

Electricity sold (MW)

3

40

2 1

30

0 –1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 20

–2

Market prices (€/MW)

50

4

10

–3 –4

0 Time periods (h)

FIG. 19.12 Microgrid-market interaction throughout the day for the deterministic case. The bars represent the energy sold (when positive) or purchased (when negative) in the market. The market price curve is added for reference.

The interaction of the microgrid and the market is shown in Fig. 19.12. During the low-price hours 1–9, 12, and 13, the electricity is purchased from the market. During the high-price hours 10, 14, and 15, the electricity is sold in the market. However, during the evening high-peak hours 18–21, not much electricity is sold in the market to make profit. This is a result of the high local load in the evening hours, as visualized in Fig. 19.13. The positive stacked bars sum the resources that supply the demand (thermal generators, storage discharge, demand response, wind turbine output, photovoltaics output, and electricity purchased in the market), while the negative stacked bars combine the resources being added on top of the base demand (nonflexible and base flexible demand), such as storage charge, returned demand response, and electricity sold in the market. The return of the reduced demand appears as small negative bar in hours 1 through 7 (DRR 1) and hours 7–9, 11–13, and 15 (DRR 2). The excess electricity is used either for charging the storage (in hours with low market prices) or selling in the market (in hours with high market prices). The demand in hour 19 is barely satisfied with the available generation and low storage discharge, while the storage is discharged at the maximum rate in hour 20, in order to maximize the sold electricity in this peak-price hour. Insufficient local capacity in hour 21 is supplemented with purchased electricity from the market. Since the generation cost of the thermal generators is lower than the market prices in hours 22–24, their excess output is sold in the market. Throughout the day, the microgrid sells 13.7 MWh and purchases 23.3 MWh from the market.

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14 12 10

Power (MW)

8 6 4 2 0 –2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

–4 –6 –8 Time periods (h) Wind + PV

Gen. 1 + 2

Discharge

Purchased

Charge

DRR 1 + 2

Sold

Demand

DR 1 + 2

FIG. 19.13 Power balance throughout the day for the deterministic case.

19.3.2 Results of the Stochastic Simulation The stochastic simulation is performed using five wind turbine and five photovoltaics output scenarios given in Table 19.3. This table contains also minimum and maximum values over all wind and photovoltaic scenarios used later on for robust and interval optimizations. The last row shows the probability of each stochastic scenario. Note that the first scenario for both wind and photovoltaics output is actually the deterministic scenario used in the previous section. The first scenario therefore has the highest probability of the five scenarios in Table 19.3. The expected generation cost is €4233.2, which, after adding the €437.8 of the market profit, results in the overall profit of €3795.4. The resulting market actions in comparison to the deterministic formulation are shown in Fig. 19.14. The market volumes are very similar in all time periods. The small differences are the result of using multiple weighted scenarios as opposed to the single scenario in the deterministic case. Since each of the 25 scenarios (5 wind times 5 photovoltaic scenarios) results in different generation, storage, and demand response schedules, these results are not shown here. Almost identical as in the deterministic case, the microgrid sells 13.7 MWh and purchases 23.3 MWh from the market throughout the day.

19.3.3 Results of the Robust Simulation When running the robust simulation, instead of using scenarios, we use minimum and maximum output values over all scenarios at each hour from

Wind Turbine Output Scenarios

Photovoltaics Output Scenarios

S.1

S.2

S.3

S.4

S.5

Min

Max

S.1

S.2

S.3

S.4

S.5

Min

Max

1

1.3

1.25

1.4

1.15

1.1

1.1

1.4

0

0

0

0

0

0

0

2

1.2

1.2

1.45

1.25

1.15

1.15

1.45

0

0

0

0

0

0

0

3

1.4

1.1

1.3

1.6

1.25

1.1

1.6

0

0

0

0

0

0

0

4

0.8

0.77

0.95

0.77

0.9

0.77

0.95

0

0

0

0

0

0

0

5

0.7

0.5

0.6

0.66

0.59

0.5

0.66

0

0

0

0

0

0

0

6

1.1

0.94

1

0.95

1.15

0.94

1.15

0

0

0

0

0

0

0

7

0.8

0.75

0.95

0.93

0.88

0.75

0.95

0.05

0.08

0.1

0.02

0.11

0.02

0.11

8

0.5

0.4

0.55

0.6

0.75

0.4

0.75

0.1

0.14

0.2

0.08

0.17

0.08

0.2

9

0.4

0.4

0.6

0.56

0.62

0.4

0.62

0.2

0.22

0.28

0.22

0.14

0.14

0.28

10

0.7

0.6

0.66

0.77

0.55

0.55

0.77

0.25

0.2

0.3

0.18

0.31

0.18

0.31

11

0.9

0.95

1.04

0.8

0.85

0.8

1.04

0.2

0.26

0.35

0.3

0.18

0.18

0.35

12

0.5

0.55

0.7

0.4

0.66

0.4

0.7

0.4

0.44

0.5

0.34

0.48

0.34

0.5

13

0.6

0.7

0.75

0.55

0.6

0.55

0.75

0.4

0.3

0.41

0.38

0.44

0.3

0.44

14

1

0.8

1.1

1.05

0.76

0.76

1.1

0.6

0.64

0.69

0.5

0.55

0.5

0.69

15

1.1

0.85

0.96

0.9

0.8

0.8

0.96

0.45

0.55

0.59

0.6

0.5

0.45

0.6

16

0.8

0.96

0.85

0.9

0.92

0.8

0.96

0.4

0.49

0.55

0.36

0.45

0.36

0.55

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TABLE 19.3 Uncertain Generation Scenarios (S.1–S.5) and Their Probabilities (Prob.)

0.7

0.66

0.74

0.8

0.84

0.66

0.84

0.3

0.29

0.4

0.28

0.33

0.28

0.4

18

0.3

0.5

0.44

0.25

0.55

0.25

0.55

0.15

0.2

0.22

0.13

0.1

0.1

0.22

19

0.5

0.45

0.55

0.4

0.37

0.37

0.55

0

0

0

0

0

0

0

20

0.2

0.4

0.4

0.15

0.18

0.15

0.4

0

0

0

0

0

0

0

21

0.4

0.32

0.35

0.45

0.5

0.32

0.5

0

0

0

0

0

0

0

22

0.4

0.3

0.45

0.6

0.5

0.3

0.6

0

0

0

0

0

0

0

23

0.5

0.5

0.55

0.4

0.6

0.4

0.6

0

0

0

0

0

0

0

24

0.6

0.55

0.7

0.45

0.4

0.4

0.7

0

0

0

0

0

0

0

Prob.

0.3

0.15

0.18

0.2

0.17

0.31

0.22

0.15

0.15

0.17

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6 5

Electricity sold (MW)

4 3 2 1 0 –1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

–2 –3 –4 Time periods (h) Deterministic case

Stochastic case

FIG. 19.14 Market actions throughout the day for the stochastic and the deterministic cases. Positive values refer to electricity sold, and negative to electricity purchased in the market.

Table 19.3. The conservatism of the solution is controlled by the continuous robustness parameter that can range from 0 (maximum wind turbine or photovoltaics output) to 1 (minimum wind turbine or photovoltaics output) at each hour. Therefore, when the robustness parameter is equal to 0, the obtained solution is even less conservative than the one obtained using the deterministic formulation because the maximum output values of renewable generation are equal or greater than the values from the deterministic scenario (compare values for columns “S.1” and “Max” in Table 19.3). We include the analysis of the robust simulation results with budget of uncertainty equal to 0 (i.e., the most optimistic case), and equal to 1 (i.e., the pv most pessimistic case). For Γ w t ¼ 0 and Γ t ¼ 0, the overall profit is €3671.6, which is higher than both in the deterministic and stochastic formulations. The market profit is €592.6, and generation cost is €4264.2. Generation cost is identical as in the deterministic case, but the market profit is higher due to the higher expected realization of renewable outputs. Overall, 11.4 MWh is purchased (compared with 13.7 MWh in the deterministic case) and 25.4 MWh is sold (as compared with 23.3 MWh in the deterministic case) in the market. On pv the other hand, if Γ w t ¼ 1 and Γ t ¼ 1, the overall profit is €3890.1. Again, the generation cost is €4264.2, but highly conservative expectations of uncertain renewable generation result in lower market profit, €374.1. Conservative microgrid market actions result in 15.4 MWh of purchased and only 21.6 MWh of sold electricity. This is visualized in Fig. 19.15, the robust case Γ ¼ 0 for both wind and photovoltaics purchases less and sells more electricity than the deterministic case, and the robust case Γ ¼ 1 for both wind and photovoltaics purchases more and sells less electricity than the deterministic case.

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229

60

5

3

40

2 30 1 0 –1

20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Market prices (€/MW)

Electricity sold (MW)

50 4

10 –2 –3

Time periods (h) Deterministic case

Robust case (G = 0)

Robust case (G = 1)

Market prices

0

FIG. 19.15 Market actions throughout the day for the robust cases and the deterministic case.

Generator outputs, demand response actions, and storage operation are identical as in the deterministic case (Figs. 19.9–19.11). This means that the only difference between these three cases is the market participation: robust case (Γ ¼ 0) expects higher outputs of renewable sources and purchases less electricity in the market than the deterministic case, while the robust case (Γ ¼ 1) expects lower outputs of renewable sources and thus purchases more electricity in the market than the deterministic case.

19.3.4 Results of the Interval Simulation Interval simulation results in the overall profit of €3821.1, which is slightly worse than €3795.1 obtained using the deterministic formulation. Both of these formulations use the same scenario in the objective function, but the interval formulation imposes feasibility on the entire range of uncertainty, which results in a more conservative schedule. The generation cost is €4191.3, which is lower than in the deterministic case. This can be seen when comparing generator outputs of the interval and the deterministic cases in Fig. 19.16. Although both generators start at the same time periods as in the deterministic case, their respective outputs are lower in certain time periods, which reduces the thermal generator operating costs. Fig. 19.17 compares the demand response actions of the interval and the deterministic cases. Flexible load 1 behaves exactly in the same way, while flexible load 2 is more active in the interval case (1.62 MWh of energy reduced as opposed to 1.5 MWh in the deterministic case). Storage operation for the interval and the deterministic cases is compared in Fig. 19.18. Storage charging and discharging cycles for the interval case are not

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4.5

Generator output (MW)

4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Time periods (h)

Generator 1 (interval case)

Generator 2 (interval case)

Generator 1 (deterministic case)

Generator 2 (deterministic case)

FIG. 19.16 Comparison of generator outputs for the interval and the deterministic cases.

0.4

Demand response (MW)

0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 –0.1 –0.2 –0.3

Time periods (h)

Flexible load 1 (interval case)

Flexible load 2 (interval case)

Flexible load 1 (deterministic case)

Flexible load 2 (deterministic case)

FIG. 19.17 Comparison of demand response actions for the interval and the deterministic cases.

as deep as for the deterministic case. Overall, 5.9 MWh are charged in the interval case as opposed to 7.3 MWh in the deterministic case. Similarly, 5.0 MWh are discharged in the interval case as opposed to 6.2 MWh in the deterministic case. As analyzed in Figs. 19.16–19.18, the interval formulation results in different generator operation, demand response actions, and storage operation than the deterministic formulation. Considering the power balance, these are the only

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3.5

State of charge (MWh)

3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Time periods (h) SoC (interval case)

SoC (deterministic case)

Max. and min. SoC

FIG. 19.18 Comparison of the storage state of charge for the interval and the deterministic cases.

6

60

5 Electricity sold (MW)

3

40

2 30 1 0 –1

20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Market prices (€/MW)

50 4

10 –2 –3

0 Time periods (h) Deterministic case

Robust case (G = 0)

Robust case (G = 1)

Market prices

FIG. 19.19 Market actions throughout the day for the interval and the deterministic cases.

three degrees of freedom to move in between the lower bound, the central forecast, and the upper bound in the interval formulation. When observing generator outputs of the lower bound scenario (minimum wind turbine and photovoltaics output), they are identical to the ones in the deterministic case. This brings the conclusion that the ability of the schedule to cover the imposed uncertain area and flexibility to move in between the bounds results in extra cost equal to the difference in the objective functions of the interval and the deterministic cases. This is further elaborated in Fig. 19.19, which compares market actions for both

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the interval and the deterministic formulations. The figure shows that the interval schedule results in more purchased electricity, 15.7 MWh compared with 13.7 MWh in the deterministic case, and less sold electricity, 21.8 MWh compared with 23.3 MWh in the deterministic case.

19.4 CONCLUSIONS Although microgrid operation is related to the control layer and detailed modeling of its components, a longer-term optimization is needed to provide setpoints for microgrid operation and maximization of its market performance. This longterm scheduling model uses simplified representation of the microgrid elements. On top of this, it needs to consider the uncertain nature of some of the input parameters. The uncertain modeling practice presented in this chapter provides an appropriate decision-making platform and minimizes the risk of not meeting the day-ahead schedule. All the models from Section 19.3 are coded in GAMS and solved using CPLEX solver. The GAMS code of all the models is available in Appendices A–D at the end of this book.

ACKNOWLEDGMENTS Hrvoje Pandzˇic is with the Faculty of Electrical Engineering and Computing University of Zagreb, Croatia. His work is a result of projects microGRId Positioning (uGRIP), supported in part by the Croatian Environmental Protection and Energy Efficiency Fund, and Smart Integration of RENewables (SIREN), funded by the Croatian TSO-HOPS and Croatian Science Foundation under the contract I-2583-2015.

REFERENCES [1] J. Rocabert, A. Luna, F. Blaabjerg, P. Rodrguez, Control of power converters in AC microgrids, IEEE Trans. Power Electron. ISSN 0885-8993, 27 (11) (2012) 4734–4749, https:// doi.org/10.1109/TPEL.2012.2199334. [2] T. Dragicˇevic, Model predictive control of power converters for robust and fast operation of AC microgrids, IEEE Trans. Power Electron. ISSN 0885-8993, PP (2017) 1, https://doi.org/ 10.1109/TPEL.2017.2744986. [3] T. Dragicˇevic, X. Lu, J.C. Vasquez, J.M. Guerrero, DC microgrids #x2014; Part I: a review of control strategies and stabilization techniques, IEEE Trans. Power Electron. ISSN 0885-8993, 31 (7) (2016) 4876–4891, https://doi.org/10.1109/TPEL.2015.2478859. [4] E. Barklund, N. Pogaku, M. Prodanovic, C. Hernandez-Aramburo, T.C. Green, Energy management in autonomous microgrid using stability-constrained droop control of inverters, IEEE Trans. Power Electron. ISSN 0885-8993, 23 (5) (2008) 2346–2352, https://doi.org/ 10.1109/TPEL.2008.2001910. [5] A. Parisio, E. Rikos, L. Glielmo, A model predictive control approach to microgrid operation optimization, IEEE Trans. Control Syst. Technol. ISSN 1063-6536, 22 (5) (2014) 1813–1827, https://doi.org/10.1109/TCST.2013.2295737.

Control of Smart Grid Architecture Chapter

19

233

[6] GAMS Users Guide [Online], Available from: https://www.gams.com/latest/docs/UG_MAIN. html (Accessed 5 March 2018). [7] H. Pandzˇic, T. Qiu, D.S. Kirschen, Comparison of state-of-the-art transmission constrained unit commitment formulations, 2013 IEEE Power Energy Society General Meeting, ISSN 1932-5517, 2013, pp. 1–5, https://doi.org/10.1109/PESMG.2013.6672719. [8] H. Pandzˇic, I. Kuzle, Energy storage operation in the day-ahead electricity market, 2015 12th International Conference on the European Energy Market (EEM), ISSN 2165-4077, 2015, pp. 1–6, https://doi.org/10.1109/EEM.2015.7216754. [9] S. Meyn, P. Barooah, A. Busˇic, J. Ehren, Ancillary service to the grid from deferrable loads: the case for intelligent pool pumps in Florida, 52nd IEEE Conference on Decision and Control, ISSN 0191-2216, 2013, pp. 6946–6953, https://doi.org/10.1109/CDC.2013.6760990. [10] J.R. Birge, F. Louveaux, Introduction to Stochastic Programming, second ed., Springer Publishing Company, Incorporated, New York, NY, 2011. ISBN 1461402360, 9781461402367. [11] R. Weron, Electricity price forecasting: a review of the state-of-the-art with a look into the future, Int. J. Forecast. ISSN 0169-2070, 30 (4) (2014) 1030–1081, https://doi.org/10.1016/j.ijforecast. 2014.08.008http://www.sciencedirect.com/science/article/pii/S0169207014001083. [12] Y. Dvorkin, Y. Wang, H. Pandzˇic, D. Kirschen, Comparison of scenario reduction techniques for the stochastic unit commitment, 2014 IEEE PES General Meeting—Conference Exposition, ISSN 1932-5517, 2014, pp. 1–5, https://doi.org/10.1109/PESGM.2014.6939042. [13] A. Ben-Tal, L. El Ghaoui, A.S. Nemirovski, Robust Optimization, Princeton University Press, Princeton, NJ, 2009. [14] D. Bertsimas, M. Sim, Robust discrete optimization and network flows. Math. Program. ISSN 1436-4646, 98 (1) (2003) 49–71, https://doi.org/10.1007/s10107-003-0396-4. [15] T. Dragicˇevic, H. Pandzˇic, D. Sˇkrlec, I. Kuzle, J.M. Guerrero, D.S. Kirschen, Capacity optimization of renewable energy sources and battery storage in an autonomous telecommunication facility, IEEE Trans. Sustain. Energy ISSN 1949-3029, 5 (4) (2014) 1367–1378, https:// doi.org/10.1109/TSTE.2014.2316480. [16] Y. Wang, Q. Xia, C. Kang, Unit commitment with volatile node injections by using interval optimization, IEEE Trans. Power Syst. ISSN 0885-8950, 26 (3) (2011) 1705–1713, https://doi. org/10.1109/TPWRS.2010.2100050. [17] H. Pandzˇic, Y. Dvorkin, T. Qiu, Y. Wang, D.S. Kirschen, Toward cost-efficient and reliable unit commitment under uncertainty, IEEE Trans. Power Syst. ISSN 0885-8950, 31 (2) (2016) 970–982, https://doi.org/10.1109/TPWRS.2015.2434848.