How are Microgrid Networks Optimized to Prevent Voltage and Frequency Droop?

## **Short Answer**

With the advent of multiple small generators and energy storage devices in the modern power network, innovative solutions to the unique problems and opportunities of these new systems must be found.  One of these problems (or opportunities) is the issue of how to correct voltage and frequency droop in
distributed Microgrids.  This article describes some of the methods that can be used to help mitigate this issue.

### **Smartgrid**

Smartgrids are collections of various power producing, storage, and distribution systems that are controlled in a manner to optimize the system in a number of different ways.  Micorgrids are smaller potentially self-contained systems that may consist of Distributed Generators (DGs), storage units, and local distribution lines. Microgrids interface with the Maingrid through a Point of Common Connection (PCC).  The DGs within the Microgrid may consist of Photovoltaic (PV) cells, wind generators, small fuel burning units, or other power producing units. The energy storage units may consist of Uninterruptable Power Supplies (UPSs), capacitor banks, batteries, or electrical cars connected to the power network.  Within Microgrids, the distribution system is usually transmitted at a relatively low voltage in comparison to the voltages that can be transmitted throughout the Maingrid.

In the case that there is a fault or
generation failure in the Maingrid, Microgrids have the capability to isolate
themselves from the Maingrid and perhaps maintain power within them by a
process referred to as Islanding.

Loads within the system can come
online or go offline and when this occurs, it will cause fluctuations within
the power system.  This is seen as
changes in the voltage or frequency levels.
These changes are referred to as droop.
In order to counteract the droop introduced by the changing power
requirements, a control system that has the ability to adjust the power
provided by DGs and storage units may be implemented.

### **Types of Power**

The power in a power system consists
of two types of power, Real power (P) that is consumed by resistive loads, and
Reactive power (Q) which is consumed by inductive or capacitive loads.  The sum of these two powers is called the
Apparent Power (S).

# S
P
Q

(
1
)

{\displaystyle S=P+Q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)}

If power is transmitted across a lossless line (one without a resistive load) the real and reactive power can be related to the voltages on either side of the line and to the power angle by (2) and (3).

# P

V

1

V

2

X

s
i
n
(
θ
)

(
2
)

{\displaystyle P=\frac{V\_{1}V\_{2}}{X}sin(\theta) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)}

# Q

V

2

X

(

V

2

−

V
1

c
o
s
(
θ
)
)

(
3
)

{\displaystyle Q=\frac{V\_{2}}{X}(V\_{2}-V\_1cos(\theta)) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)}

Since the power angle

θ

{\displaystyle \theta}

 will be small,
we can say that

s
i
n
(
θ
)
≈
θ

{\displaystyle sin(\theta)\approx\theta}

 and

c
o
s
(
θ
)
≈
1

{\displaystyle cos(\theta)\approx 1}

.  This allows (2) and
(3) to be rewritten as (4) and (5).

θ
≈

P
X

V

1

V

2

(
4
)

{\displaystyle \theta\approx \frac{PX}{V\_{1}V\_{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)}

(

V

2

−

V

1

)
≈

Q
X

V

2

(
5
)

{\displaystyle (V\_{2}-V\_{1})\approx \frac{QX}{V\_{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)}

### **Droop Control**

 From (4) it is seen that the real power P has a large influence on the power angle

θ

{\displaystyle \theta}

. Since the frequency is dependent on the power angle

θ

{\displaystyle \theta}

, the frequency droop can be controlled by the varying the real power. Likewise by (5) it is seen that the voltage difference between

(

V

2

−

V

1

)

{\displaystyle (V\_{2}-V\_{1})}

 is heavily influenced by the reactive power Q. So the voltage droop in a system can be controlled by changing the reactive power.

The control equations for frequency and voltage droop are shown in (6) and (7). One can think of the base frequency and base voltage as the target values, and the system values as the values for the frequency and voltage at a particular time.

# f

f

0

−

k

p

(
P
−

P

0

)

(
6
)

{\displaystyle f = f\_{0} - k\_{p} (P - P\_{0}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)}

# V

V

0

−

k

q

(
Q
−

Q

0

)

(
7
)

{\displaystyle V = V\_{0} - k\_{q} (Q - Q\_{0}) \, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)}

Where

f

{\displaystyle f\,}

 is the system frequency

f

0

{\displaystyle f\_{0} \, }

 is the base frequency

k

p

{\displaystyle k\_{p} \, }

 is the frequency droop control setting

P

{\displaystyle P \,}

 is the active power of the unit

P

0

{\displaystyle P\_{0} \, }

 is the base active power of the unit

V

{\displaystyle V \, }

 is the voltage at the measurement location

V

0

{\displaystyle V\_{0} \, }

 is the base voltage

Q

{\displaystyle Q \, }

 is the reactive power of the unit

Q

0

{\displaystyle Q\_{0} \, }

 is the base reactive power of the unit

k

q

{\displaystyle k\_{q} \, }

 is the voltage droop control setting

Within Microgrids, the various DGs which generally produce dc voltage, will interface through the power grid through inverters. Inverters are devices that convert dc voltage to ac voltage. Also, the energy storage devices will interface with the grid and either supply or absorb power from the grid. Theses storage devices will require an inverter when supplying power to the grid and an ac to dc converter when absorbing power from the grid. These inverters and converters can be controlled either locally by sensing the frequencies and voltages at their location, or controlled remotely by a Supervisory Control and Data Acquisition System (SCADA).

### **Example of a Distributed Grid**

A house with an example power profile over a 24 hour period is shown in Figure 2. The red line indicates the power that is being used by the house. The regular high power low power levels are indicative of the relatively high power use of an air conditioning unit cycling during the summer time. It is seen that the power usage spikes in early evening when the residents return home from the day. The light green line indicates the production of power from a PV array and the cyan line indicates the power output of a wind generator. The blue line is the grid power. The grid power (often hidden by the red line) is positive during the night and early morning. During the day, the excess power produced by the DG systems will at times provide power to the grid. This is shown when the grid power goes negative. The power is supplied to the grid through an inverter that is connected to the PCC of the house.

Depending in the type of control system utilized, distributed local control or hierarchical centralized control; different modeling systems may be used. In general for distributed control using Nash game algorithm provides relatively good results, but in centralized control, the Stackleberg game algorithm provides superior results.

## **Long Answer**

### **Game Selection**

1. **Cooperative games** are used where binding contracts are possible to reach a pareto-superior position for both players. (No fix power power output and no fixed load demand)
2. **Non Cooperative Game**: Non-cooperative games are used when negotiation and enforcement of contracts are not possible. (Independently adjusting power according to need).
3. Main types of Non Cooperative games

* In non-cooperative games where players make their decisions simultaneously, the game is a **Nash game**.
* **Nash game** is applied to capture stability notions.

* In non-cooperative games where players make decisions sequentially, then the game is called a **Stackelberg Game.** Stackelberg game is important in networks with hierarchy. It states that if the leader knows the optimal response of the follower, then it can play the Stackelberg strategy to optimize its objective or performance index.

### **Simple Stackelberg Model**

* One player (leader) has dominate influence over another.
* Typically there are two stages:
* One player moves first.
* Then the other follows in the second stage.
* Can be generalized to have multiple groups of players.
* Static games in both stages.

### **Main Theme**

Leader plays by backward induction, based on the anticipated behaviour of his/her follower.

### **Game Solution**

In Figure 2, the predicted PV generation, load profiles and the current hour choices of *β0*; and *Pμi* are utilized; and the average values of the parameters for the upcoming hours are calculated and used to estimate the performance indices from hour *(k+1)* to *N*. Thus, the performance indices are modeled as :

###
* *Jμ*(*βi*(*k*), *Pμ*(*k*))=*β*(*k*)*Pμi*(*k*) + (*N - K*)*βiavg*(*k + 1→N*)*Pμiavg*(*k + 1→N)]*                  (8)

*Jt*(*βi*(*k*),*Pμ*(*k*)) = *aiPGi*(*k*)+*β*(*k*)*Pμi*(*k*)+(*N - K*)[*aiavg*(*k+1→N*)*PGiavg*(*k+1→N)*)+*βiavg*(*k+1→N*)*Pμiavg*(*k+1→N))*

To solve the game problem, the following steps are carried out to calculate performance indicies *Jμ* and *Jt* to find the matrix between the main grid and a microgrid.

1. The active power *Pμ(k)* flowing from the microgrid (on the *i*th bus) to the main bus grid at k hour is given by (10)

2. Assume that the storage level of a microgrid should return to its initial value after *N* hours, then the average active power from hour *(k+1)* to *N*, *Pμavg(k+1→N)* is given by:

###

where avg[Pμi(k+1→N)] is the predicted average power from historical data without considering the storage.

3. Conventional generation *PGi(k)* (on the the ith bus) at hour k is given by:

###
* *PGi(k) = PDi(k)-Pμi(k).*                                                                                   (9)

And the predicted average generation from hour(k+1) to N is given by:

###
* *PGiavg(k+1→N) = PDavg(k+1→N)-Pμiavg(k+1→N)*                                          (10)

where *PDavg (k+1→N)* is the predicted average demand from hour(K+1) to N.

###

4. Using *PGi* is computed by:

where *PGi*\* is the optimal operation of the power of the conventional generator. *β 0* is a base price and *η i*(*k*) is a variable, which the main grid perturbs to find different energy price offers to play the game.

5. Therefore,

###

Hence, a matrix can be constructed as in Table 1:

1. For each η i (k, l ) (k, j ) for j = 1,2,....M1 is corresponding to *Pμi*(*k*) can be found such that :

###
* *Jμi*  [ *ηi*(*k,j*),*Pμi*(*k*)]is maximized.                                                                       (11)

###
and that *Pμi(k)* is denoted as *PμiS*[*ηi*(*k,j*)].

2. The Stackelberg solution of the main grid is *ηi*(*k,l*) for some l such that:

###
* Jt [*ηi* (k,l), *PμiS* (*ηi* (*k, l* ) ) ] *≤ Jt* [ *ηi* (*k, j* ), *PμiS* (*k, j* ) ) ]                                       (12)

To illustrate this algorithm, a simple example is provided as folows:

### Example

Suppose that the matrix game is shown Table 2.

The Stackelberg solution with the main grid as the leader is obtained as follows:

###
1. For *ηi*(*k*)=*1, Pμi*(*k*) = *1* maximizes *Jμi*because

###
* *Jμi*(*1,1*) = *10* is greater than *Jμi*(*10,8*) = *8*.

###
* For *ηi*(*k*)=*1.2*, *Pμi*(*k*) = *0.8* maximizes *Jμi* because

###
* *Jμi*(*1.2,0.8*) = *7* is greater than *Jμ*(*1.2,1*) = *6*

###
2. As in Table 2

###
* *Jt*(*1.2,0.8*) = *5* is less than *Jt*(*1,1*) = *9*

###
* the Stackelberg solution of the main grid is *η*(*k*) = *1.2*

## **References**

1. ↑ A. Maknouninejad, W. Lin, H. G. Harno, Z.Qu, M. A. Simaan, Cooperative control for self-organizing microgrids and game strategies for optimal dispatch of distributed renewable generations, Energy Systems, vol. 3, issue 1, pp. 23-60, January 2012.
2. ↑ 2.0 2.1 2.2 K. De Brabandere, B. Bolsens, J. Van den Keybus, A. Woyte, J. Driesen, and R. Belmans, A voltage and frequency droop control method for parallel inverters, IEEE Transactions on Power Electronics, vol. 22, no. 4, pp. 1107-1115, July 2007.
3. ↑ “Smart Grid Simulations”, www.mathworks.com. Accessed May 16, 2016. [Online]. Available: http://www.mathworks.com/matlabcentral/fileexchange/54734-smart-grid-simulations/content/Step1\_LoadData.m
4. ↑  H. Xin, Z. Qu, J. Seuss, A. Maknouninejad, A self-organizing strategy for power flow control of photovoltaic generators in a distribution network, IEEE Transactions on Power Systems vol. 26, pp. 1462–1473, 2011.
5. ↑ M. Simaan, J. Cruz, Additional aspects of the Stackelberg strategy in nonzero-sum games. J. Optim. Theory Appl. Vol. 11, pp. 613–626, 1973.