Optimization Problems in Microgrids

**Optimization Problems in Microgrids**

Jae-Won Lee, Michael Pfister

Seoul National University

Power System Optimization Course,

**I. Introduction**

Due
to increasing environmental awareness, economic incentives and technology
progress there has been an active integration of distributed energy resources
(DERs) including photovoltaics (PVs), wind turbines (WTs), fuel cells (FCs),
etc. which has the advantages of low investment cost compared to traditional
generators, system loss reduction and lower emissions in the power system
sector. In order to accommodate these resources reliably and securely in the
distribution network, the concept of microgrids (MGs) has been actively
researched throughout the past few years. The MG is defined as an aggregation
of DERs, electrical loads, storages, and generations interconnected among
themselves and with distribution network as well [2]. It is a small scale power
system with the ability of self-supply and islanding, which provides a
distributed local intelligence for the power system to supply loads in a
reliable and economic manner [1]. An energy management system (EMS) controls the operation of the MG by coordinating all its distributed generators, distributed
energy storages and controllable loads. Control has to comply with different
requirements as reliability, sustainability, cost-effectiveness and
environmental friendliness. A MG control system aims at meeting its utility
demands and if not islanded operated also to participate in the market. The
highest priority of the control is to maintain system stability and reacts to
disturbances. Secondly it optimizes the power allocation according to an
objective function while complying with constraints [20]. A typical EMS
looks like the figure shown below.

*Figure 1. Smart energy management*
system (SEMS) [8]

Most
of the MG operating cost optimization problems have an objective function that
is similar to the form shown below.

*Figure 2. Operating cost function*
of a typical MG [8]

T\_c represents the total operating
cost, T is the total number of hours considered, L is the total number of
generators and DERs within the MG, u(t) is on/off state of the i th generator,
P\_G is the active power output of the generator i at time t. B\_G is the bidding
price of the i th generator at time t, and S\_G represents the startup and
shutdown cost of a generator. M is the total number of storages within the
system, and P\_s and B\_s are the active power output of the j th storage and the
bidding price of jth storage. Most of the EMS systems have very similar
objective function. For the constraints, there are power balance, active power
output constraints of the generators and storages, and charge and discharge
rate of ESSs. Although the considered constraints vary from literature to
literature, these are the most commonly considered constraints.

MGs
provide a high flexibility since new energy generators or storages can easily
be added. Consumers can become independent of big electricity suppliers since MGs
are often owned locally by its direct consumers. Overall MGs with sophisticated
energy management systems will be likely to play an important role in the
future in increasing reliability, cost effectiveness and sustainability of
energy supply to end consumers [20].

To analyze the research flow in the subject of MGs, this paper aims to
provide an overview of the MG optimization problems. The topics covered in this
paper are operation and emission cost minimization, optimal scheduling of MGs,
and the effect of integrating numerous MGs on the distribution network.

The paper is organized as follows:
in section two, papers dealing with the operation cost minimization of a MG are
discussed. In section three, papers solving the optimization problem taking
into account both operation cost and emission cost are reviewed. In section
four, papers dealing with scheduling of MGs considering its islanding
probability are examined. Following in section five, operation cost
minimization of islanded MGs are discussed. In section six, papers that analyze
the effect of multi-MG in distribution network are discussed, ending with a
conclusion in section seven.

**II. Operation cost minimization of a**
grid-connected Microgrid

One of the first topics researched in the field of MG was its operation
cost optimization when it is connected to the main grid or the utility.
Although the objective functions and constraints of each paper differ slightly,
they are very similar. The objective function of papers is the operation cost
minimization of the MG, which includes fuel cost of generators and energy
storage systems (ESSs) within the MG and the power exchange cost from the main
grid. Some papers also include the operation and maintenance costs of the units
within their system. Also, constraints of the optimization problem include
power balance, minimum and maximum outputs of the DERs and ESSs, minimum and
maximum flow between the MG and the main grid, minimum up and down time of generators,
and the charge and discharge rate of ESSs. There are some papers that consider
the heat balance constraint; however, it is not a major concern in solving this
problem.

In
[8], the operation of a MG is optimized using a smart EMS. The EMS used in this
paper is a composition of a PV weather-based forecasting module, an ESS
operating module, and a MG operation optimization module. The EMS manages the
optimal charging state of ESS, economic load dispatch, and the dispatch of
DERs. In order to solve this optimization problem, a matrix real-coded genetic
algorithm (MRC-GA) is used to realize an economically optimal load dispatch
schedule for three different operation modes. The first mode is the islanded
operation of the MG where the electrical load and supply is balanced locally,
where the overproduction of non-dispatchable resources are stored in the ESS.
In the second mode, the EMS operates does not export any power to the main
grid, while minimizing the power purchase from the main grid to meet the local
demand. This strategy is used to prevent congestions in the main grid during
peak demand hours since the MG almost fully able to supply all of its demand by
itself. The third case describes a full participation of the MG in the market.
The EMS maximizes the profit of the MG. In order to realize the described
scenarios, the power balance constraint is modified accordingly to include or
exclude the main grid power exchange variable.

In [9], the advantage of integrating a communication link in unit
commitment stage is analyzed. In order to explore these benefits, power sharing
schemes for different power sources within a MG is considered. There are four
power sharing schemes described in this paper, which are linear, nonlinear,
dynamic, and optimized. The linear power sharing scheme operates the units
within a system so that the sources provide an equal amount of normalized power
output when there is a frequency deviation. The second scheme, nonlinear power
sharing, the frequency droop curves of the power sources are not linear,
allowing the operator to design the droop curve for the generators according to
its most efficient point of operation. In dynamic power sharing scheme, the
y-intercept (frequency) of each generator droop curves are moved to modify the
amount of power each source contributes to the overall change in demand.
Lastly, in optimized power sharing scheme, it is a high nonlinear power sharing
scheme where the power outputs of each generator are normalized to a unit power
setting, and the outputs of generators are considered simultaneously in order
to determine the optimal operation of the resources for the entire range of
power demand. The input of the optimization problem is the normalized power
setting of each unit within the MG, and it outputs total delivered power and
the total running cost of the system. Then, the optimal power setting is found,
so that the cost is minimal. The constraints include power balance, reserve,
and heat. A penalty function is added to the cost function in the case where
more heat is delivered than the required amount.

In [4], an efficient online optimal MG operation strategy is
implemented, which respects time-varying request and complex constraints. The
goal is to minimize operational costs while providing energy for the forecasted
amount of load. The problem is formulated using mixed-integer linear
programming coupled with a feedback mechanism tackling forecast errors and
disturbances in the MG operation, which can be the production fluctuation of non-dispatchable
resources or load or energy price volatility. The loads are divided into two
categories, controllable and critical, to integrate demand response to the
operation system. Furthermore, for every sampling interval (15 minutes), the
controller makes decisions whether to start or stop the generators, the output
level of each unit, whether the ESS should be charged or discharged, if the
power should be bought or sold to the main grid, and if the load curtailment is
necessary. The optimization function minimizes the cost from energy production,
startup and shutdown cost of generators, and penalty cost from load
curtailment. The constraints include minimum and maximum output of DERs and
ESSs, ramp up and down rate limits of the units within the system, and the
bounds on controllable load curtailments. Experimental results show that the
implemented mixed integer programming with feedback control algorithm saves
money compared to general practice.

In [13], MG with PV, ESS, FC, and controllable load is modeled. The
storage is composed of a battery bank and a hydrogen energy storage system. FC
and hydrogen are very promising technologies which will likely gain more
importance. An optimal control is sought by optimizing the operation cost of a
grid-connected MG, taking into account the value of generated energy, cost of
energy storage, and the depreciation cost of the components. The constraints
considered are operational constraints of the units within the system and power
balance. Then, the optimization problem is formulated using mixed-integer
predictive control problem, which is simplified into nonlinear optimization
problem with a sampling time. The optimization problem maximizes the profit of
MG by maximizing sales and minimizing the purchase from the main grid, while
minimizing the depreciation cost of its components from excessive overuse. The
operational constraints include a limitation of the hydrogen storage for safe
electrolysis operation of the fuel cells. For ESS, the state of charge is set
by a minimum and maximum and the applied or drained current is limited. In
addition, the proposed controller accurately regulates short-time variations in
PV production by exchanging energy not only with its storages but also with the
grid. Long-term deviations are handled by using the hydrogen energy storage.

In [17], MG operation is optimized
using a robust optimization approach taking into account the wind turbine
output uncertainty. The study is simulated in a grid-connected and islanded
operation mode. The wind power uncertainty is modeled using ARIMA model. In the
grid-connected mode, the MG is able to earn revenue by selling its reserve
capacity and excess power to the spot market. Also, MG operator purchases power
from the main grid if the spot market prices are lower than the MG’s generation
cost. The overall goal of this paper is to minimize the operation cost of the
MG. The constraints considered include power balance, transmission capacity,
and minimum and maximum power outputs of the units within the system. The
deviation between the forecasted and real-time load is compensated by reserve
capacity, which can be the dispatchable power units within the MG or the power
from the main grid. For the islanded operation of the MG, power exchange cost variable
is deleted from the objective function, and load curtailment cost is added. In
this mode, real time power balance constraints are met by the reserve capacity.
Also, a demand shedding constraint is added, which show that the load
curtailment will be less than or equal to the day-ahead forecasted load. The
power balance and imbalance constraints are same as the grid-connected mode.
Then, a comparison between deterministic, stochastic, and the proposed robust
optimization is performed. In both grid-connected and islanded mode, the robust
optimization approach provides better results. An interesting point is that the
best result for the robust optimization is obtained when the wind parameter is
set as uncertain for a whole day.

**III. Emission and operation cost minimization of a**
grid-connected Microgrid

Following the operation cost optimization section, in this section, the
papers that optimize both operation and emission cost of a MG is discussed. The
articles in this section solve a multi-objective optimization problem.

*Figure 3. Objective function considering*
both operation and emission costs [11]

C\_g
represents the operating cost of generator i producing power P\_i, and C\_e
represents the emission cost of generator i producing power P\_i. Emission cost
is modeled with a fixed coefficient, as shown in the equation below.

*Figure 4. Emission cost function [11]*

K\_e
is the emissions coefficient for i th unit, in (kg/kWh), and M\_e is the greenhouse
gas emission cost ($/kg).

Most of the papers in this section
consider the emission costs from the generator, storage, and the main grid,
with the similar constraints from the previous section.

In [7], operation and emission cost of a renewable-based MG were
optimized while satisfying power balance and minimum and maximum power
generation constraints of the units within the MG. The unit commitment problem
of the MG is formulated and solved using a mixed-integer programming using a
convex optimization perspective. Emphasis is put on the intermittency behavior
of RES and the decision if the MG can operate autonomously or needs a grid
connection. The problem statement if the MG is self-sufficient  in supplying local demand is approached
probability-based.

Considered constraints comprise
minimum and maximum output power of DG, minimum on-time after being switched on
and minimum down-time after being switched off, ramp rates, emission limits,
operating reserves and a probability of self-sufficiency which shows the
probability that the MG can successfully operate in islanded mode by being
completely independent from the grid.

Results show no loss of optimality
compared to existing algorithms and confirm the essential contribution of ESS
in building an independent MG. Also measures for the optimal ESS size of a MG
are provided. The problem formulation bases unit commitment successfully on
probability of intermittency of FRES, emission limits on and forecast errors in
RES production.

In [3], operation of a MG with renewable energy sources and back-up
power sources, such as micro-turbine, FC, and ESS is optimized using adaptive
modified particle swarm optimization algorithm. The difference of this approach
compared to the traditional particle swarm optimization is that the
conventional particle swarm optimization depends heavily on its parameter and
it can easily settle in local optima. In order to address this problem, a
pareto front of optimal solutions are obtained, and fuzzy self-adaptive
mechanism and chaotic local search method is used. This approach provides
non-dominated pareto-optimal solutions with fast convergence and low
computational time. The operational costs considered in this paper are the fuel
costs of DERs, startup and shutdown costs of the units, and the power exchange
cost between the MG and the main grid. In the emission cost function, carbon,
sulfur, and nitro oxide costs from the generators, ESSs, and main grid are
considered. The constraints of the optimization problem include power balance,
minimum and maximum generation of the generator, storage, and the main grid.
Furthermore, charge and discharge rate constraint of ESS is also considered.
The results verify that the proposed method is superior to the traditional
particle swarm optimization, with dynamic stability and excellent convergence
of the swarms. Moreover, the pareto optimal front solutions provide various
operation options for the system operator to choose.

In [12], the operation and emission cost
are minimized considering separately grid-connected and islanded mode of the
MG. The problem is set up to minimize the fuel, depreciation, and emission
cost, while satisfying the load balance and system security constraint.
Considered MG model includes WTs, micro-turbines, PVs, and FCs. In this paper,
in order to analyze the tradeoff between operation cost and emission cost, a
weighting factor is applied to the emission cost for the operator to determine
the importance of environmental constraint. The constraints of the problem
include power balance, minimum and maximum unit output, and line flow. The
results show that operation in grid connection is beneficial for the total
cost. The total costs of the islanded mode operation simulation are three times
higher than in grid-connected operation mode. Also with diminishing the
weighting of the environmental factor in the optimization function the total
costs decrease.

In [5], scheduling of different power sources, PV, FC, ESS, WT, and
micro-turbine, within a MG is solved. Two objective functions are solved, where
one minimizes the total operation cost and the other minimizes the total
emission caused by the generation units. For the solving method, a
lexicographic optimization (optimizing the range of the objective function) and
hybrid augmented-weighted epsilon-constraint method (solves the problem of
generating dominated or inefficient solutions) have been used to solve the
multi-objective optimization, which outputs pareto optimal solutions. Then, the
fuzzy satisfying method is used to find the best solution. In previous
literatures, the weighted sum method is often used to convert multi-objective
problems into single objective problem, which only produced efficient extreme
solutions. Also the objective function must be scaled properly in the weighted
sum method. These two problems are over come in the epsilon-constraint
technique by producing non-extreme efficient solutions and not requiring
scaling of the objective functions. The simulation results verified that the
proposed method requires less computation time while producing better solutions
compared to other methods.

In [11], a chaotic quantum genetic algorithm is used to solve the
environmental economic dispatch of a smart MG. The objective function minimizes
the fuel cost and emission cost. For the fuel cost, each participating
generator’s fuel and operation and maintenance costs are considered, while a
linear emission cost is considered in the emission cost function. For the
constraints, power balance and minimum and maximum output of DERs are
considered. Previously, when a genetic algorithm method is used to solve the
operation optimization problem, it takes a long time to solve the problem, and
the global solution was not guaranteed. However, in the proposed method,
quantum bits were used to encode the chromosomes. This algorithm uses the
quantum probability vector encoding mechanism and adopts the genetic algorithm
crossover update strategy to effectively improve the global search ability and
escape the local optimal solution trap by using chaotic algorithms, so that it
can achieve the real objective of the global optimal solutions (chaotic system
possesses inherent diversity, which allows the solution to not fall into the
local optima during the optimization process). The author compares different
solving methods (DP, GA, EP, QGA) with the proposed solving algorithm, where
the proposed method achieved the minimum operation cost.

In [10], the operation of a MG consisting
of WT, diesel generator, micro-turbine, PV, FC, and ESS is optimized. The costs
considered in this paper are fuel, startup and shutdown cost of generators,
operation and maintenance cost of the units, and the emission cost of each
generator. This problem is modeled and solved using modified game theory to
solve the nonlinear constrained multi-objective optimization problem. Player
one tries to minimize emission while player two's goal is to minimize the
operating cost while the load is met and constraints are complied with. The
optimization is performed for three scenarios with increasing complexity. The
first scenario describes an islanded operation. In the cost function the fuel
consumption rate of different generators, fuel costs and operation-maintenance
costs are considered. The constraints are keeping power balance and being
inside generation capacity constraints. In the second scenario there is a
battery storage added which alters the power balance constraint. In addition to
the initial cost function now start-up costs are also considered. In the third
scenario the MG is grid-connected. The cost function contains now an adding
term representing from the grid bought energy in the case when the load demand
can not be covered by the MG's self-production. On the other side there is now
also potential for profit when market prices are higher than production costs.
Results show that in comparison with other optimization techniques the used
multi-objective game theory approach delivers the best results.

In [21], an EMS for the MG operator to determine optimal operating
strategy has been designed. The considered MG system includes PV, ESS, WT, and
FC. Then, a sensitivity analysis of the investment on ESS capacity and demand
has been conducted to determine the optimal investment in generators and ESSs.
The optimization problem has been solved using the General Algebraic Modeling
System (GAMS) program, specifically the CPLEX solver. The decision variables in
the optimization problem include the output levels of PV, WT, biomass, gas
turbine, and FC. Then, the maximum profit is calculated using the simple
equation where the cost of each generating units are subtracted from the
revenues produced by each units. For the constraints, unit capacity,
transmission capacity, power balance, and heat balance have been considered.
Furthermore, the emission rates of each unit within the MG are also considered
to account for the environmental effect. The results show that a greater
battery storage capacity requires greater investment, and so installation of
additional storage devices might not be profitable when a high fixed cost is
considered.

In [19], an optimal EMS is designed, which optimizes the operation and
emission cost of a MG. The author claims that the uncertainties of resources within
a MG have not been considered in previous literatures, hence, to account for
the unpredictability and uncertainty of WT and PV outputs and load demand, a
scenario based stochastic programming has been used. The generated scenarios
are reduced using a backward method, and the uncertainties have been modeled
using a Lattice Monte Carlo simulation method. The operation cost function
includes the bidding of each resource and the startup and shutdown costs of the
units. For the mission cost function, carbon, sulfur, and nitro oxide emission
costs of the DER, ESS, and the main grid have been considered. The constraints
include power balance, active power limits of units, spinning reserve
requirements, and charge and discharge rate limit related to the storage
device. The optimization problem is solved using improved
teaching-learning-based optimization, which utilizes self-adaptive
probabilistic modification strategy to avoid trapping in local optima. The best
advantage of the proposed algorithm is quick transfer of the information
between agents which gives more ability to the proposed algorithm in finding
the global optima irrespective of the complexity of the problem.

**IV. Optimal scheduling of a Microgrid considering islanded**
operation

In this section, the optimal scheduling of the islanded MGs and
operation of MGs considering islanding are reviewed. Although some papers in
previous section have considered the islanded operation, the articles in this
section specifically determine the optimal operating parameters of MG or
dispatch of generators so that it is able to operate safely even when it is islanded. The main change in the optimization problem is that
the after islanding, the objective function changes from operating cost minimization
to power balance mismatch minimization. In order to operate the system reliably
and securely, demand response function of the
loads is more actively considered.

In [18], a dispatch problem for the DERs within a MG has been solved.
The fuel cost has been minimized during the grid-connected operation while
ensuring stable operation after islanding in case of faults in the main grid.
The author utilizes adjustable droop control of the participating generators as
a constraint in the optimization problem to ensure a stable islanded operation.
The objective function is to minimize the total generation cost, and the
constraints include power balance, spinning reserve requirement, and inter-area
flow limit. The constraints are formulated using both fixed and adjustable
droop, where it was tested on a system with 15 DERs, and it showed that the MG
could be operated economically and safely during both grid-connected mode and
islanded mode when the droop constraints were considered.

In [1], a centralized MG optimal scheduling was solved considering
multi-period islanding constraints. The optimization problem was divided into a
master problem and a sub problem. The master problem aimed to minimize day
ahead grid-connected operation cost, considering generation cost and power
exchange cost, where the sub problem solved the optimal islanded operation of
the MG. In order to ensure a safe operation after islanding cuts were made,
where the unit commitment and ESS schedule was revised until enough capacity
was procured to be islanded. However, in the case where there were not enough
dispatchable units within the system, demand response was utilized to account
for insufficient generation, and an inconvenience factor (form of penalty
function) was added to the objective function. The primary application of ESS
in this paper is to coordinate with generators to guarantee the generation
adequacy of MG. Furthermore, to operate safely after islanding, T-tau islanding
criterion was used where tau is the number of consecutive hours that MG can
operate in islanded mode. The problem was formulated using mixed integer
programming, and the simulation results verified that although there was a
slight increase in the operation cost, the MG was able to be operated more
reliably.

**V. Operation cost minimization of an islanded Microgrid**

In previous section, the optimal scheduling of generators within a MG,
considering its islanding probability was considered. In this section, papers
discussing the economic operation of islanded or standalone MGs are reviewed.

In [6], operation of a standalone MG considering the lifetime
characteristics of ESS is optimized. The operation of WT, PV, diesel generator,
and ESS are considered. The multi-objective optimization problem aims to
minimize power generation cost and increase the battery lifetime. To find these
optimal operation parameters, several costs are taken into account: battery
life loss cost, operation and maintenance cost, fuel cost, and environmental
cost. Optimization is realized by use of non-dominated sorting genetic
algorithm. The different generation sources are modeled taking into account
specific behavior of WT, PV, ESS, and diesel power generation. The optimal goal
of the multi-objective function is to find a pareto optimal front. One
objective is to minimize power generation cost, which is realized by higher
utilization or renewables and using less power from the diesel generator. Minimizing
the life loss of the battery is the second objective, which can be achieved by
maintaining a high level of state of charge. The constraints considered are
power balance and minimum and maximum output levels of the units. Accordingly,
the results show that in the case of excessive renewable energy output, the
battery should be operated at a state of charge stop state of 0.61. In the case
of shortage of renewable energy production, battery management relies much more
on the usage of diesel generators and therefore a high state of charge stop of
0.9 is needed. This real time operation makes sure that the renewable are fully
utilized while the batteries are effectively managed, thus increasing battery
life and reducing generation cost.

In [15], the operation cost of an islanded MG consisting of a diesel
generator, micro-turbine, WT, PV, ESS, and an inverter/rectifier is optimized.
The operation cost function considers the fuel costs and operation and
maintenance costs of the units. The main contribution of this paper is the
modeling of WT, PV, and its controller, AC/DC converter and the central
converter. The optimization problem is formulated using multi-cross learning
based chaotic differential evolution algorithm which comes with several
advantages including learning capability, high exploitation, and enhanced
convergence behavior. In the objective function, fuel, startup and shutdown,
operation and maintenance, and gas emission costs are considered. For the
constraints, ESS properties, ramp up and down rate limits of diesel generator
and micro-turbine, and power balance are considered. The simulation has been
conducted in islanded mode, and the results have been compared with 21 other
optimization methods. The proposed method shows superiority compared to other
methods in terms of effectiveness and robustness.

**VI. Microgrids on distribution network**

In this section, the papers dealing with the effect of having a numerous
MGs in the distribution network are discussed.

In [2], a fuzzy self-adaptive particle swarm optimization algorithm is
proposed and implemented to dispatch the generations in a typical MG
considering its operation cost and emission cost. The cost function of the
optimization problem considers the fuel cost and startup and shutdown cost of
the generators and ESSs, and also considering the power exchange cost between
the main grid. Also, the emission cost function includes carbon, sulfur, and
nitro oxide emissions from DERs, storages, and the main grid. For the
constraints, power balance, active power constraints of the units, and charge
and discharge rate limits of storage devices are considered. Previously,
conventional particle swarm optimization was used very often to solve the
optimization problems in MGs.  However,
it had the disadvantages of depending heavily upon its learning and weighting
factors, and having the possibility of being trapped in local optima. Hence, in
this paper, a fuzzy self-adaptive approach is adopted to mitigate this problem.
The superior performance of the proposed algorithm is shown in comparison with
those of other evolutionary optimization methods, such as conventional particle
swarm optimization and genetic algorithm, and the results show that although
the suggested algorithm increases the operation cost slightly, it is able to
manage the emission to a certain level.

In
[16], the effect of integrating multiple MGs in the distribution network is
analyzed, especially in the aspect of loss and emission reduction. In order to
analyze this effect, a typical low voltage network with DERs operating
coordinated as a MG and a typical mid-voltage network comprising of many low
voltage networks are used. As generating units are located closer to the load
side, the network losses will be reduced. To analyze this effect, power flow
analysis has been used to calculate the system loss. Furthermore, since the
penetration of MGs is not very high, it is assumed that the unit commitment of
the main grid will not change, however, due to the generation capability of the
MGs, its economic dispatch results may change, changing the emission output of
the generators. The estimation of the avoided emissions due to MGs operation is
calculated based on the marginal emissions curve of the upstream network.

**VII. Conclusion**

In this paper, previous and current
literatures dealing with optimization problems in MGs have been reviewed. Some
of the topics discussed were MG operation cost minimization, MG operation and
emission cost minimization, MG scheduling considering islanding probability,
etc.

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