*3.2.2. Problem specific decision*

Another set of decisions to make is specific to the problem to solve and is presented further below.

• Neighbourhood structure

When thinking about the problem of optimal filtering power sizing, the choice of the way to move from a current solution to another one is questioned. This means that a neighbourhood is to be defined. A relevant study is proposed in [23].

In the present application, the generator of random changes in the configuration is based on a varying neighbourhood as the algorithm progresses. The amplitude of the filtering current is then randomly selected in a solution space according to (11):

$$J\mu^i = \beta(T) \cdot J\mu^i \tag{11}$$

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 33

<sup>h</sup> ( ) hk limit k 1

*N* <sup>=</sup> <sup>é</sup> <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> é ù D = <sup>ê</sup> <sup>ú</sup> ê ú ë û <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

*k*

*Nnodes*

å (12)

(13)

<sup>1</sup> min min . v - v ²

Instead, when the voltage conditions are satisfied, the existence of a global optimum is clearly confirmed. The voltage annealing process is limited to the success of a local search which already guarantees a solution that works. Then, it automatically switches to a simulated current annealing which optimisation process is naturally applied to the harmonic currents injected by the filters connected to the grid. In that second instance, the objective is to minimise the total filtering current *jtotal* that preserves the voltage requirements. The resultant cost function is therefore defined by (13) with respect to the constraint (14) on each nodal harmonic voltage *vhk*. The returned value of the total filtering

power is then minimal while the maximal nodal voltage meets the expected limits.

min j min j total

It can be appreciated that the minimum filtering power is obtained for the locations defined by the list of nodes *(ListFilter)* initially proposed. The optimal placement on the distribution network is then determined by testing all the possible combinations of the filters connections. In practice, the number of configurations is often small due to the limited number of busbars able to accept harmonic filters. As a result, an optimal process is not required to determine the 'best' configuration corresponding to the minimum value of the filtering power. Besides, the minimum power is not necessarily the best solution retained by an electrical engineer who must take into account many other technical and economical considerations. It seems then better to give the optimal solution for each configuration and

Tested on several real power systems, the above optimisation technique is presently implemented into a software package developed in C language. The example of an aboard ship power system is proposed further below so as to point out the benefit of the SA process

The electrical ship network of figure 3 is composed of six busbars with three voltage levels: 690V, 400V and 230V. Four 1.8MVA generators supply the 690V busbar *(TPF1)*. The main powerful loads including the electric propulsion system are then connected directly to it. The propulsion system is made up of two variable speed drives *(MP\_BD, MP\_TD)* with a twelve-pulse structure. Five low voltage busbars, i.e. one 400V busbar on the port side and another one on the starboard side in addition with three 230V busbars, distribute the energy

to a very practical issue regarding harmonics and power quality in electrical systems.

Î <sup>é</sup> <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> é ù <sup>ê</sup> úê ú <sup>=</sup> ë û <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

å *hq q ListFilter*

v v k 1,.., N h k limit k £ "= nodes (14)

*v*

to leave the engineer to select the best strategic choice afterwards.

**4. Application to a ship power system** 

**4.1. Description of the power system** 

nodes v i k = 1, .., N i : index of the current state j : index of the new state β : weight factor depending on the temperature parameter T = T or T ìï ï ï ï ïï í ï ï ï ï ïïî

Consequently, as the temperature declines, the weight coefficient β is adjusted as a function of *Tv* or *Ti* depending on whether the SA process is applied to the harmonic voltages or to the filtering currents and the neighbourhood is gradually restricted, which ensures the success of the convergence process towards the expected global optimum.

• Cost function

Then, a cost function that models the current problem to solve is needed. As it will be calculated at every step of the algorithm, this objective function must be also easy and fast to calculate.

As the proposed algorithm (Figure 2) involves a double SA, a test procedure is systematically performed in order to assess whether or not the annealing process should be applied to the voltage or to the current according to the gap observed between the maximal nodal voltage newly calculated and the required limits. If the nodal voltage is thus more than the specified requirements, a voltage annealing is applied; conversely, if the standard is met, a current annealing is conducted. Then, the following stages of the algorithm are those of a conventional SA scheme like previously described in section 2.

In consequence, two distinctive objective functions can be referred depending on the nature of the SA process executed. When the problem shows no filtering solution, it means that no injected current goes to providing remedies for a reduction of the harmonic voltages below the standards. The objective is then specifically directed at decreasing as far as possible the nodal harmonic voltages to closely approximate the specified limits. Thus, the appropriate cost function (12) is defined as the standard deviation between the harmonic voltages and their threshold values over all the electrical system's nodes.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 33

$$\min\left[\Delta\nu\right] = \min\left|\sqrt{\frac{1}{N}\sum\_{k=1}^{N\_{\text{nuk}}} \left(\mathbf{v\_{hk}} \cdot \mathbf{v\_{lim}}\right)^{2}}\right|\tag{12}$$

Instead, when the voltage conditions are satisfied, the existence of a global optimum is clearly confirmed. The voltage annealing process is limited to the success of a local search which already guarantees a solution that works. Then, it automatically switches to a simulated current annealing which optimisation process is naturally applied to the harmonic currents injected by the filters connected to the grid. In that second instance, the objective is to minimise the total filtering current *jtotal* that preserves the voltage requirements. The resultant cost function is therefore defined by (13) with respect to the constraint (14) on each nodal harmonic voltage *vhk*. The returned value of the total filtering power is then minimal while the maximal nodal voltage meets the expected limits.

$$\min \left[ \mathbf{j}\_{\text{total}} \right] = \min \left| \sum\_{q \in ListFilter} \left| \mathbf{j}\_{\text{h}q} \right| \right| \tag{13}$$

$$\mathbf{v\_{hk}} \le \mathbf{v\_{h\text{init}}} \quad \forall \text{ } \mathbf{k} = \mathbf{1} \text{...} \text{.} \text{N}\_{\text{nodes}} \tag{14}$$

It can be appreciated that the minimum filtering power is obtained for the locations defined by the list of nodes *(ListFilter)* initially proposed. The optimal placement on the distribution network is then determined by testing all the possible combinations of the filters connections. In practice, the number of configurations is often small due to the limited number of busbars able to accept harmonic filters. As a result, an optimal process is not required to determine the 'best' configuration corresponding to the minimum value of the filtering power. Besides, the minimum power is not necessarily the best solution retained by an electrical engineer who must take into account many other technical and economical considerations. It seems then better to give the optimal solution for each configuration and to leave the engineer to select the best strategic choice afterwards.

#### **4. Application to a ship power system**

32 Simulated Annealing – Single and Multiple Objective Problems

Another set of decisions to make is specific to the problem to solve and is presented further

When thinking about the problem of optimal filtering power sizing, the choice of the way to move from a current solution to another one is questioned. This means that a

In the present application, the generator of random changes in the configuration is based on a varying neighbourhood as the algorithm progresses. The amplitude of the filtering current

() *j i Jh h k k* = ⋅ *b T J* (11)

β : weight factor depending on the temperature parameter T = T or T

of *Tv* or *Ti* depending on whether the SA process is applied to the harmonic voltages or to the filtering currents and the neighbourhood is gradually restricted, which ensures the

Then, a cost function that models the current problem to solve is needed. As it will be calculated at every step of the algorithm, this objective function must be also easy and fast to calculate.

As the proposed algorithm (Figure 2) involves a double SA, a test procedure is systematically performed in order to assess whether or not the annealing process should be applied to the voltage or to the current according to the gap observed between the maximal nodal voltage newly calculated and the required limits. If the nodal voltage is thus more than the specified requirements, a voltage annealing is applied; conversely, if the standard is met, a current annealing is conducted. Then, the following stages of the algorithm are those

In consequence, two distinctive objective functions can be referred depending on the nature of the SA process executed. When the problem shows no filtering solution, it means that no injected current goes to providing remedies for a reduction of the harmonic voltages below the standards. The objective is then specifically directed at decreasing as far as possible the nodal harmonic voltages to closely approximate the specified limits. Thus, the appropriate cost function (12) is defined as the standard deviation between the harmonic voltages and

v i

is adjusted as a function

β

neighbourhood is to be defined. A relevant study is proposed in [23].

is then randomly selected in a solution space according to (11):

Consequently, as the temperature declines, the weight coefficient

of a conventional SA scheme like previously described in section 2.

their threshold values over all the electrical system's nodes.

success of the convergence process towards the expected global optimum.

nodes

i : index of the current state j : index of the new state

k = 1, .., N

ìï ï ï ï ïï í ï ï ï ï ïïî

• Cost function

*3.2.2. Problem specific decision* 

• Neighbourhood structure

below.

Tested on several real power systems, the above optimisation technique is presently implemented into a software package developed in C language. The example of an aboard ship power system is proposed further below so as to point out the benefit of the SA process to a very practical issue regarding harmonics and power quality in electrical systems.

#### **4.1. Description of the power system**

The electrical ship network of figure 3 is composed of six busbars with three voltage levels: 690V, 400V and 230V. Four 1.8MVA generators supply the 690V busbar *(TPF1)*. The main powerful loads including the electric propulsion system are then connected directly to it. The propulsion system is made up of two variable speed drives *(MP\_BD, MP\_TD)* with a twelve-pulse structure. Five low voltage busbars, i.e. one 400V busbar on the port side and another one on the starboard side in addition with three 230V busbars, distribute the energy

everywhere aboard ship. According to their rated voltage, they supply the onboard equipment: two winches *(MT\_BD, MT\_TD)*, the lighting system, several UPS units and battery chargers. Every load is modelled at each node by an equivalent linear impedance or by an equivalent source of current for the converters and the fluorescent lighting. The detailed specifications of the power system are given in the tables 8, 9, 10 of the appendix.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 35

harmonic order (h) 5 7 11 13 IEC – Vh (%) 6 5 3.5 3 shipyard - Vh(%) on 690V 3.5 3.5 3 3 shipyard - Vh(%) on 400V & 230V 3.5 3.5 1.2 1.2

n° busbar 1 2 3 4 5 6 Vh=11 (%Vn) 4.88 5.40 5.43 5.46 5.40 5.49

n° busbar 1 2 3 4 5 6 Vh=5 (%Vn) 3.28 3.74 3.88 4.59 4.06 5.41

The two most stringent configurations will be presented and discussed further below. The first one relates to the study of the 11th harmonic order when the ship is travelling at full speed with three diesel engine generators running (configuration A). The second one involves the study of the fifth harmonic order when the ship is on berth with only one diesel engine generator operating (configuration B). Given the harmonic currents introduced by the non-linear loads of the electrical system as mentioned in the table 9 of the appendix, the simulation of the power system for the both configurations above shows the results of the table 2 and the table 3 respectively. It can be appreciated that the resulting voltage levels for the 11th harmonic order extend far beyond the limits set by the requirements. When simulating the harmonic voltages for the 5th order when the ship is in dock, the specified limits are not exceeded on the 690V busbar only. A filtering schedule is however required on the other busbars. The choice of the most suitable placement is then questioned: is it better to plan filtering on the 400V busbar or on the 230V busbar? The proposed optimisation

As displayed in figure 3, the electrical system shows six nodes and thus offers six possible placements for active or passive filters. With six possible locations, the number of filters can vary from one to six. A complete analysis to determine the most suitable placement(s) to select requires sixty-three case studies. With the support of the implemented software, all the combinations are examined in 9.3 seconds of CPU time with a personal computer fitted out with an Intel© CoreDuo T8100, 2.1GHz processor. Among the list of the possible filtering solutions, the most relevant ones will be presented and argued in the following section.

In order to ensure that the optimisation procedure works well, several variables have been saved at each temperature of the SA process as the search for the optimal solution moves forward. The progress of the filtering power and the nodal voltages is then reported at the end of the computation procedure in order to control the performance of the convergence

**Table 1.** Harmonic voltage limits

**4.2. Study statement** 

**Table 2.** Harmonic voltage: 11th order in configuration A

**Table 3.** Harmonic voltage: 5th order in configuration B

procedure makes it possible to get answers to that critical question.

towards the global optimal solution.

**Figure 3.** Single line diagram of the aboard ship network

The above-mentioned non linear loads inject harmonic currents in the network, which induces unfavourable resultant voltages. The IEC standard limits harmonic voltages according to the specifications given in the table 1. Nevertheless, the shipyard imposes even lower levels in order to guarantee a better power quality on the low voltage distribution system which might supply sensitive equipment.

Optimal Sizing of Harmonic Filters in Electrical Systems: Application of a Double Simulated Annealing Process 35


**Table 1.** Harmonic voltage limits

34 Simulated Annealing – Single and Multiple Objective Problems

**Figure 3.** Single line diagram of the aboard ship network

system which might supply sensitive equipment.

The above-mentioned non linear loads inject harmonic currents in the network, which induces unfavourable resultant voltages. The IEC standard limits harmonic voltages according to the specifications given in the table 1. Nevertheless, the shipyard imposes even lower levels in order to guarantee a better power quality on the low voltage distribution

everywhere aboard ship. According to their rated voltage, they supply the onboard equipment: two winches *(MT\_BD, MT\_TD)*, the lighting system, several UPS units and battery chargers. Every load is modelled at each node by an equivalent linear impedance or by an equivalent source of current for the converters and the fluorescent lighting. The detailed specifications of the power system are given in the tables 8, 9, 10 of the appendix.


**Table 2.** Harmonic voltage: 11th order in configuration A


**Table 3.** Harmonic voltage: 5th order in configuration B
